Chapter 10 Mensuration (Concepts)

Perimeter of Plane Figures

Mensuration is a vital branch of mathematics that focuses on measuring geometric figures. It allows us to quantify properties like length, area, and volume. In this section, we will start by exploring the concept of the perimeter of plane figures. A plane figure is a two-dimensional shape that lies completely on a flat surface (a plane), such as a square, rectangle, triangle, or circle. A closed plane figure is one that has no open ends, its boundary forms a complete loop.

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What is Perimeter?

The Perimeter of a closed plane figure is defined as the total length of its boundary. Imagine walking along the edge of a playground, tracing the outline of a picture frame with your finger, or putting a decorative border around a piece of fabric. The total distance covered in one complete round along the boundary is the perimeter of that figure.

For any closed figure made up of straight line segments (which is called a polygon), the perimeter is simply found by adding the lengths of all its sides.

Rule: Perimeter of any polygon = Sum of the lengths of all its sides.

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Perimeter of a Rectangle

A rectangle is a common four-sided closed figure (quadrilateral). Key properties of a rectangle are that its opposite sides are equal in length and parallel, and all four internal angles are right angles ($90^\circ$). We usually refer to the longer side as the length and the shorter side as the breadth.

Let the length of the rectangle be denoted by $l$ and its breadth be denoted by $b$. A rectangle has two sides of length $l$ and two sides of breadth $b$.

Using the general rule that perimeter is the sum of all side lengths, the perimeter of a rectangle is:

Perimeter $= \text{length} + \text{breadth} + \text{length} + \text{breadth}$

$= l + b + l + b$

Since addition is commutative and associative, we can rearrange and group the terms:

$= (l + l) + (b + b)$

$= 2l + 2b$

We can factor out the common term '2' from both terms:

$= 2 \times (l + b)$

Thus, the formula for the perimeter of a rectangle is:

$\text{Perimeter of a rectangle} = 2 \times (\text{length} + \text{breadth})$

This formula tells us that to find the perimeter of a rectangle, you add its length and breadth, and then multiply the sum by 2.

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Perimeter of a Square

A square is a very special type of rectangle. It is a quadrilateral where all four sides are equal in length, and all four interior angles are right angles ($90^\circ$).

Let the length of each side of the square be denoted by $s$. Since all four sides are equal, the length of each side is $s$.

Using the general rule that perimeter is the sum of all side lengths, the perimeter of a square is:

Perimeter $= \text{side} + \text{side} + \text{side} + \text{side}$

$= s + s + s + s$

Since we are adding the same quantity ($s$) four times, this can be written as multiplication:

$= 4 \times s$

Thus, the formula for the perimeter of a square is:

$\text{Perimeter of a square} = 4 \times \text{side}$

This formula means that if you know the length of one side of a square, you can find its perimeter by multiplying that length by 4.

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Perimeter of a Triangle

A triangle is the simplest possible polygon, having three sides and three angles. Triangles can vary greatly in the lengths of their sides:

• Scalene Triangle: All three sides have different lengths.
• Isosceles Triangle: Two sides have equal length.
• Equilateral Triangle: All three sides have equal length.

Regardless of the type of triangle, its perimeter is found by adding the lengths of its three sides.

Let the lengths of the three sides of a triangle be $a, b,$ and $c$.

The perimeter of the triangle is the sum of the lengths of its three sides:

$\text{Perimeter of a triangle} = a + b + c$

For an equilateral triangle with side length $s$, the perimeter would be $s+s+s = 3 \times s$.

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Perimeter of Other Polygons

The fundamental rule for finding the perimeter remains the same for any polygon, no matter how many sides it has: add up the lengths of all its sides.

Rule: The perimeter of any polygon = Sum of the lengths of all its sides.

Example: Find the perimeter of a pentagon (a 5-sided polygon) with side lengths 3 cm, 4 cm, 2 cm, 5 cm, and 3 cm.

Perimeter $= 3 \text{ cm} + 4 \text{ cm} + 2 \text{ cm} + 5 \text{ cm} + 3 \text{ cm} = 17 \text{ cm}$.

Perimeter of a Regular Polygon

A regular polygon is a special type of polygon where all its sides are equal in length, and all its interior angles are equal in measure. Examples are an equilateral triangle (3 sides), a square (4 sides), a regular pentagon (5 sides), a regular hexagon (6 sides), a regular octagon (8 sides), and so on.

