Surface Area of Standard Solids
Surface Area: Definition and Concept for Solids
The surface area of a three-dimensional (3D) object, or solid, is the total area that the outer surface of the object occupies. It is a measure of the total exposed area of the solid.
Imagine the solid object is completely covered by a thin film or wrapper. The amount of material needed to make this film or wrapper is equal to the surface area of the object. For example:
- If you are painting a room, the area you need to paint (walls, ceiling) is related to the total surface area of the room (considered as a cuboid, minus the floor and openings).
- The amount of material needed to manufacture a closed container (like a tin can or a cardboard box) is determined by its total surface area.
- The rate at which heat transfers from an object often depends on its surface area.
Mathematically, for solids with flat faces (polyhedrons), the surface area is simply the sum of the areas of all its individual faces.
For solids with curved surfaces (like cylinders, cones, spheres), the surface area includes the area of the curved part of the surface, plus the area(s) of any flat base(s).
The diagram illustrates this concept for a cuboid. If you could 'unfold' the cuboid, its surface would flatten out into a 2D shape called a net. The total area of this net is the surface area of the cuboid.
Since surface area is a measure of the extent of a two-dimensional surface, it is always expressed in square units. Common units include square millimetres ($\text{mm}^2$), square centimetres ($\text{cm}^2$), square metres ($\text{m}^2$), square kilometres ($\text{km}^2$), square inches ($\text{in}^2$), square feet ($\text{ft}^2$), etc. The unit used should be consistent with the units of the lengths or dimensions of the solid.
Understanding the concept of surface area is fundamental for calculating how much material is needed to cover or make a solid object.
Total Surface Area (TSA) and Lateral/Curved Surface Area (LSA/CSA)
When calculating the surface area of solid figures, it is often useful to distinguish between the area of the side surfaces and the area(s) of the base(s).
1. Total Surface Area (TSA)
The Total Surface Area (TSA) of a solid figure is the sum of the areas of all its surfaces (faces). This includes the area of all lateral faces (sides) and the area(s) of all base(s) (top and bottom, if applicable).
Think of the TSA as the total area you would cover if you painted the entire exterior of the solid, including any bases it rests upon or any top surfaces.
2. Lateral Surface Area (LSA) / Curved Surface Area (CSA)
This refers specifically to the area of the side surfaces of the solid, intentionally excluding the area of its base(s).
-
Lateral Surface Area (LSA):
This term is typically used for solids that have flat side faces, such as prisms (like cubes, cuboids, triangular prisms) and pyramids. The LSA is the sum of the areas of all the lateral faces (the faces that are not bases).
For a cuboid resting on one of its faces, the LSA would be the area of its four vertical 'walls', excluding the top and bottom faces.
-
Curved Surface Area (CSA):
This term is used for solids that have a curved lateral surface, such as cylinders, cones, and hemispheres. The CSA is the area of the curved part of the surface only, excluding any flat base(s) or flat top surfaces.
For a cylinder, the CSA is the area of the curved wall, excluding the top and bottom circular bases. For a cone, it's the area of the slanted, curved part, excluding the circular base.
The choice between LSA and CSA depends on whether the side surface(s) are flat or curved. Both terms represent the area of the sides excluding the base(s).
Relationship between TSA and LSA/CSA
The relationship between the total surface area and the lateral/curved surface area depends on whether the solid has one or two bases, and whether the bases are included in the specific LSA/CSA definition being used (standard definitions exclude bases). The general relationship is:
$\mathbf{TSA = LSA \$\$ (or \$\$ CSA) + Area \$\$ of \$\$ Base(s)}$
... (1)
The number of bases depends on the type of solid:
- For solids with two identical and parallel bases (like prisms and cylinders):
$\mathbf{TSA = LSA/CSA + 2 \times (Area \$\$ of \$\$ one \$\$ Base)}$
... (1a)
- For solids with only one base (like pyramids and cones):
$\mathbf{TSA = LSA/CSA + Area \$\$ of \$\$ Base}$
... (1b)
- For a sphere: It has no bases. The entire surface is curved. Thus, LSA/CSA is not a standard term, and the total surface area is simply the area of the curved surface.
$\mathbf{TSA = CSA}$
- For a hemisphere (half of a sphere): It has one flat circular base and a curved surface.
$\mathbf{TSA = CSA + Area \$\$ of \$\$ Circular \$\$ Base}$
Note that the term 'lateral surface area' is generally used for polyhedrons, while 'curved surface area' is used for solids with curved sides. For the purpose of the formulas, they both represent the 'area of the sides excluding the bases'.
Visual Examples
Here are visual representations of the distinction between lateral/curved surface area and total surface area for common solids:
| Solid | Diagram | LSA / CSA (Area of side surface(s)) | TSA (Area of all surfaces) |
|---|---|---|---|
| Cuboid (Rectangular Prism) |
|
Area of the 4 rectangular side faces ('walls'). | Area of the 4 side faces + Area of Top face + Area of Bottom face. |
| Cylinder (Right Circular) |
|
Area of the curved surface. | Area of the curved surface + Area of Top circular base + Area of Bottom circular base. |
| Cone (Right Circular) |
|
Area of the curved surface. | Area of the curved surface + Area of Circular Base. |
| Sphere |
|
N/A (or same as TSA) | Area of the entire curved surface. |
| Hemisphere |
|
Area of the curved hemispherical surface. | Area of the curved surface + Area of the flat circular base. |
It is important to understand exactly which surfaces are included when calculating LSA, CSA, or TSA for any given problem.
