Geometry and MeasuresSurface Area and Volume

of a Prism

Keywords

Prism

Cross-section

volume

To do now task

Name at least 5 different 3D shapes

Wednesday 19th October 2011

Learning objectives:To know, understand and be able to calculate the volume and surface area of a prism

Three-dimensional shapes

Some examples of three-dimensional shapes include:

A triangular prism

A square-based pyramid

A cylinderA cube

A sphere A tetrahedron

What is the definition of a 3D shape?

Why?

Cross-sections of prisms

A prism is a 3-D shape that has a constant cross-section along its length.

For example, this hexagonal prism has the same hexagonal cross-section throughout its length.

What is meant by cross section?

Where would you find it?

Volume of a prismThe volume of a prism is found by multiplying the area of its cross-section A by its length l (or by its height if it is standing on its cross-section).

V = Al

A

l

V = Ah

A

h

or

What is the volume of this L-shaped prism?

Volume of a prism made from cuboids

6 cm

5 cm

3 cm

4 cm

3 cmWe can think of the shape as two cuboids joined together.

Volume of the green cuboid

= 6 × 3 × 3 = 54 cm3

Volume of the blue cuboid

= 3 × 2 × 2 = 12 cm3

Total volume

= 54 + 12 = 66 cm3

Remember, a prism is a 3-D shape with the same cross-section throughout its length.

Volume of a prism

We can think of this prism as lots of L-shaped surfaces running along the length of the shape.

Volume of a prism

= area of cross-section × length

If the cross-section has an area of 22 cm2 and the length is 3 cm,

Volume of L-shaped prism = 22 × 3 = 66 cm3

3 cm

Which method is easier to use?

Why?

Area formulae of 2-D shapes (cross sections)

You should know the following formulae:

b

hArea of a triangle = bh

12

Area of a parallelogram = bh

Area of a trapezium = (a + b)h12

b

h

a

h

b

What is the volume of this triangular prism?

Volume of a prism

5 cm

4 cm

7.2 cm

Area of cross-section = ½ × 5 × 4 = 10 cm2

Volume of prism = 10 × 7.2 = 72 cm3

Work setPage 305Exercise 11CQuestion 1 only10 Minutes time limit

We can find the formula for the surface area of a cuboid as follows.

Surface area of a cuboid =

Formula for the surface area of a cuboid

h

lw

2 × lw Top and bottom

+ 2 × hw Front and back

+ 2 × lh Left and right side

= 2lw + 2hw + 2lh

How can we find the surface area of a cube of length x?

Surface area of a cube

x

All six faces of a cube have the same area.

The area of each face is x × x = x2

Therefore,

Surface area of a cube = 6x2

Can we link the faces of a 3D to the surface area?

How?

Why?

What is the surface area of this L-shaped prism?

Surface area of a prism

6 cm

5 cm

3 cm

4 cm

3 cm To find the surface area of this shape we need to add together the area of the two L-shapes and the area of the six rectangles that make up the surface of the shape.

Total surface area

= 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15

= 110 cm2

5 cm

6 cm

3 cm6 cm

3 cm3 cm

3 cm

It can be helpful to use the net of a 3-D shape to calculate its surface area.

Using nets to find surface area

Here is the net of a 3 cm by 5 cm by 6 cm cuboid.Write down the area of each face.

15 cm2 15 cm2

18 cm2

30 cm2 30 cm2

18 cm2

Then add the areas together to find the surface area.

Surface Area = 126 cm2

Work setPage 306Question 2 onwardsLevel C