Year 8 Interactive Maths - Second Edition

## Prisms

Prisms are three-dimensional objects having end faces that are:

Prisms have identical cross-sections if a plane cuts them parallel to the ends.

For example, a cuboid is a rectangular prism.  The ends of a cuboid are rectangular and it has identical rectangular cross-sections when cut by a plane parallel to the ends.

Prisms are named according to the shape of their base (or cross-section).

The following three-dimensional objects are not prisms, as they do not have identical cross-sections when cut by a plane parallel to the base or in the case of the sphere, parallel to a point on the sphere.

If the ends of a solid (i.e. a three-dimensional object) are not regular polygon but have identical cross-sections when cut by a plane parallel to one of the ends, then the solid is said to be an irregular prism.

## Volume of a Prism

In this section, we will consider the volume of a cube, cuboid, cylinder and triangular prism.

If the rectangular box were filled with 1 cm cubes, there would be:

As there are 3 layers,

Now note that the area of the box's base is given by:

From the above discussion, we can derive a formula for the volume of a rectangular box as follows:

###### In general:

The volume, V, of a prism is given by

where A is the area of the base (or cross-section) of the prism and h is the height.

The volume of the following solids are often required to solve real world problems involving quantity, capacity, mass and strength of materials including liquids.

A cube of side-length l units has a volume of V cubic units given by

A cuboid with length l units, width w units and height h units has a volume of V cubic units given by

A cylinder with radius r units and height h units has a volume of V cubic units given by

A triangular prism whose length is l units, and whose triangular cross-section has base b units and height h units, has a volume of V cubic units given by

#### Example 2

Find the volume of a cube of side 5 cm.

#### Example 3

Find the volume of a brick 10 cm by 5 cm by 4 cm.

#### Example 4

Find the volume of the following cylinder:

##### Solution:

So, the volume of the cylinder is 10,857.34 cm3.

#### Example 5

Find the volume of the following triangular prism:

##### Solution:

So, the volume is 10,716 cm3.