Multiple Correct Answers MCQs for Sub-Topics of Topic 7: Mensuration

Question 1. Which of the following statements correctly describe concepts in Mensuration?

(A) Mensuration is the study of shapes and their properties.

(B) Perimeter is the measure of the boundary length of a $2\text{D}$ shape.

(C) Area measures the space occupied by a $3\text{D}$ solid.

(D) Volume is a $2\text{D}$ measurement.

Question 2. A rectangular field has length $20\ \text{m}$ and width $15\ \text{m}$. Which of the following statements are true?

(A) Its perimeter is $35\ \text{m}$.

(B) Its area is $300\ \text{m}^2$.

(C) If fenced, the length of fencing needed is $70\ \text{m}$.

(D) Its perimeter is measured in square metres.

Question 3. Consider the units of measurement for perimeter and area. Which of the following pairings are correct?

(A) Perimeter: metres ($\text{m}$)

(B) Area: square centimetres ($\text{cm}^2$)

(C) Perimeter: kilometres ($\text{km}$)

(D) Area: cubic metres ($\text{m}^3$)

Question 4. If the area of a square is $81\ \text{cm}^2$, which of the following statements are correct?

(A) The side length of the square is $9\ \text{cm}$.

(B) The perimeter of the square is $36\ \text{cm}$.

(C) The area is measured in linear units.

(D) The diagonal of the square is $9\sqrt{2}\ \text{cm}$.

Question 5. Mensuration concepts are used in which of the following real-life scenarios in India?

(A) Calculating the amount of paint needed for a wall.

(B) Determining the length of boundary needed for a farm.

(C) Calculating the capacity of a water tank.

(D) Finding the weight of an object.

Question 6. Which of the following are valid units for measuring area?

(A) Acre

(B) Hectare

(C) Square kilometre ($\text{km}^2$)

(D) Cubic foot ($\text{ft}^3$)

Question 7. Converting between units is important in mensuration. Which of the following conversions are correct?

(A) $1\ \text{m} = 100\ \text{cm}$

(B) $1\ \text{km} = 1000\ \text{m}$

(C) $1\ \text{m}^2 = 100\ \text{cm}^2$

(D) $1\ \text{km}^2 = 100\ \text{hectares}$

Question 8. If the perimeter of an equilateral triangle is $18\ \text{cm}$, which statements are true?

(A) Each side of the triangle is $6\ \text{cm}$.

(B) The triangle is a plane figure.

(C) Its area can be calculated using $\frac{\sqrt{3}}{4} \times (\text{side})^2$.

(D) Its area is measured in $\text{cm}$.

Question 9. Area is a property of which types of geometric figures?

(A) Plane figures

(B) Solid figures (specifically, their surfaces)

(C) Lines

(D) Points

Question 10. Which of the following statements are generally true for closed plane figures?

(A) They have a well-defined perimeter.

(B) They enclose a specific area.

(C) They can have curved boundaries.

(D) They are always polygons.

Question 1. A rectangle has sides $10\ \text{cm}$ and $6\ \text{cm}$. Which of the following statements about its perimeter are correct?

(A) The perimeter is $2(10+6)\ \text{cm}$.

(B) The perimeter is $10 \times 6\ \text{cm}$.

(C) The perimeter is $32\ \text{cm}$.

(D) If it was a square with the same perimeter, its side would be $8\ \text{cm}$.

Question 2. Consider a parallelogram with adjacent sides $a$ and $b$. Which expressions represent its perimeter?

(A) $a+b+a+b$

(B) $2a+2b$

(C) $2(a+b)$

(D) $ab$

Question 3. A triangular park has sides $13\ \text{m}$, $14\ \text{m}$, and $15\ \text{m}$. Which of the following are correct?

(A) Its perimeter is $42\ \text{m}$.

(B) Its semi-perimeter is $21\ \text{m}$.

(C) The unit of perimeter is square metres.

(D) The perimeter is the sum of the lengths of its sides.

Question 4. If a regular hexagon has a perimeter of $54\ \text{cm}$, which statements are correct?

(A) It has $6$ equal sides.

(B) Each side length is $9\ \text{cm}$.

(C) Its area can be found by dividing it into $6$ equilateral triangles.

(D) The sum of its interior angles is $360^\circ$.

Question 5. The perimeter of a figure made of a rectangle with a semicircle on one of its sides includes which lengths?

(A) The length of the diameter of the semicircle.

(B) The lengths of the sides of the rectangle that form the outer boundary.

(C) The length of the arc of the semicircle.

(D) The length of the side of the rectangle covered by the semicircle.

Question 6. Which of the following units can be used to measure the perimeter of a plot of land in India?

(A) Metres ($\text{m}$)

(B) Kilometres ($\text{km}$)

(C) Centimetres ($\text{cm}$)

(D) Square feet ($\text{ft}^2$)

Question 7. If the perimeter of a square is equal to the perimeter of a rectangle, which of the following is necessarily true?

(A) The area of the square is equal to the area of the rectangle.

(B) The side length of the square equals the length of the rectangle.

(C) The length of the square is equal to the average of the length and width of the rectangle.

(D) The shapes have the same boundary length.

Question 8. A wire of length $L$ is used to form a closed plane figure. Which of the following shapes could it be bent into?

(A) An equilateral triangle with perimeter $L$.

(B) A square with perimeter $L$.

(C) A circle with circumference $L$.

(D) An open figure like a line segment of length $L$.

Question 9. For any triangle with side lengths $a, b, c$, which statements are true?

(A) Its perimeter is $a+b+c$.

(B) The sum of any two sides is greater than the third side ($a+b > c$, $b+c > a$, $c+a > b$).

(C) The perimeter is always a positive value.

(D) If $a=b=c$, it is an equilateral triangle and its perimeter is $3a$.

Question 10. If the perimeter of a rhombus is $P$, which statements are correct?

(A) All four sides of the rhombus are equal in length.

(B) The length of each side is $P/4$.

(C) The perimeter is $4s$, where $s$ is the side length.

(D) The diagonals bisect each other at right angles.

Question 1. A square plot has a side length of $20\ \text{m}$. Which of the following statements about its area are correct?

