Surface Area and Volume

source: antiprism.com
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.

 

Show Answer

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.

 

Show Answer

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}