Surface Area Formulas for Cubes and Cuboids
Cubes and cuboids are fundamental three-dimensional shapes with flat rectangular faces. Their surface areas are calculated by summing the areas of these faces. The formulas for Lateral Surface Area (LSA) and Total Surface Area (TSA) are derived from their dimensions.
Cuboid
A cuboid (or rectangular prism) is a solid figure bounded by six rectangular faces. Its dimensions are typically referred to as length, breadth, and height.
Let the dimensions of the cuboid be:
- Length = $l$
- Breadth (or Width) = $b$
- Height = $h$
Lateral Surface Area (LSA) of Cuboid
The Lateral Surface Area (LSA) of a cuboid is the sum of the areas of its four vertical faces (the 'walls'), excluding the top and bottom faces. Assuming the cuboid is resting on one of its rectangular faces ($l \times b$), the four lateral faces are:
- Front face: Rectangle with dimensions $l \times h$. Area = $l \times h$.
- Back face: Rectangle with dimensions $l \times h$. Area = $l \times h$.
- Left side face: Rectangle with dimensions $b \times h$. Area = $b \times h$.
- Right side face: Rectangle with dimensions $b \times h$. Area = $b \times h$.
The LSA is the sum of these four areas:
$\text{LSA}_{\text{cuboid}} = (l \times h) + (l \times h) + (b \times h) + (b \times h)$
... (1)
Combine the like terms:
$\text{LSA}_{\text{cuboid}} = 2(l h) + 2(b h)$
... (2)
Factor out the common factor $2h$:
$\mathbf{LSA_{\text{cuboid}} = 2h(l + b)}$
... (3)
This formula can also be interpreted as the perimeter of the base rectangle ($2(l+b)$) multiplied by the height ($h$).
Total Surface Area (TSA) of Cuboid
The Total Surface Area (TSA) of a cuboid is the sum of the areas of all six faces. This includes the four lateral faces plus the top and bottom faces.
- Area of Top face: Rectangle with dimensions $l \times b$. Area = $l \times b$.
- Area of Bottom face: Rectangle with dimensions $l \times b$. Area = $l \times b$.
The TSA is the sum of the LSA and the areas of the two bases:
$\text{TSA}_{\text{cuboid}} = \text{LSA}_{\text{cuboid}} + \text{Area(Top Face)} + \text{Area(Bottom Face)}$
... (4)
Substitute the formula for LSA (3) and the areas of the top and bottom faces:
$\text{TSA}_{\text{cuboid}} = 2h(l + b) + (l \times b) + (l \times b)$
[Substituting LSA and base areas into (4)]
$= 2lh + 2bh + 2lb$
Rearrange the terms and factor out the common factor 2:
$\mathbf{TSA_{\text{cuboid}} = 2(lb + bh + hl)}$
... (5)
This formula indicates that the TSA is twice the sum of the areas of the three pairs of identical faces: top/bottom ($l \times b$), front/back ($l \times h$), and left/right ($b \times h$).
Cube
A cube is a special type of cuboid where all six faces are congruent squares. This means that the length, breadth, and height are all equal.
Let the length of each edge (or side) of the cube be '$a$'.
So, for a cube: $l = a$, $b = a$, and $h = a$.
Lateral Surface Area (LSA) of Cube
The LSA of a cube is the sum of the areas of its four vertical faces. Since each face is a square with side length '$a$', the area of each face is $a \times a = a^2$.
$\text{LSA}_{\text{cube}} = \text{Area(Face 1)} + \text{Area(Face 2)} + \text{Area(Face 3)} + \text{Area(Face 4)}$
$= a^2 + a^2 + a^2 + a^2$
... (6)
$\mathbf{LSA_{\text{cube}} = 4a^2}$
... (7)
Alternatively, we can use the LSA formula for a cuboid, substituting $l=a$, $b=a$, and $h=a$ into formula (3):
$\text{LSA}_{\text{cube}} = 2h(l + b) = 2a(a + a) = 2a(2a) = 4a^2$
[Using (3) with $l=b=h=a$]
Total Surface Area (TSA) of Cube
The TSA of a cube is the sum of the areas of all six faces. Since all six faces are identical squares, each with area $a^2$, the total surface area is 6 times the area of one face.
$\text{TSA}_{\text{cube}} = \text{Area(Face 1)} + ... + \text{Area(Face 6)}$
$= a^2 + a^2 + a^2 + a^2 + a^2 + a^2$
... (8)
$\mathbf{TSA_{\text{cube}} = 6a^2}$
... (9)
Alternatively, we can use the TSA formula for a cuboid, substituting $l=a$, $b=a$, and $h=a$ into formula (5):
$\text{TSA}_{\text{cube}} = 2(lb + bh + hl) = 2(a \cdot a + a \cdot a + a \cdot a)$
[Using (5) with $l=b=h=a$]
$= 2(a^2 + a^2 + a^2) = 2(3a^2) = 6a^2$
The formulas for the surface areas of cubes and cuboids are essential for many practical applications involving boxes, rooms, tanks, etc.