(A) The area is $20 \times 20\ \text{m}^2$.

(B) The area is $400\ \text{m}^2$.

(C) If the side was doubled, the area would become $800\ \text{m}^2$.

(D) The unit of area is linear.

Question 2. A parallelogram has a base of $12\ \text{cm}$ and a height of $8\ \text{cm}$. Which statements about its area are true?

(A) The area is $12 \times 8\ \text{cm}^2$.

(B) The area is $96\ \text{cm}^2$.

(C) If the base remains $12\ \text{cm}$ and the height is doubled, the area becomes $192\ \text{cm}^2$.

(D) The area is $\frac{1}{2} \times \text{base} \times \text{height}$.

Question 3. A triangle has a base of $10\ \text{m}$ and a corresponding height of $6\ \text{m}$. Which statements are correct regarding its area?

(A) The area is $\frac{1}{2} \times 10 \times 6\ \text{m}^2$.

(B) The area is $30\ \text{m}^2$.

(C) If the base was halved and height remained the same, the area would be $15\ \text{m}^2$.

(D) The area is measured in cubic units.

Question 4. A rectangular room needs flooring. The length is $5\ \text{m}$ and the width is $4\ \text{m}$. Which calculations are relevant to finding the cost of flooring?

(A) Calculate the perimeter of the room.

(B) Calculate the area of the room in square metres.

(C) Multiply the area by the rate of flooring per square metre.

(D) Convert the dimensions to centimetres before calculating the area.

Question 5. If a square and a rectangle have the same area, which of the following can be true?

(A) They have the same perimeter.

(B) They can have different perimeters.

(C) If the square has side $s$ and the rectangle has length $l$ and width $w$, then $s^2 = lw$.

(D) If they also have the same perimeter, they must be congruent.

Question 6. A parallelogram and a rectangle stand on the same base and between the same parallel lines. Which property is true?

(A) They have the same height.

(B) They have the same area.

(C) They have the same perimeter.

(D) The area of the parallelogram is base $\times$ height.

Question 7. Which formulas can be used to find the area of a square with side length $s$?

(A) $s^2$

(B) side $\times$ side

(C) $(\frac{1}{2} \times \text{diagonal})^2$

(D) $\frac{1}{2} \times d_1 \times d_2$ where $d_1, d_2$ are diagonals

Question 8. The area of a trapezium with parallel sides $a$ and $b$ and height $h$ is given by $\frac{1}{2}(a+b)h$. How can this formula be understood?

(A) A trapezium can be divided into two triangles with bases $a$ and $b$ and the same height $h$ (Incorrect division).

(B) A trapezium can be divided into a rectangle and two right triangles.

(C) The formula is equivalent to multiplying the average of the parallel sides by the height.

(D) The formula comes from dividing the trapezium into two triangles using a diagonal.

Question 9. If the area of a rectangle is $A$ and its length is $l$, which expressions represent its width $w$?

(A) $w = A/l$

(B) $w = \sqrt{A}$ (only if it's a square)

(C) $w = A - l$

(D) $w = \frac{A}{l}$

Question 10. A farmer in Punjab has a rectangular field that is $100\ \text{m}$ long and $80\ \text{m}$ wide. Which statements are correct?

(A) The area of the field is $8000\ \text{m}^2$.

(B) The area can be expressed in hectares.

(C) The perimeter of the field is $360\ \text{m}$.

(D) The area is $8\ \text{hectares}$.

Question 1. Heron's formula for the area of a triangle is $\sqrt{s(s-a)(s-b)(s-c)}$. Which statements are correct about this formula?

(A) It requires knowing the base and height of the triangle.

(B) It can be used when the lengths of all three sides ($a, b, c$) are known.

(C) $s$ represents the semi-perimeter, calculated as $s = \frac{a+b+c}{2}$.

(D) The term $(s-a)(s-b)(s-c)$ must be positive for a valid triangle.

Question 2. A triangle has sides $7\ \text{cm}$, $8\ \text{cm}$, and $9\ \text{cm}$. Which of the following are correct steps or results in finding its area using Heron's formula?

(A) The perimeter is $24\ \text{cm}$.

(B) The semi-perimeter $s = 12\ \text{cm}$.

(C) $s-a = 5\ \text{cm}$, $s-b = 4\ \text{cm}$, $s-c = 3\ \text{cm}$.

(D) The area is $\sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720}\ \text{cm}^2$.

Question 3. For an equilateral triangle with side length $a$, which of the following can be used to calculate its area?

(A) Using base $a$ and height $\frac{\sqrt{3}}{2}a$: Area $= \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{4}a^2$.

(B) Using Heron's formula with $a=b=c$ and $s = \frac{3a}{2}$.

(C) Area $= \sqrt{\frac{3a}{2}(\frac{3a}{2}-a)(\frac{3a}{2}-a)(\frac{3a}{2}-a)} = \sqrt{\frac{3a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}} = \sqrt{\frac{3a^4}{16}} = \frac{\sqrt{3}}{4}a^2$.

(D) Area $= \frac{1}{2} \times \text{base} \times \text{side length}$.

Question 4. An isosceles triangle has equal sides of length $17\ \text{cm}$ and a base of $16\ \text{cm}$. Which of the following are correct?

(A) The semi-perimeter is $\frac{17+17+16}{2} = 25\ \text{cm}$.

(B) Using Heron's formula, $s-a=8$, $s-b=8$, $s-c=9$.

(C) The area is $\sqrt{25 \times 8 \times 8 \times 9} = \sqrt{25 \times 64 \times 9} = 5 \times 8 \times 3 = 120\ \text{cm}^2$.

(D) The height to the base is $15\ \text{cm}$.

Question 5. The sides of a triangle are $a, b, c$. Which condition(s) must be met for these sides to form a valid triangle?

(A) $a+b > c$

(B) $b+c > a$

(C) $c+a > b$

(D) $a+b+c > 0$

Question 6. If the sides of a triangle are $5\ \text{cm}$, $12\ \text{cm}$, and $13\ \text{cm}$, which statements are true?

(A) This is a right-angled triangle.

