Measurement 2: Surface Area and Volume
Mathematics
Topic Six

Previous - Measurement 1: Area and Perimeter

Surface Area

The surface area of a figure is the total area of all the sides of the figure.

Triangular Prism (Right Angle Triangle)

Figure 1

To find the surface area of the prism above, follow the steps below:

Divide the figure into smaller shapes, find the area of each smaller shape and then sum the areas of each smaller shape.  A right-angle triangle prisim can be divided into five smaller shapes which are two right angle triangles and three rectangles.

Example

Find the surface area of the prism below.

Figure 2

Area of right angle triangle = 1/ 2 base  x  height = 1/ 2 x 4 x 3 = 6

As there are 2 of these right angle triangles, we have 6 x 2 = 12

Area of rectangle 1:  4 x 7=  28

Area of rectangle 2:  5 x 7=  35

Area of rectangle 3:  3 x 7= 21

Surface area = 12 + 28  + 35 + 21 = 96cm2

Pyramid

A pyramid is a solid formed by connecting a polygonal base and a point, called the apex.

The surface area of a pyramid = Area of the base + 1/ 2 Perimeter of Base x Length of Side

Example

Find the surface area of the following pyramid

Surface area of pyramid = Area of base + 1/ 2 perimeter of base x length

Figure 3

Area of 6 x 6 = 36cm2

Perimeter = 1/ 2  4(6) = 12cm

Surface area = 36 + 12 x 12 =  36 + 144 = 180 cm2

Cylinder

Figure 4

A cylinder can be divided into three parts: two circles and a curved surface area. 

So the surface area = 2πr2 + 2πrh

The first term is the area of the two circles and the second term is the perimeter of the cylinder.

Example

Find the surface area of a cylinder with a radius of 3 cm, and a height of 2 cm.

Solution

SA = 2 × π × r2  +   2 × π × r × h

SA = 2 × 3.14 × 32  +   2 × 3.14 × 3 × 2

SA = 6.28 × 9  +   6.28 × 3 x 2

SA = 56.52 + 37.68

Surface area = 94.2 cm2 

Cube

A cube has 6 sides or faces of the same surface area. Therefore, if the area of a square is side times side, then the surface area is S times S times 6.

Cuboid

A cuboid is an object with six flat faces and all angles are right angles with all its faces being rectangles.

Figure 5

The surface area of a cuboid, A =  2Lw + 2 Lh + 2wh

where,

L is the length of the cuboid

w is the width and

h is the height of the cuboid.

Example

Find the surface area of a cuboid with length of 5cm, width 6cm and height 7cm.

Solution

A =  2Lw + 2 Lh + 2wh

= 2 x 5 x 6 + 2 x 5 x 7 + 2 x 6 x 7

= 60 + 70 + 84

= 214cm2

Sphere

A sphere is a three-dimensional object such as a ball or the globe with every point on the surface halfway from the center.

Figure 6

The surface area of a sphere, A = 4πr2

where, r is the radius of the sphere.

Example

Find the surface area of a sphere with radius of 9cm.

Solution

Surface Area of a sphere, A = 4 πr2

4 x 3.14 x 9 x  9 = 1018 cm2

Volume

Volume is the amount of 3-dimensional space an object occupies.

Cube

Figure 7

The volume of a cube , V = L3

where, L is the length of a side of the cube.

Example

The volume of a cube with side of 4cm = 4 x 4 x 4 = 64cm3

Cuboid

Figure 8

The volume of a cuboid, V = Lwh

where, L is the length of the cuboid,

w is the width and h is the height of the cuboid.

Sphere

The volume of a sphere, V = 4/3 πr3

where, r is the radius of the sphere.

Example

The volume of a sphere with radius of 7cm = 4/ 3 x 22/ 7 x 7 x 7 x 7 = 4/ 3 (22/ 7) 7x7x7 = 1436.9cm3

Triangular Prism (Right Angle Triangle)

 

Figure 9

The volume of a triangular prism, V =  Area of one of the triangles times length

= 1/ 2 base x height x length of the triangular prism

Example

Find the volume of a triangular prism with base of 6cm, length of 8cm and height of 7cm.

Solution

Volume = 1/  2 (6) x 7 x 8 

= 3 x 7 x 8 = 168cm3

Pyramid

The volume of a pyramid = 1/ 3 Area x height.

Example

Find the volume of a pyramid with length of 9cm and height of 14cm.

Solution

V = 1/ 3 (9 x 9) x 14 =

    = 27 x 14 = 378cm3

Cylinder

The volume of a cylinder, V =  πr2h

where, πr2 is the area of the circular cross-section and h is the height of the cylinder.

Example

Find the volume of a cylinder with radius of 7 cm and height of 14cm.

Solution

Volume = 22/ 7 x 7 x 7 x 14

            = 154 x 14 = 2155cm3

Distance, Speed and Time

There are mathematical problems that require students to be able to calculate distance, time and speed.

Distance = Speed x Time

Time = Distance/Speed

Speed = Distance/Time

Distance

Example

Joe drove from his home to the mall at an average speed of 50 kilometres an hour and took 4 hours to do so.  Calculate the distance from his home to the mall.

Solution

Distance = Speed x Time = 50 x 4 = 200 kilometres.

Speed

Example

Sherry walked to the park located 80 kilometers in 5 hours.  Calculate her speed per hour.

Speed = Distance/Time = 80/5 = 16 kilometers per hour.

Time

A lady travelled a distance of 45 kilometers at a speed of 5 kilometers per hour.  Calculate the time she took.

Speed = Distance/ Speed = 45/5 = 9 hours

Next - Algebra 1: Algebraic Expressions and Indices