Surface Area Formulas for Cylinders
We usually deal with a Right Circular Cylinder, which is a cylinder where the two circular bases are exactly parallel to each other, and the axis connecting the center of the top base to the center of the bottom base is perpendicular to the planes of the bases. This is the most common type of cylinder studied in mensuration.
To calculate the surface area of a right circular cylinder, we need two primary dimensions:
- Radius ($r$): The radius of each of the circular bases. Both bases are identical.
- Height ($h$): The perpendicular distance between the centers of the two circular bases.
The surface of a cylinder consists of two flat circular surfaces (the top and bottom bases) and one curved lateral surface that connects the boundaries of the bases.
Curved Surface Area (CSA) of a Cylinder
The Curved Surface Area (CSA) of a cylinder is the area of its lateral surface only. It excludes the areas of the top and bottom bases. This is the area of the 'tube' or 'wall' part of the cylinder.
Derivation of the Formula:
Imagine you have a label wrapped perfectly around a cylindrical can. The area of this label is the curved surface area of the cylinder. If you carefully peel off this label and flatten it out, you will find that it forms a perfect rectangle.
Let's examine the dimensions of this resulting rectangle:
- The length of the rectangle corresponds to the distance around the circular base of the cylinder. This distance is the circumference of the base circle, which is $2 \pi r$.
- The breadth (or width) of the rectangle corresponds to the height of the cylinder, $h$.
The area of this rectangle is equal to its length multiplied by its breadth. This area is precisely the curved surface area of the cylinder.
$\text{CSA}_{\text{cylinder}} = \text{Area of the rectangle} = \text{Length} \times \text{Breadth}$
... (1)
Substitute the dimensions of the rectangle ($2\pi r$ and $h$):
$\text{CSA}_{\text{cylinder}} = (2 \pi r) \times h$
[Substituting length and breadth into (1)]
$\mathbf{CSA_{\text{cylinder}} = 2 \pi r h}$
... (2)
Formula:
The formula for the Curved Surface Area (CSA) of a right circular cylinder is:
$\textbf{CSA of Cylinder} = \mathbf{2 \pi \times Radius \times Height}$
Or simply, $\mathbf{CSA = 2 \pi r h}$. The unit of CSA is square units (e.g., $\text{cm}^2$, $\text{m}^2$), consistent with it being an area.
Total Surface Area (TSA) of a Cylinder
The Total Surface Area (TSA) of a cylinder is the sum of the areas of all its surfaces. For a closed cylinder, this includes the curved surface area and the areas of both the top and bottom circular bases.
Derivation of the Formula:
The total surface area is the sum of the curved surface area and the areas of the two bases:
$\text{TSA}_{\text{cylinder}} = \text{CSA}_{\text{cylinder}} + \text{Area of Top Base} + \text{Area of Bottom Base}$
... (3)
The top and bottom bases are identical circles, each with radius $r$. The formula for the area of a circle is $\pi r^2$.
Area of Top Base $= \pi r^2$
Area of Bottom Base $= \pi r^2$
Substitute the formula for CSA (2) and the areas of the two bases into equation (3):
$\text{TSA}_{\text{cylinder}} = (2 \pi r h) + (\pi r^2) + (\pi r^2)$
[Substituting CSA and base areas into (3)]
Combine the areas of the two bases:
$= 2 \pi r h + 2 \pi r^2$
... (4)
To simplify the formula, factor out the common terms, which are $2 \pi r$:
$\mathbf{TSA_{\text{cylinder}} = 2 \pi r (h + r)}$
... (5)
Formula:
The formula for the Total Surface Area (TSA) of a right circular cylinder is:
$\textbf{TSA of Cylinder} = \mathbf{2 \pi \times Radius \times (Height + Radius)}$
Or simply, $\mathbf{TSA = 2 \pi r (h + r)}$. The unit is square units (e.g., $\text{cm}^2$, $\text{m}^2$).
Note that if the cylinder is open at the top (like a cylindrical container without a lid), its total surface area would be CSA + Area of Bottom Base = $2 \pi r h + \pi r^2 = \pi r (2h + r)$. Always carefully determine which surfaces are included in the required area.
Examples
Example 1. Find the curved surface area and the total surface area of a right circular cylinder whose height is $10$ cm and base radius is $7$ cm. (Use $\pi = \frac{22}{7}$)
Answer:
Given:
Right circular cylinder.
Height, $h = 10$ cm.
Base radius, $r = 7$ cm.
Value of $\pi = \frac{22}{7}$.