(B) Its area can be calculated as $\frac{1}{2} \times 5 \times 12 = 30\ \text{cm}^2$.

(C) Using Heron's formula, $s=15$, $s-a=10$, $s-b=3$, $s-c=2$. Area $= \sqrt{15 \times 10 \times 3 \times 2} = \sqrt{900} = 30\ \text{cm}^2$.

(D) Heron's formula is only for non-right-angled triangles.

Question 7. The perimeter of a triangle is $32\ \text{cm}$. If two sides are $8\ \text{cm}$ and $11\ \text{cm}$, which statements are correct?

(A) The third side is $32 - (8+11) = 13\ \text{cm}$.

(B) The side lengths are $8\ \text{cm}$, $11\ \text{cm}$, $13\ \text{cm}$.

(C) The semi-perimeter $s = 16\ \text{cm}$.

(D) The area can be calculated using Heron's formula with $s=16, s-a=8, s-b=5, s-c=3$.

Question 8. For a triangle with sides $a, b, c$ and area $A$, if $A = \sqrt{s(s-a)(s-b)(s-c)}$, what is true about $s$? ($\theta$ is an angle, $h$ is height)

(A) $s$ is always positive.

(B) $s$ is a measure of length.

(C) $s$ is half the perimeter.

(D) $s$ is related to the angle $\theta$ like $s = r \theta$ (Incorrect context).

Question 9. A triangular plot in a village has sides $25\ \text{m}$, $60\ \text{m}$, $65\ \text{m}$. Which calculations are appropriate for finding its area?

(A) Find semi-perimeter $s = (25+60+65)/2 = 75\ \text{m}$.

(B) Use Heron's formula: Area $= \sqrt{75(75-25)(75-60)(75-65)}$.

(C) Note that $25^2 + 60^2 = 625 + 3600 = 4225 = 65^2$, so it's a right triangle.

(D) Use the right triangle area formula: Area $= \frac{1}{2} \times 25 \times 60\ \text{m}^2 = 750\ \text{m}^2$.

Question 10. Heron's formula is particularly useful when:

(A) The triangle is equilateral.

(B) The triangle is right-angled.

(C) Only the three side lengths are easily available.

(D) It's difficult to find the height corresponding to a base.

Question 1. For a general quadrilateral with a diagonal $d$ and perpendiculars from opposite vertices $h_1$ and $h_2$, which formula(s) give its area?

(A) Area $= \frac{1}{2} d (h_1 + h_2)$

(B) Area $= \text{Area of triangle 1} + \text{Area of triangle 2}$

(C) Area $= d \times (h_1 + h_2)$

(D) Area $= \frac{1}{2} d \times h_1 + \frac{1}{2} d \times h_2$

Question 2. A rhombus has diagonals $d_1$ and $d_2$. Which statements about its area are correct?

(A) The diagonals are perpendicular bisectors of each other.

(B) The area can be calculated as $\frac{1}{2} d_1 d_2$.

(C) The rhombus can be divided into four congruent right-angled triangles.

(D) The area is equal to the product of its diagonals.

Question 3. A kite has diagonals $24\ \text{cm}$ and $10\ \text{cm}$. Which statements are true?

(A) The area of the kite is $\frac{1}{2} \times 24 \times 10\ \text{cm}^2$.

(B) The area is $120\ \text{cm}^2$.

(C) The diagonals are perpendicular.

(D) The area formula $\frac{1}{2} d_1 d_2$ applies to all quadrilaterals.

Question 4. To find the area of a quadrilateral using Heron's formula, which approach can be used?

(A) Apply Heron's formula directly to the four sides of the quadrilateral (Incorrect).

(B) Divide the quadrilateral into two triangles using a diagonal.

(C) Calculate the area of each triangle formed by the diagonal using Heron's formula.

(D) Sum the areas of the two triangles to get the area of the quadrilateral.

Question 5. A field is in the shape of a quadrilateral ABCD, where AB=$7\text{m}$, BC=$6\text{m}$, CD=$12\text{m}$, DA=$15\text{m}$, and AC=$10\text{m}$. To find the area, you could:

(A) Find the area of triangle ABC using Heron's formula.

(B) Find the area of triangle ADC using Heron's formula.

(C) Sum the areas of triangle ABC and triangle ADC.

(D) Use the formula $\frac{1}{2} \times \text{diagonal} \times (\text{sum of perpendiculars})$.

Question 6. For a regular polygon with $n$ sides, which statements about its area are true?

(A) It can be divided into $n$ congruent isosceles triangles with a common vertex at the center.

(B) The area of each such triangle can be calculated if the side length and the distance from the center to the midpoint of a side (apothem) are known.

(C) The area of the polygon is $n$ times the area of one of these triangles.

(D) The area formula $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$ can be used.

Question 7. Which of the following quadrilaterals always have diagonals that bisect each other?

(A) Parallelogram

(B) Rectangle

(C) Rhombus

(D) Trapezium

Question 8. The area of a rectangle with sides $l$ and $w$ is $lw$. How is this related to the area of a parallelogram?

(A) A rectangle is a special type of parallelogram where the height equals one of the sides.

(B) The area of a parallelogram is base $\times$ height.

(C) If a rectangle and a parallelogram have the same base and height, they have the same area.

(D) The area of a parallelogram is always less than the area of a rectangle with the same base and adjacent side.

Question 9. A field is divided into a rectangle and two triangles. To find the total area of the field, you should:

(A) Calculate the area of the rectangle.

(B) Calculate the areas of the two triangles.

(C) Add the areas of the rectangle and the two triangles.

(D) Find the perimeter of the entire field.

Question 10. For a rhombus with side length $s$ and one diagonal $d_1$, how can you find the length of the other diagonal $d_2$?

(A) Use the property that diagonals bisect each other at right angles.

(B) Apply the Pythagorean theorem to one of the four right triangles formed by the diagonals: $(\frac{d_1}{2})^2 + (\frac{d_2}{2})^2 = s^2$.

(C) $d_2 = \frac{2 \times \text{Area}}{d_1}$.

(D) $d_2 = \sqrt{s^2 - (\frac{d_1}{2})^2}$.