If a regular polygon has $n$ sides, and each side has a length denoted by $s$, then its perimeter can be calculated using a simple formula derived from the general rule. Since all $n$ sides have the same length $s$, the sum of the lengths is $s$ added to itself $n$ times.

$\text{Perimeter of a regular polygon} = \underbrace{s + s + s + ... + s}_{n \text{ times}}$

$= n \times s$

Thus, the formula for the perimeter of a regular polygon is:

$\text{Perimeter of a regular polygon} = \text{Number of sides} \times \text{Length of one side}$

$= n \times s$

Example: Find the perimeter of a regular hexagon (a 6-sided regular polygon) with a side length of 7 cm.

Here, the number of sides $n=6$, and the length of one side $s=7$ cm.

Perimeter $= n \times s = 6 \times 7 \text{ cm} = 42 \text{ cm}$.

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Area of Plane Figures

In the previous section on Perimeter, we learned about the length of the boundary of a closed plane figure. Now, we will look at another important measurement related to plane figures: the amount of surface they cover. This measure is called the area. Mensuration deals with both the perimeter and the area (and later, volume for 3D shapes) of geometric figures.

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What is Area?

The Area of a closed plane figure is the measure of the amount of flat surface enclosed within its boundary. Think of the surface of a table, the floor of a room, or the space covered by a piece of paper. The area tells us how much two-dimensional space a figure occupies.

Area is always measured in square units. This is because we are measuring a two-dimensional space (length $\times$ length). Common units of area include square centimetres ($cm^2$), square metres ($m^2$), square kilometres ($km^2$), etc. If the dimensions are in feet, the area is in square feet ($ft^2$).

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Area on a Grid (Counting Squares)

One fundamental way to understand the concept of area is by visualising a figure placed on a grid made up of unit squares. A unit square is a square with a side length of 1 unit (e.g., 1 cm $\times$ 1 cm, 1 metre $\times$ 1 metre). The area of a unit square is 1 square unit ($1 \text{ unit}^2$).

To find the area of a figure on a grid, we can count the number of unit squares that the figure covers. This method works well for both regular shapes and irregular shapes, although for irregular shapes, it often provides an approximate area.

Here are the general rules for counting squares on a grid to estimate area:

• Count the number of full squares that are completely inside the boundary of the figure. Each full square counts as 1 square unit of area.
• Count the number of squares that are more than half covered by the figure. Each of these is usually counted as one full square (this is an approximation).
• Count the number of squares that are exactly half covered by the figure. Each half square counts as $\frac{1}{2}$ (or 0.5) square unit of area.
• Ignore the squares that are less than half covered by the figure.

The total area is the sum of the areas from the counts of these different types of squares.

For irregular shapes, this method gives an estimation of the area. For regular shapes like rectangles and squares, counting squares on a grid leads directly to formulas for calculating the exact area.

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Area of a Rectangle

Let's consider a rectangle with length '$l$' and breadth '$b$' placed on a grid of unit squares.

If the length of the rectangle is $l$ units and the breadth is $b$ units, we can think of the rectangle as being made up of $b$ rows of unit squares, with each row containing $l$ unit squares. Alternatively, it has $l$ columns of unit squares, with each column containing $b$ unit squares.

To find the total number of unit squares inside the rectangle, we can multiply the number of rows by the number of squares in each row, or the number of columns by the number of squares in each column.

Total number of unit squares = Number of rows $\times$ Number of squares per row

$= b \times l$

Or, Total number of unit squares = Number of columns $\times$ Number of squares per column

$= l \times b$

Since the area is measured by the number of unit squares covered, and the multiplication of numbers is commutative ($l \times b = b \times l$), the area of the rectangle is $l \times b$ square units.

Thus, the formula for the area of a rectangle is:

$\text{Area of a rectangle} = \text{length} \times \text{breadth}$

If the length and breadth are given in centimetres, the area will be in $cm^2$. If they are in metres, the area will be in $m^2$.

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Area of a Square

A square is a special case of a rectangle where the length and the breadth are equal. Both dimensions are simply referred to as the side length, denoted by $s$.

Using the formula for the area of a rectangle, $\text{Area} = \text{length} \times \text{breadth}$, we replace both 'length' and 'breadth' with 'side' ($s$) since they are equal in a square.

$= s \times s$

This can be written using exponents as $s^2$ (s squared).

$\text{Area of a square} = s^2$

So, to find the area of a square, you multiply the length of one side by itself.

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