To Find:
1. Curved Surface Area (CSA).
2. Total Surface Area (TSA).
Solution:
Part 1: Calculate the Curved Surface Area (CSA)
Using the formula for CSA of a cylinder, $CSA = 2 \pi r h$. Using formula (2) derived above:
"$CSA = 2 \pi r h$"
Substitute the given values for $\pi$, $r$, and $h$:
"$CSA = 2 \times \frac{22}{7} \times 7 \$ \text{cm} \times 10 \$ \text{cm}$"
[Substituting values]
Simplify the calculation by cancelling the common factor 7 in the denominator and numerator:
"$CSA = 2 \times \frac{22}{\cancel{7}_1} \times \cancel{7}_1 \times 10 \$ \text{cm}^2$"
"$CSA = 2 \times 22 \times 10 \$ \text{cm}^2$"
"$CSA = 44 \times 10 \$ \text{cm}^2$"
"$\mathbf{CSA = 440 \$\$ cm^2}$"
The curved surface area of the cylinder is 440 square centimetres ($\text{cm}^2$).
Part 2: Calculate the Total Surface Area (TSA)
Using the formula for TSA of a cylinder, $TSA = 2 \pi r (h + r)$. Using formula (5) derived above:
"$TSA = 2 \pi r (h + r)$"
Substitute the given values for $\pi$, $r$, and $h$:
"$TSA = 2 \times \frac{22}{7} \times 7 \$ \text{cm} \times (10 \$ \text{cm} + 7 \$ \text{cm})$"
[Substituting values]
First, calculate the sum inside the parenthesis:
"$TSA = 2 \times \frac{22}{7} \times 7 \$ \text{cm} \times (17 \$ \text{cm})$"
[Calculating $10+7$]
Simplify by cancelling the common factor 7:
"$TSA = 2 \times \frac{22}{\cancel{7}_1} \times \cancel{7}_1 \times 17 \$ \text{cm}^2$"
"$TSA = 2 \times 22 \times 17 \$ \text{cm}^2$"
"$TSA = 44 \times 17 \$ \text{cm}^2$"
Perform the multiplication:
$\begin{array}{cc}& & 4 & 4 \\ \times & & 1 & 7 \\ \hline & 3 & 0 & 8 \\ & 44 & \times \\ \hline & 7 & 4 & 8 \\ \hline \end{array}$"$\mathbf{TSA = 748 \$\$ cm^2}$"
Alternate Method for TSA: Use the relationship TSA = CSA + Area of 2 Bases.
We already found CSA = $440 \$ \text{cm}^2$.
Area of one base circle $= \pi r^2 = \frac{22}{7} \times (7 \$ \text{cm})^2 = \frac{22}{7} \times 49 \$ \text{cm}^2$.
Area of one base $= \frac{22}{\cancel{7}_1} \times \cancel{49}^7 \$ \text{cm}^2 = 22 \times 7 \$ \text{cm}^2 = 154 \$ \text{cm}^2$"
Area of two bases $= 2 \times (\text{Area of one base}) = 2 \times 154 \$ \text{cm}^2 = 308 \$ \text{cm}^2$.
TSA = CSA + Area of two bases $= 440 \$ \text{cm}^2 + 308 \$ \text{cm}^2$.
"$\mathbf{TSA = 748 \$\$ cm^2}$"
[Adding 440 and 308]
Both methods give the same result. The total surface area is 748 square centimetres ($\text{cm}^2$).
Surface Area Formulas for Cones
We usually deal with a Right Circular Cone, which is a cone where the circular base is perfectly flat, and the apex (the pointed top vertex) is located directly above the center of the base. The perpendicular distance from the apex to the center of the base is called the height ($h$).
To calculate the surface area of a right circular cone, we need its dimensions:
- Radius ($r$): The radius of the circular base.
- Perpendicular Height ($h$): The distance from the apex to the center of the base, measured along the axis of the cone.
- Slant Height ($l$): The distance from the apex to any point on the circumference of the base. This is the length along the sloped surface of the cone. All points on the circumference are equidistant from the apex in a right circular cone.
The surface of a cone consists of a flat circular base and a curved lateral surface.
Relationship between Radius, Height, and Slant Height
In a right circular cone, the radius ($r$), the perpendicular height ($h$), and the slant height ($l$) form a right-angled triangle. The height is perpendicular to the base radius, and the slant height is the hypotenuse of this triangle.
Using the Pythagorean theorem ($a^2 + b^2 = c^2$), where $a=r$, $b=h$, and $c=l$:
$\mathbf{l^2 = r^2 + h^2}$
[Pythagoras Theorem]
Therefore, if you know any two of these dimensions, you can find the third. To find the slant height if $r$ and $h$ are given:
$\mathbf{l = \sqrt{r^2 + h^2}}$
... (1)
This formula is often needed to calculate the slant height before finding the surface areas if only radius and height are provided.
Curved Surface Area (CSA) of Cone
The Curved Surface Area (CSA) of a cone is the area of its sloped lateral surface only, excluding the area of the circular base. This is the area of the 'hat' part of the cone.
Derivation of the Formula:
Imagine cutting the curved surface of a cone along its slant height from the apex to the base, and then 'unrolling' it. The unrolled curved surface forms a sector of a circle.
Let's examine the properties of this resulting sector:
- The radius of this sector (the straight edges) is equal to the slant height of the cone, $l$.
- The arc length of this sector (the curved edge) is equal to the distance around the base of the original cone, which is the circumference of the base circle, $2 \pi r$.
The area of a circular sector can be calculated using the formula $A_{\text{sector}} = \frac{1}{2} \times (\text{arc length}) \times (\text{radius of sector})$.