Question 1. Which of the following correctly describe the relationship between the circumference ($C$), radius ($r$), and diameter ($d$) of a circle?

(A) $C = 2 \pi r$

(B) $C = \pi d$

(C) $d = 2r$

(D) $\pi = \frac{C}{d}$

Question 2. Which of the following formulas correctly represent the area ($A$) of a circle?

(A) $A = \pi r^2$ where $r$ is the radius.

(B) $A = \pi (d/2)^2$ where $d$ is the diameter.

(C) $A = \frac{C \times r}{2}$ where $C$ is the circumference and $r$ is the radius.

(D) $A = 2 \pi r$

Question 3. A circular playground has a radius of $7\ \text{m}$. (Use $\pi = \frac{22}{7}$). Which statements are correct?

(A) Its circumference is $2 \times \frac{22}{7} \times 7 = 44\ \text{m}$.

(B) Its area is $\frac{22}{7} \times 7^2 = 154\ \text{m}^2$.

(C) If its radius was doubled, its circumference would be $88\ \text{m}$.

(D) If its radius was doubled, its area would be $308\ \text{m}^2$.

Question 4. If the circumference of a circle is $110\ \text{cm}$, which of the following calculations or results are correct? (Use $\pi = \frac{22}{7}$)

(A) $2 \pi r = 110$

(B) $r = \frac{110 \times 7}{2 \times 22} = \frac{110 \times 7}{44} = \frac{10 \times 7}{4} = \frac{5 \times 7}{2} = 17.5\ \text{cm}$.

(C) The diameter is $35\ \text{cm}$.

(D) The area is $\pi \times (17.5)^2$.

Question 5. If the area of a circle is $308\ \text{cm}^2$, which of the following steps or results are correct? (Use $\pi = \frac{22}{7}$)

(A) $\pi r^2 = 308$

(B) $r^2 = \frac{308 \times 7}{22} = \frac{14 \times 7 \times 22}{22} = 14 \times 7 = 98$.

(C) $r = \sqrt{98} = 7\sqrt{2}\ \text{cm}$.

(D) The diameter is $14\sqrt{2}\ \text{cm}$.

Question 6. The value of $\pi$ (pi) is approximately $3.14$ or $\frac{22}{7}$. Which statements about $\pi$ are correct?

(A) It is the ratio of a circle's circumference to its diameter.

(B) It is a constant value for all circles.

(C) It is a rational number ($\pi = \frac{22}{7}$ is an approximation).

(D) It is an irrational number.

Question 7. If the radius of a circle is $r$, which statements relating circumference ($C$) and area ($A$) are true?

(A) $A = \pi r^2$

(B) $C = 2\pi r$

(C) $C = \frac{2A}{r}$

(D) $A = \frac{C^2}{4\pi}$

Question 8. A circular field needs to be fenced. Which measurement is required to determine the amount of fencing needed?

(A) Area

(B) Circumference

(C) Radius

(D) Diameter

Question 9. If the circumference of a circle is equal to the perimeter of a square, which of the following comparisons of their areas might be true?

(A) The area of the circle is greater than the area of the square.

(B) The area of the square is greater than the area of the circle.

(C) The areas are equal.

(D) It is impossible for their circumferences/perimeters to be equal if their areas are different.

Question 10. A wheel travels a distance by rotating. The distance covered in one revolution is equal to its:

(A) Diameter

(B) Radius

(C) Circumference

(D) Area

Question 1. A sector of a circle is defined by:

(A) A central angle

(B) Two radii

(C) The intercepted arc

(D) A chord

Question 2. For a sector of a circle with radius $r$ and central angle $\theta$ (in degrees), which of the following formulas are correct?

(A) Area of sector $= \frac{\theta}{360} \times \pi r^2$

(B) Length of arc $= \frac{\theta}{360} \times 2\pi r$

(C) Area of sector $= \frac{1}{2} \times \text{arc length} \times r$

(D) Perimeter of sector $= 2r + \text{arc length}$

Question 3. A segment of a circle is the region bounded by:

(A) An arc

(B) A chord

(C) Two radii

(D) A tangent

Question 4. The area of a minor segment of a circle can be calculated by:

(A) Area of corresponding sector $-$ Area of corresponding triangle formed by radii and chord.

(B) Area of corresponding sector $+$ Area of corresponding triangle formed by radii and chord.

(C) Area of the circle $-$ Area of the major segment.

(D) $\frac{\theta}{360} \pi r^2 - \frac{1}{2} r^2 \sin \theta$ (where $\theta$ is in degrees).

Question 5. A circle has a radius of $14\ \text{cm}$. A sector has a central angle of $60^\circ$. (Use $\pi = \frac{22}{7}$). Which statements are correct?

(A) The length of the arc of this sector is $\frac{60}{360} \times 2 \times \frac{22}{7} \times 14 = \frac{1}{6} \times 44 \times 2 = \frac{44}{3}\ \text{cm}$.

(B) The area of this sector is $\frac{60}{360} \times \frac{22}{7} \times 14^2 = \frac{1}{6} \times \frac{22}{7} \times 196 = \frac{1}{6} \times 22 \times 28 = \frac{11 \times 28}{3} = \frac{308}{3}\ \text{cm}^2$.

(C) The perimeter of the sector is $14+14+\frac{44}{3} = 28 + \frac{44}{3}\ \text{cm}$.

(D) The area of the triangle formed by the radii and chord is $\frac{1}{2} \times 14 \times 14 \times \sin 60^\circ = 98 \times \frac{\sqrt{3}}{2} = 49\sqrt{3}\ \text{cm}^2$.

Question 6. The central angle of a sector is $\theta$ (in radians). The radius is $r$. Which formulas are correct?

(A) Area of sector $= \frac{1}{2} r^2 \theta$

(B) Length of arc $= r \theta$

(C) Area of sector $= \frac{\theta}{2\pi} \times \pi r^2$

(D) Area of sector $= \frac{1}{2} \times \text{arc length} \times r$

Question 7. If the area of a sector is $\frac{1}{4}$th the area of the circle, which statements about the central angle are correct?

(A) The angle in degrees is $\frac{1}{4} \times 360^\circ = 90^\circ$.

