Surface Area of 3D Shapes
Surface Area of 3D Shapes Revision
Surface Area of 3D Shapes
The surface area of a 3D shape is a measure of the total area of all the surfaces of that shape.
When calculating the surface area of 3D shapes, they can fall into one of the following categories:
- All the surfaces are flat – e.g. cube, cuboid, prism
- Some of the surfaces are curved – cylinder, sphere, cone
Surface Area of Cubes and Cuboids
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-10.31.08-e1669632008857.png)
A cube is made of \textcolor{#00d865}{6} identical square faces, i.e. all the dimensions of the cube are the same.
Area of one face : \textcolor{#00d865}{x} \times \textcolor{#00d865}{x} = \textcolor{#00d865}{x^2}
Surface area of a cube :
\boxed{6\textcolor{#00d865}{x^2}}
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-10.58.04-e1669633388530.png)
A cuboid is made up of \textcolor{#f95d27}{6} rectangular faces, where the opposite faces are identical, i.e. we have \textcolor{#f95d27}{3} pairs of differently sized rectangles.
Area of the rectangles :
- The first rectangle has sides \textcolor{#f95d27}{x} and \textcolor{#f95d27}{y} \rightarrow \textcolor{#f95d27}{xy}
- The second rectangle has sides \textcolor{#f95d27}{x} and \textcolor{#f95d27}{z} \rightarrow \textcolor{#f95d27}{xz}
- The third rectangle has sides \textcolor{#f95d27}{y} and \textcolor{#f95d27}{z} \rightarrow \textcolor{#f95d27}{yz}
Since there are 2 of each of these rectangles, the total surface area of cuboid is:
\boxed{2(\textcolor{#f95d27}{xy}+\textcolor{#f95d27}{xz}+\textcolor{#f95d27}{yz})}
Surface Area of a Cylinder
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-16.12.02.png)
A cylinder is made up of 2 circles and one rectangle, shown in the diagram to the left.
Area of a circle : \textcolor{#10a6f3}{\pi r^2}
Circumference of a circle : \textcolor{#10a6f3}{2 \pi r}
\textcolor{#10a6f3}{r} \rightarrow radius of the circle
Area of rectangle : circumference \times height of cylinder = \textcolor{#10a6f3}{2 \pi rh}
Surface area of a cylinder :
\boxed{\textcolor{#10a6f3}{2 \pi rh} + \textcolor{#10a6f3}{2 \pi r^2}}
Surface Area of a Prism
A prism is a 3D shape that has identical faces at both ends, i.e. the shape has a constant cross-section (shaded faces in the diagram below) in one particular direction. Take a look at the examples below:
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-13.43.05.png)
To calculate the area of any prism you need to work out the area of the cross-section and any of the remaining faces.
Reminder: Area of a triangle : \dfrac{1}{2}bh \, Where b \rightarrow base of the triangle and h \rightarrow height of the triangle
Surface area of an (isosceles) triangular prism :
\boxed{\textcolor{#ffad05}{bh} + \textcolor{#ffad05}{2ls} + \textcolor{#ffad05}{bl}}
\textcolor{#ffad05}{l} \rightarrow length of shape
\textcolor{#ffad05}{s} \rightarrow slanted length of triangle
Surface Area of a Cone
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-11.51.58-e1669636352614.png)
A cone is made up of a circular base and a curved face joining to the circle.
Area of a circle : \textcolor{#f21cc2}{\pi r^2}
Area of the curved face : \textcolor{#f21cc2}{\pi rl}
\textcolor{#f21cc2}{r} \rightarrow radius of the circle
\textcolor{#f21cc2}{l} \rightarrow slanted height of the cone
Surface area of a cone :
\boxed{\textcolor{#f21cc2}{\pi r^2} + \textcolor{#f21cc2}{\pi rl}}
Surface Area of a Sphere
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-12.04.14.png)
Surface area of a sphere :
\boxed{\textcolor{#aa57ff}{4\pi r^2}}
\textcolor{#aa57ff}{r} \rightarrow radius of the circle
Example 1: Cube and Cuboid
The surface area of cuboid A is 108 \text{ cm}^2
The surface area of cuboid A is twice the size of the surface area of cube B
What is the width of cube B?