$\text{CSA}_{\text{cone}} = \text{Area of the sector} = \frac{1}{2} \times (\text{arc length}) \times (\text{sector radius})$
... (2)
Substitute the values for the sector's arc length ($2\pi r$) and radius ($l$):
$\text{CSA}_{\text{cone}} = \frac{1}{2} \times (2 \pi r) \times l$
[Substituting values into (2)]
Simplify the expression:
"$= \frac{1}{\cancel{2}} \times \cancel{2} \pi r l$"
$\mathbf{CSA_{\text{cone}} = \pi r l}$
... (3)
If the slant height $l$ is not given, you can substitute the formula for $l$ from equation (1):
$\mathbf{CSA_{\text{cone}} = \pi r \sqrt{r^2 + h^2}}$
... (3a)
Formula:
The formula for the Curved Surface Area (CSA) of a right circular cone is:
$\textbf{CSA of Cone} = \mathbf{\pi \times Radius \times Slant \$\$ Height}$
Or simply, $\mathbf{CSA = \pi r l}$. The unit is square units (e.g., $\text{cm}^2$, $\text{m}^2$).
Total Surface Area (TSA) of Cone
The Total Surface Area (TSA) of a cone is the sum of the areas of all its surfaces. For a closed cone, this includes the curved surface area and the area of the single circular base.
Derivation of the Formula:
The total surface area is the sum of the curved surface area and the area of the base:
$\text{TSA}_{\text{cone}} = \text{CSA}_{\text{cone}} + \text{Area of Base}$
... (4)
The base is a circle with radius $r$. The area of a circle is $\pi r^2$.
Area of Base $= \pi r^2$
Substitute the formula for CSA (3) and the area of the base into equation (4):
$\text{TSA}_{\text{cone}} = (\pi r l) + (\pi r^2)$
[Substituting CSA and base area into (4)]
Factor out the common term $\pi r$:
$\mathbf{TSA_{\text{cone}} = \pi r (l + r)}$
... (5)
If the slant height $l$ is not given, you can substitute the formula for $l$ from equation (1):
$\mathbf{TSA_{\text{cone}} = \pi r (\sqrt{r^2 + h^2} + r)}$
... (5a)
Formula:
The formula for the Total Surface Area (TSA) of a right circular cone is:
$\textbf{TSA of Cone} = \mathbf{\pi \times Radius \times (Slant \$\$ Height + Radius)}$
Or simply, $\mathbf{TSA = \pi r (l + r)}$. The unit is square units (e.g., $\text{cm}^2$, $\text{m}^2$).
Examples
Example 1. Find the curved surface area and the total surface area of a cone with a base radius of $12$ cm and a slant height of $20$ cm. (Use $\pi = 3.14$)
Answer:
Given:
Right circular cone.
Base radius, $r = 12$ cm.
Slant height, $l = 20$ cm.
Value of $\pi = 3.14$.
To Find:
1. Curved Surface Area (CSA).
2. Total Surface Area (TSA).
Solution:
Part 1: Calculate the Curved Surface Area (CSA)
Using the formula for CSA of a cone, $CSA = \pi r l$. Using formula (3) derived above:
"$CSA = \pi r l$"
Substitute the given values for $\pi$, $r$, and $l$:
"$CSA = 3.14 \times 12 \$ \text{cm} \times 20 \$ \text{cm}$"
[Substituting values]
Perform the multiplication:
"$CSA = 3.14 \times (12 \times 20) \$ \text{cm}^2$"
"$CSA = 3.14 \times 240 \$ \text{cm}^2$"
"$\mathbf{CSA = 753.60 \$\$ cm^2}$"
The curved surface area is 753.60 square centimetres ($\text{cm}^2$).
Part 2: Calculate the Total Surface Area (TSA)
Using the formula for TSA of a cone, $TSA = \pi r (l + r)$. Using formula (5) derived above:
"$TSA = \pi r (l + r)$"
Substitute the given values for $\pi$, $r$, and $l$:
"$TSA = 3.14 \times 12 \$ \text{cm} \times (20 \$ \text{cm} + 12 \$ \text{cm})$"
[Substituting values]
First, calculate the sum inside the parenthesis:
"$TSA = 3.14 \times 12 \$ \text{cm} \times (32 \$ \text{cm})$"
[Calculating $20+12$]
Perform the multiplication:
"$TSA = 3.14 \times (12 \times 32) \$ \text{cm}^2$"
"$TSA = 3.14 \times 384 \$ \text{cm}^2$"
[Calculating $12 \times 32$]
"$\mathbf{TSA = 1205.76 \$\$ cm^2}$"
Alternate Method for TSA: Use the relationship TSA = CSA + Area of Base.
We found CSA = $753.60 \$ \text{cm}^2$.
Area of base circle $= \pi r^2 = 3.14 \times (12 \$ \text{cm})^2 = 3.14 \times 144 \$ \text{cm}^2$.
Area of base $= 452.16 \$ \text{cm}^2$"
TSA = CSA + Area of base $= 753.60 \$ \text{cm}^2 + 452.16 \$ \text{cm}^2$.