(B) The angle in radians is $\frac{1}{4} \times 2\pi = \frac{\pi}{2}$.

(C) The angle is a right angle.

(D) The arc length of the sector is $\frac{1}{4}$th the circumference of the circle.

Question 8. Consider a semicircle with radius $r$. Which statements are true?

(A) It is a sector with central angle $180^\circ$ or $\pi$ radians.

(B) Its area is $\frac{1}{2}\pi r^2$.

(C) The length of its arc is $\pi r$.

(D) Its perimeter is $\pi r + 2r$.

Question 9. The area of a sector with angle $\theta$ (in degrees) and radius $r$ is $A_s$. The area of the circle is $A_c$. Which relationship is always true?

(A) $A_s = \frac{\theta}{360} A_c$

(B) $\frac{A_s}{A_c} = \frac{\theta}{360}$

(C) $A_s < A_c$ (for $0 < \theta < 360$)

(D) The arc length of the sector is proportional to the central angle.

Question 10. A chord divides a circle into two segments, minor and major. Which statements are correct?

(A) The area of the circle is the sum of the areas of the minor segment and the major segment.

(B) The perimeter of a segment is the sum of the length of the arc and the length of the chord.

(C) The area of the major segment is the area of the circle minus the area of the minor segment.

(D) A diameter divides the circle into two equal segments (semicircles).

Question 1. A figure is formed by joining a square of side $s$ and an equilateral triangle of side $s$ along one side. Which statements are correct?

(A) The perimeter of the combined figure is $3s+s+s+s = 6s$.

(B) The perimeter consists of three sides of the triangle and three sides of the square.

(C) The area of the combined figure is the sum of the area of the square and the area of the triangle.

(D) Area $= s^2 + \frac{\sqrt{3}}{4}s^2$.

Question 2. A path of uniform width $w$ runs outside around a rectangular park of length $l$ and width $b$. Which calculations are involved in finding the area of the path?

(A) Calculate the area of the inner rectangle (park): $lw$.

(B) Calculate the dimensions of the outer rectangle (park + path): length $l+2w$, width $b+2w$.

(C) Calculate the area of the outer rectangle: $(l+2w)(b+2w)$.

(D) Subtract the area of the inner rectangle from the area of the outer rectangle.

Question 3. A semicircular lawn is attached to one side of a rectangular building. To find the total area of the lawn and the perimeter of its boundary (excluding the building wall), which measurements are needed?

(A) The dimensions of the rectangular building.

(B) The diameter of the semicircle (which is the length of the side of the building it's attached to).

(C) The radius of the semicircle.

(D) The area of the rectangular building.

Question 4. A circular garden of radius $R$ has a concentric circular path inside it with width $w$. To find the area of the path, which steps are correct?

(A) The radius of the inner circle is $R-w$.

(B) The area of the outer circle is $\pi R^2$.

(C) The area of the inner circle is $\pi (R-w)^2$.

(D) The area of the path is the area of the outer circle minus the area of the inner circle.

Question 5. A square piece of cloth has a circular hole cut out from the center. To find the area of the remaining cloth, which information is needed and how is it used?

(A) The side length of the square ($s$).

(B) The radius ($r$) or diameter of the circular hole.

(C) Calculate the area of the square ($s^2$).

(D) Calculate the area of the circle ($\pi r^2$) and add it to the area of the square.

Question 6. Two crossing paths run in the middle of a rectangular park, parallel to its sides. The length of the park is $L$, width $W$, and the width of each path is $w$. Which statements are correct about the area of the paths?

(A) The area of the path along the length is $Lw$.

(B) The area of the path along the width is $Ww$.

(C) The area of the central square where paths overlap is $w^2$.

(D) The total area of the paths is $Lw + Ww - w^2$.

Question 7. To calculate the perimeter of a composite figure, you need to:

(A) Identify all the individual shapes that make up the figure.

(B) Calculate the perimeter of each individual shape.

(C) Sum the lengths of all the boundary segments of the combined figure.

(D) Exclude any internal lines or segments that are not part of the outer boundary.

Question 8. A circular lawn has a radius of $10\ \text{m}$. Inside it, a square flower bed is made such that its vertices are on the circumference of the circle. Which of the following calculations or properties are correct?

(A) The diagonal of the square is equal to the diameter of the circle ($20\ \text{m}$).

(B) If the diagonal of the square is $d$, its side length $s = d/\sqrt{2}$.

(C) The side length of the square is $20/\sqrt{2} = 10\sqrt{2}\ \text{m}$.

(D) The area of the remaining lawn is Area of circle $-$ Area of square.

Question 9. A design is made from four quadrants of a circle of radius $r$ joined to form a larger shape. Which descriptions or calculations are relevant?

(A) Each quadrant is a sector with a $90^\circ$ central angle.

(B) The area of each quadrant is $\frac{1}{4}\pi r^2$.

(C) The perimeter of each quadrant is $2r + \frac{1}{4}(2\pi r)$.

(D) The perimeter of the composite figure depends on how the quadrants are joined.

Question 10. Consider a figure formed by placing a rectangle on top of a square, such that one side of the rectangle matches the top side of the square. To find the perimeter of the combined figure, you need to sum the lengths of:

(A) Three sides of the square.

(B) Two lengths and one width of the rectangle.

(C) Three sides of the square and three sides of the rectangle.

(D) The bottom side of the square, the two vertical sides of the square, the two vertical sides of the rectangle, and the top side of the rectangle.

Question 1. Which of the following are examples of three-dimensional shapes?

(A) Cube

(B) Cylinder

(C) Sphere

(D) Square

Question 2. Identify the components that make up the boundary of a polyhedron.

(A) Faces

(B) Edges

(C) Vertices

(D) Curved surfaces

Question 3. Which statements are true about a cuboid?

(A) It has $6$ rectangular faces.

(B) It has $12$ edges.

(C) It has $8$ vertices.

(D) All its faces are squares.

Question 4. Which of the following solid shapes have at least one curved surface?

(A) Sphere

(B) Cylinder

(C) Cone

(D) Cube

Question 5. A prism is a solid with:

(A) Two identical and parallel polygonal bases.