[3 marks]
Surface area of cube B = 108 \div 2 = 54 \text{ cm}^2
Let’s say cube B has dimensions x \text{ cm}
Then we have,
6x^2=54
x^2=9
x=3
The width of cube B is 3 \text{ cm}
Example 2: Cylinders
Work out the surface area of a cylinder with radius 3 \text{ cm} and height 10 \text{ cm}
Give your answer in terms of \pi
[2 marks]
Surface area of a cylinder: 2\pi r^2 + 2\pi rh
Substituting r=3 and h=10,
Surface area of a cylinder: 2\pi \times 3^2 + 2\pi \times 3 \times 10 = 78\pi \text{ cm}^2
Example 3: Cone
A cone with radius 4 \text{ cm}, has a surface area of 40\pi \text{ cm}^2
Work out the slanted height of the cone
[3 marks]
Surface area of a cone: \pi r^2 + \pi rl
Substituting r=4,
\pi \times 4^2 + \pi \times 4l = 40\pi
16 \pi + 4l \pi = 40\pi
4l \pi = 24\pi
l = 6
The slanted height of the cone is 6 \text{ cm}.
Example 4: Hemisphere
Shown below is a hemisphere with radius 5 \text{ cm}.
Calculate the surface area of the hemisphere.
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-13.12.53.png)
Area of a circle with radius 5 \text{ cm} : \pi \times 5^2 = 25 \pi \text{ cm}^2
Area of a sphere with radius 5 \text{ cm} : 4 \times \pi \times 5^2 = 100 \pi \text{ cm}^2
Area of a hemisphere with radius 5 \text{ cm} : \dfrac{100 \pi}{2} + 25 \pi= 75 \pi \text{ cm}^2
Example 5: Triangular Prism
Work out the surface area of the triangular prism below. Give your answer to two decimal places.
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-14.01.07.png)
![](https://mmerevise.co.uk/app/uploads/2022/11/Screenshot-2022-11-28-at-14.12.54.png)
To calculate the area of the two identical rectangles remaining, we first need to calculate the length of the slanted side of the triangle. To do this we use Pythagoras:
s^2=2^2+3^2 = 13
s = \sqrt{13}
Area of the side rectangles : \sqrt{13} \times 8 = 8\sqrt{13} \text{ cm}^2
Surface area of triangular prism : 8\sqrt{13} + 8\sqrt{13} + 32 + 6 + 6 = 44 + 16\sqrt{13} or 101.69 \text{ cm}^2
Surface Area of 3D Shapes Example Questions
Question 1: A cylinder with diameter 12 \text{ cm} has a surface area of 168 \pi \text{ cm}^2.
Work out the height of the cylinder.
[3 marks]
Radius of the cylinder : 6 \text{ cm}
Surface area of a cylinder : 2 \pi rh + 2 \pi r^2
Substituting r=6,
2\pi \times 6^2 + 2\pi \times 6 \times h = 168\pi
72\pi + 12\pi h = 168\pi
12\pi h = 96\pi
h = 8
The height of the cylinder is 8 \text{ cm}
Question 2:
Work out the surface area of the cuboid.
[3 marks]
Surface area of a cuboid : 2(xy+xz+yz)
Substituting x=5 , z=6 and y=10,
2(5\times 10+5\times 6+6\times 10) = 280 \text{ cm}^2Question 3: A sphere has surface area of 36 \pi \text{ cm}^2.
Work out the diameter of the sphere.
[3 marks]
Surface area of a sphere : 4\pi r^2
4\pi r^2=36\pi
r^2=9
r=3
The sphere has a diameter of 6 \text{ cm}
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