# 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**

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}}

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**

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:

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**

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**

**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.

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.

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}^2**Question 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}