(B) Lateral faces that are rectangles or parallelograms.

(C) Faces that meet at a single vertex (Incorrect for prism).

(D) A circular base and a vertex (Incorrect for prism).

Question 6. A pyramid is a solid with:

(A) A polygonal base.

(B) Triangular lateral faces.

(C) Lateral faces that meet at a common vertex (apex).

(D) Two identical and parallel polygonal bases (Incorrect for pyramid).

Question 7. Euler's formula $F + V - E = 2$ applies to which type of solids?

(A) All solid shapes

(B) Polyhedrons

(C) Shapes with flat faces only

(D) Cubes and pyramids

Question 8. For a square pyramid, which statements about its faces, vertices, and edges are correct?

(A) It has $5$ faces ($1$ square base, $4$ triangular lateral faces).

(B) It has $5$ vertices ($4$ base vertices, $1$ apex).

(C) It has $8$ edges ($4$ base edges, $4$ lateral edges).

(D) Applying Euler's formula: $5+5-8 = 2$.

Question 9. A cylinder has:

(A) Two circular bases.

(B) A curved lateral surface.

(C) Edges where the bases meet the curved surface.

(D) Vertices.

Question 10. A sphere has:

(A) One curved surface.

(B) No edges.

(C) No vertices.

(D) A flat base (Incorrect).

Question 1. Surface area is measured in square units. Which of the following are correct units for surface area?

(A) $\text{cm}^2$

(B) $\text{m}^2$

(C) $\text{km}^2$

(D) $\text{cm}^3$

Question 2. For a solid cuboid with dimensions $l, w, h$, which formulas are correct?

(A) TSA $= 2(lw + wh + hl)$

(B) LSA $= 2(l+w)h$

(C) Area of top face $= lw$

(D) Area of one side face (width $\times$ height) $= wh$

Question 3. For a solid cylinder with radius $r$ and height $h$, which formulas are correct?

(A) CSA $= 2\pi r h$

(B) Area of one base $= \pi r^2$

(C) TSA $= 2\pi r h + 2\pi r^2$

(D) TSA $= \pi r (2h + r)$

Question 4. For a solid cone with radius $r$, height $h$, and slant height $l$, which formulas are correct?

(A) $l = \sqrt{r^2 + h^2}$

(B) CSA $= \pi r l$

(C) Area of base $= \pi r^2$

(D) TSA $= \pi r^2 + 2\pi r l$

Question 5. For a sphere with radius $r$, which formulas are correct?

(A) Surface Area $= 4\pi r^2$

(B) Surface Area $= \pi d^2$ where $d=2r$

(C) Volume $= \frac{4}{3}\pi r^3$

(D) Surface Area $= (\text{Circumference of great circle})^2 / \pi$

Question 6. For a solid hemisphere with radius $r$, which formulas are correct?

(A) CSA $= 2\pi r^2$

(B) Area of base $= \pi r^2$

(C) TSA $= 2\pi r^2 + \pi r^2 = 3\pi r^2$

(D) TSA is half the surface area of a full sphere ($4\pi r^2 / 2 = 2\pi r^2$).

Question 7. A cylindrical tank has base diameter $10\ \text{m}$ and height $15\ \text{m}$. It is open at the top. Which calculations are needed to find the area of the metal sheet required? (Use $\pi = 3.14$)

(A) Radius of base $= 5\ \text{m}$.

(B) Area of base $= \pi \times 5^2\ \text{m}^2 = 25\pi\ \text{m}^2$.

(C) CSA $= 2\pi \times 5 \times 15\ \text{m}^2 = 150\pi\ \text{m}^2$.

(D) Total area of sheet = CSA $+$ Area of top base.

Question 8. A conical tent has base radius $7\ \text{m}$ and height $24\ \text{m}$. Which calculations are needed to find the area of the canvas required? (The base is on the ground and not made of canvas. Use $\pi = \frac{22}{7}$)

(A) Slant height $l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25\ \text{m}$.

(B) Area of base $= \pi \times 7^2\ \text{m}^2$.

(C) Area of canvas = CSA of cone $= \pi r l = \frac{22}{7} \times 7 \times 25\ \text{m}^2 = 550\ \text{m}^2$.

(D) Total surface area of cone includes the base area.

Question 9. A sphere of radius $3\ \text{cm}$ is painted. Which statements about its surface area are true? (Use $\pi = 3.14$)

(A) The surface area is $4\pi (3)^2\ \text{cm}^2 = 36\pi\ \text{cm}^2$.

(B) The surface area is approximately $36 \times 3.14 = 113.04\ \text{cm}^2$.

(C) If the radius was doubled to $6\ \text{cm}$, the surface area would become $4\pi (6)^2 = 144\pi\ \text{cm}^2$.

(D) Doubling the radius quadruples the surface area.

Question 10. The LSA of a cuboid represents the area of its:

(A) Top face

(B) Bottom face

(C) Four side faces

(D) All faces

Question 1. Volume is measured in cubic units. Which of the following are correct units for volume?

(A) $\text{cm}^3$

(B) Litre (L)

(C) $\text{m}^3$

(D) Hectare ($\text{ha}$)

Question 2. For a cube with side length $s$, which formulas are correct for its volume ($V$)?

(A) $V = s^3$

(B) $V = s \times s \times s$

(C) $V = (\text{Area of base}) \times \text{height}$

(D) $V = 6s^2$

Question 3. A rectangular water tank has length $4\ \text{m}$, width $2.5\ \text{m}$, and depth $1.5\ \text{m}$. Which statements are true about its capacity?

(A) Its volume is $4 \times 2.5 \times 1.5\ \text{m}^3 = 15\ \text{m}^3$.

(B) $1\ \text{m}^3 = 1000$ litres.

(C) The tank can hold $15 \times 1000 = 15000$ litres of water.

(D) Its surface area is $15\ \text{m}^3$.

Question 4. For a cylinder with base radius $r$ and height $h$, which formula(s) for its volume ($V$) are correct?