"$\mathbf{TSA = 1205.76 \$\$ cm^2}$"
[Adding 753.60 and 452.16]
Both methods give the same result. The total surface area is 1205.76 square centimetres ($\text{cm}^2$).
Example 2. Find the curved surface area and the total surface area of a cone whose height is $8$ cm and base radius is $6$ cm. (Use $\pi = 3.14$)
Answer:
Given:
Right circular cone.
Height, $h = 8$ cm.
Base radius, $r = 6$ cm.
Value of $\pi = 3.14$.
To Find:
1. Curved Surface Area (CSA).
2. Total Surface Area (TSA).
Solution:
First, we need to find the slant height ($l$) using the given radius and height. Using formula (1) derived above:
"$l = \sqrt{r^2 + h^2}$"
Substitute the values of $r$ and $h$:
"$l = \sqrt{(6 \$ \text{cm})^2 + (8 \$ \text{cm})^2}$"
[Substituting $r=6$, $h=8$]
"$l = \sqrt{36 \$ \text{cm}^2 + 64 \$ \text{cm}^2}$"
[Calculating squares]
"$l = \sqrt{100 \$ \text{cm}^2}$"
[Adding]
"$l = 10 \$ \text{cm}$"
[Taking square root]
The slant height is $10$ cm.
Part 1: Calculate the Curved Surface Area (CSA)
Using the formula $CSA = \pi r l$. Using formula (3) derived above:
"$CSA = \pi r l$"
Substitute the values for $\pi$, $r$, and $l$:
"$CSA = 3.14 \times 6 \$ \text{cm} \times 10 \$ \text{cm}$"
[Substituting values]
"$CSA = 3.14 \times 60 \$ \text{cm}^2$"
"$\mathbf{CSA = 188.40 \$\$ cm^2}$"
The curved surface area is 188.40 square centimetres ($\text{cm}^2$).
Part 2: Calculate the Total Surface Area (TSA)
Using the formula $TSA = \pi r (l + r)$. Using formula (5) derived above:
"$TSA = \pi r (l + r)$"
Substitute the values for $\pi$, $r$, and $l$:
"$TSA = 3.14 \times 6 \$ \text{cm} \times (10 \$ \text{cm} + 6 \$ \text{cm})$"
[Substituting values]
First, calculate the sum inside the parenthesis:
"$TSA = 3.14 \times 6 \$ \text{cm} \times (16 \$ \text{cm})$"
[Calculating $10+6$]
Perform the multiplication:
"$TSA = 3.14 \times (6 \times 16) \$ \text{cm}^2$"
"$TSA = 3.14 \times 96 \$ \text{cm}^2$"
[Calculating $6 \times 16$]
"$\mathbf{TSA = 301.44 \$\$ cm^2}$"
Alternate Method for TSA: Use the relationship TSA = CSA + Area of Base.
We found CSA = $188.40 \$ \text{cm}^2$.
Area of base circle $= \pi r^2 = 3.14 \times (6 \$ \text{cm})^2 = 3.14 \times 36 \$ \text{cm}^2$.
Area of base $= 113.04 \$ \text{cm}^2$"
TSA = CSA + Area of base $= 188.40 \$ \text{cm}^2 + 113.04 \$ \text{cm}^2$.
"$\mathbf{TSA = 301.44 \$\$ cm^2}$"
[Adding 188.40 and 113.04]
Both methods give the same result. The total surface area is 301.44 square centimetres ($\text{cm}^2$).
Surface Area Formulas for Spheres and Hemispheres
Spheres and hemispheres are solid shapes characterized by their continuous curved surfaces. Their surface areas are calculated using their radius and the constant $\pi$.
Sphere
A sphere is a perfectly round three-dimensional object. All points on its surface are equidistant from a fixed central point. This distance is called the radius ($r$). A sphere has no flat faces, edges, or vertices; its entire surface is curved.
Surface Area of Sphere
Since a sphere has only one surface, which is entirely curved, there is no distinction between curved surface area and total surface area. The total surface area is simply referred to as the surface area of the sphere.
The formula for the surface area of a sphere is a significant result in geometry. It was first discovered by the ancient Greek mathematician Archimedes, who showed that the surface area of a sphere is four times the area of a great circle of that sphere (a great circle is a cross-section through the center, like the equator on a globe, with area $\pi r^2$).
Alternatively, Archimedes also proved that the surface area of a sphere is exactly equal to the curved surface area of a right circular cylinder that perfectly encloses the sphere. Such a cylinder would have a radius equal to the sphere's radius ($r$) and a height equal to the sphere's diameter ($2r$).
CSA of enclosing cylinder $= 2 \pi \times (\text{cylinder radius}) \times (\text{cylinder height}) = 2 \pi \times (r) \times (2r) = 4 \pi r^2$.
Thus, the surface area of a sphere is:
$\textbf{Surface \$\$ Area \$\$ of \$\$ Sphere} = \mathbf{4 \pi r^2}$
... (1)
Note: A rigorous derivation of this formula typically involves methods from calculus (integration) and is usually beyond the scope of introductory geometry. The formula is commonly accepted and applied directly.
