# Surface Area and Volume

source: antiprism.com

Area formula are used to find surface area, by analysing and summing up all the faces of 3D solids. In addition, area formula can also be used to calculate the volume of a solid. That is because, in general, the volume of any prism is given by:

$V = Ah$

Volume = Area of Cross-Section  x  Height (or Depth)

A number of common solids are shown below:

### Rectangular Prism

Surface area = $2lb+2bh+2lh$

Volume = $lbh$

### Triangular Prism

Surface area = $2 \times \cfrac{1}{2}bh + 3 \ areas \ of \ side \ rectangles$

Volume = $\cfrac{1}{2}bh \times H$

### Cube

Surface area = $6 \times side \times side = 6{s}^{2}$

Volume = $side \times side \times side = {s}^{3}$

### Cylinder

Surface area = $2 \pi {r}^{2}+2 \pi r h$

Volume = $\pi {r}^{2} h$

### Sphere

Surface area = $4 \pi {r}^{2}$

Volume = $\cfrac{4}{3} \pi {r}^{3}$

### Cone

Surface area = $\pi r (r + s)$, where s is the length of the diagonal side

Volume = $\cfrac{1}{3} \pi {r}^{2} \times height$

### Pyramid

Surface area = area of all shapes included in the pyramid

Volume = $\cfrac{1}{3} \times area \ of \ base \times height$

## Composite 3D Shapes

3D shapes can be joined into one irregular piece of solid, however, their surface area and volume can be calculated using their basic shapes.

### Examples

Find the surface area and volume of the following 3D shapes.

1.

Surface area:

a) Top solid area:

$3 \times 9 \times 6 + 2 \times 6 \times 6 = 162 + 72 = 234{mm}^{2}$

(The top solid has 5 surface areas)

b) Bottom solid area:

$2 \times 15 \times 5 + 2 \times 9 \times 5 + ( 2 \times 15 \times 9 - 6 \times 9 ) \\ = 150 + 90 + 216 - 54 = 398{mm}^{2}$

Therefore, the area of the 3D solid:

$234 + 398 = 632{mm}^{2}$

Volume:

$15 \times 9 \times 5 + 6 \times 6 \times 9 \\ = 675 + 324 = 999{mm}^{3}$

2.

Surface area:

a) Cone:

First we could find the diagonal length of the cone's side, s. By Pythagoras' Theorem:

${s}^{2}={10}^{2}+{4}^{2} \\ {s}^{2}=100+16 \\ s=\sqrt{116} \\ s=10.77cm$

Then using the formula of the cone:

$SA=\pi r(r+s)\\ \quad \quad =\pi \cdot 4(4+10.77)\\ \quad \quad =185.61{ cm }^{ 2 }$

b) Semi-sphere

$SA=\cfrac{1}{2} \cdot 4 \pi {r}^{2} \\ SA=\cfrac{1}{2} \cdot 4 \cdot \pi \cdot {4}^{2} \\ SA=100.53{cm}^{2}$

Therefore, the total surface area:

$185.61 + 100.53 = 286.14{cm}^{2}$

Volume:

a) Cone

$V=\cfrac { 1 }{ 3 } \pi { r }^{ 2 }\times height\\ \\ V=\cfrac { 1 }{ 3 } \pi \times { 4 }^{ 2 }\times 10\\ \\ V=167.55{ cm }^{ 3 }$

b) Semi-sphere

$V=\cfrac { 1 }{ 2 } \cdot \cfrac { 4 }{ 3 } \pi { r }^{ 3 }\\ \\ V=\cfrac { 2 }{ 3 } \pi \cdot { 4 }^{ 3 }\\ \\ V=134.04{ cm }^{ 3 }$

Therefore, the total. volume:

$167.55 + 134.04 = 301.59{cm}^{3}$