(A) $V = \pi r^2 h$

(B) $V = (\text{Area of base}) \times \text{height}$

(C) $V = \frac{1}{3} \pi r^2 h$

(D) $V = \pi (\frac{d}{2})^2 h$ where $d=2r$

Question 5. A cone has base radius $r$ and height $h$. Its volume is $V_{cone} = \frac{1}{3}\pi r^2 h$. This formula suggests a relationship with the volume of a cylinder. Which statements are true?

(A) The volume of a cone is $\frac{1}{3}$rd the volume of a cylinder with the same base radius and height.

(B) If a cone and a cylinder have the same base area and height, they have the same volume.

(C) $V_{cylinder} = 3 \times V_{cone}$ for same $r$ and $h$.

(D) If a cone is inscribed inside a cylinder with the same base and height, its volume is $\frac{1}{3}$rd the cylinder's volume.

Question 6. For a sphere with radius $r$, which formulas are correct for its volume ($V$)?

(A) $V = \frac{4}{3}\pi r^3$

(B) $V = \frac{1}{6}\pi d^3$ where $d=2r$

(C) $V = \frac{4}{3}\pi (\frac{d}{2})^3$

(D) $V = 4\pi r^2$

Question 7. For a hemisphere with radius $r$, which formulas are correct for its volume ($V$)?

(A) $V = \frac{1}{2} \times (\text{Volume of a sphere with radius } r)$

(B) $V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$

(C) $V = \frac{2}{3}\pi (\frac{d}{2})^3$ where $d=2r$

(D) Its volume is half its total surface area (Incorrect concept).

Question 8. A metallic sphere of radius $r_1$ is melted and recast into another sphere of radius $r_2$. Which property remains the same?

(A) Volume

(B) Mass (assuming uniform density)

(C) Surface area (generally changes)

(D) $\frac{4}{3}\pi r_1^3 = \frac{4}{3}\pi r_2^3$, which implies $r_1=r_2$ (Incorrect reasoning if recast into a *different* shape)

Question 9. If the radius of a sphere is tripled, how do its surface area (SA) and volume (V) change?

(A) New radius $= 3r$.

(B) New SA $= 4\pi (3r)^2 = 4\pi (9r^2) = 9 \times (4\pi r^2) = 9 \times$ Original SA.

(C) New V $= \frac{4}{3}\pi (3r)^3 = \frac{4}{3}\pi (27r^3) = 27 \times (\frac{4}{3}\pi r^3) = 27 \times$ Original V.

(D) Surface area increases by a factor of $3$, volume increases by a factor of $9$.

Question 10. A cylindrical pipe has internal diameter $6\ \text{cm}$ and external diameter $8\ \text{cm}$. Its length is $10\ \text{cm}$. The volume of metal used in the pipe can be calculated by:

(A) Volume of cylinder with external radius $-$ Volume of cylinder with internal radius.

(B) Volume $= \pi (R_{ext}^2 - R_{int}^2) \times \text{length}$.

(C) External radius $R_{ext} = 4\ \text{cm}$, Internal radius $R_{int} = 3\ \text{cm}$.

(D) Volume $= \pi (4^2 - 3^2) \times 10\ \text{cm}^3 = \pi (16-9) \times 10 = 70\pi\ \text{cm}^3$.

Question 1. A solid toy is in the form of a cone mounted on a hemisphere. The radius of the hemisphere is $r$, and the height of the cone is $h$. Which statements are correct about calculating its volume?

(A) The radius of the base of the cone is also $r$ if they have the same base.

(B) Volume of cone $= \frac{1}{3}\pi r^2 h$.

(C) Volume of hemisphere $= \frac{2}{3}\pi r^3$.

(D) Total volume of the toy = Volume of cone $-$ Volume of hemisphere.

Question 2. For the toy in Question 111, to calculate its total surface area, you need to consider:

(A) The curved surface area of the cone.

(B) The curved surface area of the hemisphere.

(C) The area of the base of the cone.

(D) The total surface area of the cone and the total surface area of the hemisphere.

Question 3. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is $14\ \text{mm}$ and the diameter of the capsule is $5\ \text{mm}$. Which statements are correct?

(A) The radius of the hemisphere is $2.5\ \text{mm}$.

(B) The height of the cylindrical part is $14 - (2.5 + 2.5) = 14 - 5 = 9\ \text{mm}$.

(C) The volume of the cylinder is $\pi (2.5)^2 \times 9\ \text{mm}^3$.

(D) The volume of each hemisphere is $\frac{2}{3}\pi (2.5)^3\ \text{mm}^3$.

Question 4. To find the total surface area of the capsule in Question 113, you need to sum:

(A) The curved surface area of the cylinder.

(B) The total surface area of the two hemispheres.

(C) The curved surface area of the two hemispheres.

(D) The area of the two circular bases of the cylinder.

Question 5. A solid is made by joining a cylinder and a cone on the same base. The radius of the base is $r$. Which statements are correct about the total volume?

(A) The volume is the sum of the volume of the cylinder and the volume of the cone.

(B) If the height of the cylinder is $h_1$ and the height of the cone is $h_2$, the total volume is $\pi r^2 h_1 + \frac{1}{3}\pi r^2 h_2$.

(C) The total volume is $\pi r^2 (h_1 + \frac{1}{3}h_2)$.

(D) The surface area is the sum of the total surface areas of the cylinder and cone.

Question 6. A bird-bath is in the shape of a hemispherical depression at one end of a cylindrical stand. The height of the cylinder is $1.45\ \text{m}$ and its radius is $30\ \text{cm}$. Which statements are correct? (Convert units to meters. Use $\pi = 3.14$)

(A) Height of cylinder $= 1.45\ \text{m}$.

(B) Radius $= 0.3\ \text{m}$.

(C) Volume of cylinder $= \pi (0.3)^2 \times 1.45\ \text{m}^3$.

(D) Volume of hemispherical depression $= \frac{2}{3}\pi (0.3)^3\ \text{m}^3$.

Question 7. To find the total surface area of the bird-bath (assuming it's open at the top of the hemisphere), you need to sum:

(A) The curved surface area of the cylinder.

(B) The area of the base of the cylinder.

(C) The curved surface area of the hemisphere.

(D) The area of the top of the cylinder.