Formula:
The formula for the surface area of a sphere with radius $r$ is:
$\textbf{Surface \$\$ Area \$\$ of \$\$ Sphere} = \mathbf{4 \pi \times (Radius)^2}$
Or simply, $\mathbf{SA = 4 \pi r^2}$. The unit is square units (e.g., $\text{cm}^2$, $\text{m}^2$).
Hemisphere
A hemisphere is exactly half of a sphere. It is formed when a sphere is cut into two equal halves by a plane passing through its center. A hemisphere has a curved surface (which is half of the sphere's surface) and a flat circular base.
Let the radius be $r$. The radius of the flat circular base is the same as the radius of the original sphere.
Curved Surface Area (CSA) of Hemisphere
The Curved Surface Area (CSA) of a hemisphere is the area of its curved part only. Since it is exactly half of a sphere, its curved surface area is half the surface area of the corresponding full sphere.
$\text{CSA}_{\text{hemisphere}} = \frac{1}{2} \times (\text{Surface Area of Sphere})$
... (2)
Substitute the formula for the surface area of a sphere from (1):
$\text{CSA}_{\text{hemisphere}} = \frac{1}{2} \times (4 \pi r^2)$
[Substituting from (1) into (2)]
Simplify the expression:
$\mathbf{CSA_{\text{hemisphere}} = 2 \pi r^2}$
... (3)
Formula:
The formula for the Curved Surface Area (CSA) of a hemisphere with radius $r$ is:
$\textbf{CSA of Hemisphere} = \mathbf{2 \pi \times (Radius)^2}$
Or simply, $\mathbf{CSA = 2 \pi r^2}$. The unit is square units.
Total Surface Area (TSA) of Hemisphere
The Total Surface Area (TSA) of a solid hemisphere is the sum of the areas of all its surfaces. For a solid hemisphere, this includes the curved surface area and the area of the flat circular base.
$\text{TSA}_{\text{hemisphere}} = \text{CSA}_{\text{hemisphere}} + \text{Area of Circular Base}$
... (4)
The area of the circular base is $\pi r^2$. Substitute the formula for CSA (3) and the area of the base into equation (4):
$\text{TSA}_{\text{hemisphere}} = (2 \pi r^2) + (\pi r^2)$
[Substituting CSA and base area into (4)]
Combine the like terms ($2 \pi r^2 + \pi r^2 = 3 \pi r^2$):
$\mathbf{TSA_{\text{hemisphere}} = 3 \pi r^2}$
... (5)
Formula:
The formula for the Total Surface Area (TSA) of a solid hemisphere with radius $r$ is:
$\textbf{TSA of Hemisphere} = \mathbf{3 \pi \times (Radius)^2}$
Or simply, $\mathbf{TSA = 3 \pi r^2}$. The unit is square units.
Note that sometimes a problem might refer to a hemispherical shell (like a bowl) which might be open at the top. An open hemispherical shell only has the curved surface area ($2 \pi r^2$). Always clarify if the base is included in the area calculation required by the problem.
Examples
Example 1. Find the surface area of a sphere whose radius is $10.5$ cm. (Use $\pi = \frac{22}{7}$)
Answer:
Given:
Sphere.
Radius, $r = 10.5$ cm.
Value of $\pi = \frac{22}{7}$.
To Find:
Surface Area of the sphere.
Solution:
Using the formula for the surface area of a sphere, $SA = 4 \pi r^2$. Using formula (1) derived above:
"$SA = 4 \pi r^2$"
Substitute the given values for $\pi$ and $r$. It's helpful to write $10.5$ as a fraction: $10.5 = 10 \frac{1}{2} = \frac{21}{2}$.
"$SA = 4 \times \frac{22}{7} \times (10.5 \$ \text{cm})^2$"
[Substituting values]
"$SA = 4 \times \frac{22}{7} \times \left(\frac{21}{2} \$ \text{cm}\right)^2$"
"$SA = 4 \times \frac{22}{7} \times \frac{21}{2} \times \frac{21}{2} \$ \text{cm}^2$"
Simplify by cancelling common factors:
"$SA = \cancel{4}^{1} \times \frac{\cancel{22}^{11}}{\cancel{7}_1} \times \frac{\cancel{21}^3}{\cancel{2}_1} \times \frac{21}{\cancel{2}_1} \$ \text{cm}^2$"
[Cancel 4 with $2\times2$, Cancel 7 with 21, Cancel 22 with 2 (or cancel 4 with two 2s and 22 with nothing)]
"$SA = 1 \times 22 \times 3 \times \frac{21}{1} \$ \text{cm}^2$"
"$SA = 22 \times 63 \$ \text{cm}^2$"
"$\mathbf{SA = 1386 \$\$ cm^2}$"
The surface area of the sphere is 1386 square centimetres ($\text{cm}^2$).
Example 2. Find the curved surface area and the total surface area of a hemisphere of radius $7$ cm. (Use $\pi = \frac{22}{7}$)
Answer:
Given:
Hemisphere.
Radius, $r = 7$ cm.
Value of $\pi = \frac{22}{7}$.
