Area and Volume of Polygons and Right Prisms for the ESAT

Area of a triangle

The area of a triangle is calculated using the formula $\frac{1}{2} \times \text{base} \times \text{height}$. It is critical to ensure that the height used in this calculation is measured perpendicular to the base.

Area of a parallelogram

The area of a parallelogram is defined as $\text{base} \times \text{height}$. In this context, the height is defined as the perpendicular distance between the base and the side directly parallel to that base.

Area of a trapezium

A trapezium is a quadrilateral with at least one pair of parallel sides. Its area is found by taking $\frac{1}{2} \times \text{height} \times (\text{the sum of the parallel sides})$. As with other polygons, the height must be the perpendicular distance between the parallel sides.

Using the variables on the diagram, where $a$ and $b$ are the lengths of the parallel sides and $h$ is the perpendicular height, the formula is $\text{area} = \frac{1}{2} h (a + b)$.

Volume of a right prism

The volume of any right prism is calculated as $\text{area of cross-section} \times \text{length}$.

A right prism is a three dimensional polygon characterized by parallel, congruent ends. These ends are joined by rectangles. The length of a right prism is defined as the perpendicular distance between these congruent ends. A defining feature of a right prism is that it possesses a constant cross section when cut parallel to its ends, meaning any such cross section is congruent to the shape of the ends themselves.

Prisms are generally named after the shape of their constant cross section, such as a triangular prism or a hexagonal prism. A right prism with a rectangular cross section is specifically called a cuboid. If every edge of a cuboid is of equal length, it is known as a cube.

Worked Example: Area of a triangle

Consider a triangle ABC where the length of side AB is 12 cm and the perpendicular distance from point C to the line AB is 8 cm. What is the area of this triangle?

By applying the formula, the area is $\frac{1}{2} \times \text{base} \times \text{perpendicular height} = \frac{1}{2} \times 12 \times 8 = 48 \text{ cm}^2$.

Worked Example: Area of a parallelogram

Suppose we have a parallelogram PQRS. The side SR is 15 cm and the side QR is 12 cm. The perpendicular distance between the parallel sides SR and PQ is 8 cm. We need to find the total area and the perpendicular distance between the sides QR and PS.

1. To find the area, take SR as the base. The area is $15 \times 8 = 120 \text{ cm}^2$.
2. To find the perpendicular distance between QR and PS, denoted as $x$, take QR as the base. The area remains $120 \text{ cm}^2$, so $12 \times x = 120$. By dividing both sides by 12, we find $x = 10 \text{ cm}$.

Worked Example: Area of a trapezium

In a trapezium WXYZ, WX is parallel to ZY. Side XY is perpendicular to WX. The given lengths are WX = 6 cm, XY = 8 cm, ZY = 12 cm, and WZ = 10 cm. Find the area.

Using the formula $\text{Area} = \frac{1}{2} \times \text{height} \times (\text{sum of parallel sides})$, we substitute the known values. The height is the perpendicular side XY, which is 8 cm. The parallel sides are WX and ZY.

$\text{Area} = \frac{1}{2} \times 8 \times (6 + 12) = 4 \times 18 = 72 \text{ cm}^2$.

Note that the measurement of WZ (10 cm) is not used because it is a slant height and not required for the area formula.

Worked Example: Volume of a cuboid

A rectangular block is represented as a cuboid in the following diagram.

In a cuboid, any face can serve as the cross section. If we take the front face, the area of the cross section is $4 \times 7 = 28 \text{ cm}^2$. Given the length is 10 cm, the volume is $28 \times 10 = 280 \text{ cm}^3$.

Worked Example: Volume of a right triangular prism

A right triangular prism has a cross section that is a right angled triangle. The shorter sides of this triangle are 6 cm and 7 cm. The total length of the prism is 20 cm.

First, calculate the area of the triangular cross section: $\frac{1}{2} \times 6 \times 7 = 21 \text{ cm}^2$.

Then, find the volume by multiplying this area by the length: $21 \times 20 = 420 \text{ cm}^3$.