Question 8. A solid is formed by a cube of side $a$ with a cone placed on its top face, such that the base of the cone is inscribed in the top face of the cube. Which statements are correct?

(A) The diameter of the cone's base is $a$.

(B) The radius of the cone's base is $a/2$.

(C) The volume of the combined solid is the volume of the cube plus the volume of the cone.

(D) Volume $= a^3 + \frac{1}{3}\pi (\frac{a}{2})^2 h_{cone}$.

Question 9. To find the total surface area of the solid in Question 118, you sum:

(A) Area of $5$ faces of the cube.

(B) Curved surface area of the cone.

(C) Area of the top face of the cube ($a^2$).

(D) Area of the base of the cone ($\pi (a/2)^2$).

Question 10. A cylindrical tank is partially filled with water. A solid sphere is gently lowered into the tank until it is fully submerged. Which statements are correct?

(A) The volume of water displaced by the sphere is equal to the volume of the sphere.

(B) The level of water in the tank will rise.

(C) The volume of the sphere can be calculated from the increase in the volume of water in the cylinder.

(D) The total volume in the cylinder after submerging is the initial volume of water plus the volume of the sphere.

Question 1. A solid metallic sphere of radius $R$ is melted and recast into several smaller spheres of radius $r$. Which principle applies here?

(A) The total volume of the smaller spheres equals the volume of the original sphere.

(B) Volume is conserved during the conversion process.

(C) If there are $N$ smaller spheres, $N \times (\frac{4}{3}\pi r^3) = \frac{4}{3}\pi R^3$, so $N r^3 = R^3$, or $N = (R/r)^3$.

(D) The total surface area is also conserved.

Question 2. Earth taken out from a cylindrical well is spread out to form a rectangular platform. Which quantities are equal?

(A) Volume of earth dug out from the well.

(B) Volume of the platform.

(C) Surface area of the well (inner CSA).

(D) Base area of the well.

Question 3. A frustum of a cone is formed by cutting off the top part of a cone with a plane parallel to the base. Which statements are true about a frustum?

(A) It has two circular bases of different radii.

(B) Its lateral surface is curved.

(C) It has a common vertex (Incorrect).

(D) Its height is the perpendicular distance between the two bases.

Question 4. For a frustum of a cone with radii $r_1, r_2$ and height $h$, the slant height $l$ is given by $l = \sqrt{h^2 + (r_1-r_2)^2}$. Which other statements are correct about $l$?

(A) This formula is derived using the Pythagorean theorem on a right triangle formed by the height and the difference in radii.

(B) $l$ is the shortest distance between the circumferences of the two bases along the surface.

(C) The formula for slant height is always $l = \sqrt{h^2 + r_1^2}$.

(D) The slant height is needed to calculate the curved surface area.

Question 5. The formula for the volume of a frustum of a cone is $V = \frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2)$. How can this be conceptualized?

(A) It is the volume of the original large cone minus the volume of the small cone that was removed.

(B) If the original cone had radius $R$ and height $H$, and the removed cone had radius $r$ and height $h'$, then $V = \frac{1}{3}\pi R^2 H - \frac{1}{3}\pi r^2 h'$. ($r_1=R, r_2=r, h=H-h'$)

(C) The formula is derived from the formula for the volume of a cone using similar triangles.

(D) The volume is simply $\pi \times h \times (\text{average area of bases})$.

Question 6. The formula for the curved surface area of a frustum of a cone is CSA $= \pi (r_1 + r_2) l$. Which statements are correct?

(A) This is equivalent to the CSA of a cone with radius $(r_1+r_2)$ and slant height $l$.

(B) This formula is obtained by subtracting the CSA of the small cone from the CSA of the original cone.

(C) CSA $= \pi r_1 l_1 - \pi r_2 l_2$, where $l_1$ and $l_2$ are slant heights of the original and removed cones, respectively.

(D) This formula represents the area of the trapezoidal sector obtained by unrolling the frustum's lateral surface.

Question 7. A bucket in the shape of a frustum of a cone has radii $15\ \text{cm}$ and $25\ \text{cm}$ and height $24\ \text{cm}$. Which calculations are needed to find the amount of sheet metal required to make it (open at the top)?

(A) Calculate the slant height $l = \sqrt{24^2 + (25-15)^2} = \sqrt{576 + 100} = \sqrt{676} = 26\ \text{cm}$.

(B) Calculate the CSA of the frustum $= \pi (15+25) \times 26\ \text{cm}^2 = 40 \times 26 \pi = 1040\pi\ \text{cm}^2$.

(C) Calculate the area of the bottom base $= \pi (15)^2 = 225\pi\ \text{cm}^2$.

(D) Total area of sheet = CSA + Area of bottom base.

Question 8. A solid metallic cylinder is melted and recast into a wire of the same radius. Which statements are correct?

(A) The volume of the cylinder is equal to the volume of the wire.

(B) The wire is a very long cylinder.

(C) If the cylinder has height $H$ and the wire has length $L$, $\pi r^2 H = \pi r^2 L$, so $H=L$. (This implies the wire has the same length as the cylinder's height if the radius is the same).

(D) The surface area of the cylinder is equal to the surface area of the wire.

Question 9. A cone of height $h$ is divided into two parts by a plane parallel to the base at half its height ($h/2$). Which statements are correct about the volumes of the two parts?

(A) The upper part is a smaller cone.

(B) The lower part is a frustum.

(C) If the original cone had radius $R$, the smaller cone has radius $R/2$ (by similar triangles).

(D) The volume of the small cone is $\frac{1}{8}$th the volume of the original cone.

Question 10. For the cone divided in Question 129, what is the ratio of the volume of the small cone to the volume of the frustum?

(A) Volume of small cone = $\frac{1}{8}$ Volume of original cone.

(B) Volume of frustum = Volume of original cone $-$ Volume of small cone $= \text{Volume of original cone} - \frac{1}{8} \text{Volume of original cone} = \frac{7}{8} \text{Volume of original cone}$.

(C) The ratio of the volume of the small cone to the volume of the frustum is $\frac{1/8}{7/8} = 1:7$.

(D) The ratio is $1:8$.