To Find:
1. Curved Surface Area (CSA).
2. Total Surface Area (TSA).
Solution:
Part 1: Calculate the Curved Surface Area (CSA)
Using the formula for CSA of a hemisphere, $CSA = 2 \pi r^2$. Using formula (3) derived above:
"$CSA = 2 \pi r^2$"
Substitute the given values for $\pi$ and $r$:
"$CSA = 2 \times \frac{22}{7} \times (7 \$ \text{cm})^2$"
[Substituting values]
"$CSA = 2 \times \frac{22}{7} \times 49 \$ \text{cm}^2$"
Simplify by cancelling the common factor 7:
"$CSA = 2 \times \frac{22}{\cancel{7}_1} \times \cancel{49}^7 \$ \text{cm}^2$"
"$CSA = 2 \times 22 \times 7 \$ \text{cm}^2$"
"$CSA = 44 \times 7 \$ \text{cm}^2$"
"$\mathbf{CSA = 308 \$\$ cm^2}$"
The curved surface area of the hemisphere is 308 square centimetres ($\text{cm}^2$).
Part 2: Calculate the Total Surface Area (TSA)
Using the formula for TSA of a hemisphere, $TSA = 3 \pi r^2$. Using formula (5) derived above:
"$TSA = 3 \pi r^2$"
Substitute the given values for $\pi$ and $r$:
"$TSA = 3 \times \frac{22}{7} \times (7 \$ \text{cm})^2$"
[Substituting values]
"$TSA = 3 \times \frac{22}{7} \times 49 \$ \text{cm}^2$"
Simplify by cancelling the common factor 7:
"$TSA = 3 \times \frac{22}{\cancel{7}_1} \times \cancel{49}^7 \$ \text{cm}^2$"
"$TSA = 3 \times 22 \times 7 \$ \text{cm}^2$"
"$TSA = 66 \times 7 \$ \text{cm}^2$"
"$\mathbf{TSA = 462 \$\$ cm^2}$"
Alternate Method for TSA: Use the relationship TSA = CSA + Area of Base.
We found CSA = $308 \$ \text{cm}^2$.
Area of base circle $= \pi r^2 = \frac{22}{7} \times (7 \$ \text{cm})^2 = \frac{22}{7} \times 49 \$ \text{cm}^2 = \frac{22}{\cancel{7}} \times \cancel{49}^7 \$ \text{cm}^2 = 22 \times 7 \$ \text{cm}^2 = 154 \$ \text{cm}^2$.
TSA = CSA + Area of base $= 308 \$ \text{cm}^2 + 154 \$ \text{cm}^2$.
"$\mathbf{TSA = 462 \$\$ cm^2}$"
[Adding 308 and 154]
Both methods give the same result. The total surface area of the solid hemisphere is 462 square centimetres ($\text{cm}^2$).
Surface Area of Solid Figures (Consolidated)
This section provides a consolidated summary of the key formulas for calculating the Lateral Surface Area (LSA) or Curved Surface Area (CSA) and the Total Surface Area (TSA) for the standard solid shapes discussed in the previous sections. These formulas are essential for solving problems involving the surface area of 3D objects.
Summary Table of Surface Area Formulas
| Solid Shape | Dimensions | Lateral Surface Area (LSA) / Curved Surface Area (CSA) | Total Surface Area (TSA) |
| Cuboid | Length $l$, Breadth $b$, Height $h$ | LSA $= 2(l+b)h$ | TSA $= 2(lb + bh + hl)$ |
| Cube | Side $a$ | LSA $= 4a^2$ | TSA $= 6a^2$ |
| Right Circular Cylinder | Radius $r$, Height $h$ | CSA $= 2 \pi r h$ | TSA $= 2 \pi r (h + r)$ |
| Right Circular Cone | Radius $r$, Height $h$, Slant Height $l$ | CSA $= \pi r l$ | TSA $= \pi r (l + r)$ |
| Sphere | Radius $r$ | CSA $= 4 \pi r^2$ (Entire surface is curved) | TSA $= 4 \pi r^2$ |
| Hemisphere | Radius $r$ | CSA $= 2 \pi r^2$ (Curved part only) | TSA $= 3 \pi r^2$ (Curved part + circular base) |
Note: For the cone formulas, if the slant height ($l$) is not given but the radius ($r$) and height ($h$) are known, calculate $l$ first using the Pythagorean relationship: $l = \sqrt{r^2 + h^2}$.
Key Points Regarding Surface Area Formulas:
- LSA vs. CSA: Both terms refer to the area of the sides, excluding the base(s). LSA is for solids with flat side faces (polyhedrons like prisms, pyramids), while CSA is for solids with curved side surfaces (cylinders, cones, spheres, hemispheres).
- TSA: This always includes the areas of all surfaces enclosing the solid, including all bases.
- Units: Surface area is a measure of two-dimensional extent, so its units are always square units (e.g., $\text{cm}^2$, $\text{m}^2$). Ensure all dimensions are in consistent units before applying the formulas.
- Context is Key: Always read the problem carefully to determine exactly which surface area (lateral/curved or total) is being asked for. For solids like cylinders or hemispheres, the context (e.g., open container, solid sphere) dictates which formula or combination of areas is needed.
Mastering these formulas and understanding their components is crucial for solving problems related to the surface area of standard solid figures.