Geometry: Area and Volume for the TMUA

Area of Triangles

The most fundamental formula for the area of a triangle is derived from the fact that any triangle occupies exactly half the space of a rectangle with the same base and perpendicular height. As shown in the following diagram, the area is calculated as:

$Area = \frac{1}{2} \times base \times vertical\ height = \frac{1}{2}bh$

It is vital to distinguish between the vertical height (or perpendicular height) and the slanted height. If the top vertex is not directly above the base, the vertical height is measured as the perpendicular distance from the line containing the base to the vertex. Beyond basic geometry, the TMUA specification requires the use of trigonometry to find the area when the height is unknown but two sides and an included angle are provided:

$Area = \frac{1}{2}ab \sin C$

Area of Parallelograms and Trapezia

A parallelogram is a quadrilateral with two pairs of parallel sides. Its area formula is a direct extension of the rectangle formula. By 'cutting' a right-angled triangle from one side and shifting it to the other, a parallelogram is transformed into a rectangle with the same base and perpendicular height. Therefore:

$Area = base \times perpendicular\ height$

Note that the 'height' must always be the perpendicular distance between the parallel bases, never the length of the slanted side.

A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides. The area is found by taking the average of the two parallel lengths ($a$ and $b$) and multiplying by the perpendicular height ($h$).

$Area = h \times \frac{a + b}{2} = \frac{h}{2}(a + b)$

This formula is also the basis for the Trapezium Rule used in integration to approximate the area under a curve by dividing it into several such strips.

Volume of Cuboids and Right Prisms

A right prism is a three-dimensional solid with a constant cross-section throughout its length, where the edges connecting the two bases are perpendicular to the planes of the bases. The volume of any right prism is given by the product of the area of its cross-section and its length (or height):

$Volume = Area\ of\ cross\-section \times length$

A cuboid is a specific type of right prism where the cross-section is a rectangle. If a cuboid has length $l$, width $w$, and height $h$, its volume is:

$Volume = l \times w \times h$

For other right prisms, you must first calculate the area of the 2D base using the formulae for triangles, parallelograms, or trapezia. For example, a triangular prism with a base triangle of area $A$ and a prism length $L$ has a volume of $V = AL$.

Worked Examples

Example 1: Triangle Area A triangle has sides of $8$ cm and $10$ cm, with an included angle of $30^{\circ}$. Calculate its area. Using $Area = \frac{1}{2}ab \sin C$: $Area = \frac{1}{2} \times 8 \times 10 \times \sin 30^{\circ}$ Since $\sin 30^{\circ} = \frac{1}{2}$: $Area = \frac{1}{2} \times 80 \times \frac{1}{2} = 20$ cm$^{2}$.

Example 2: Volume of a Trapezoidal Prism A right prism has a cross-section in the shape of a trapezium. The parallel sides of the trapezium are $5$ cm and $7$ cm, and the distance between them is $4$ cm. The prism has a length of $10$ cm. Find the volume. First, find the area of the trapezium cross-section: $Area = \frac{4}{2}(5 + 7) = 2 \times 12 = 24$ cm$^{2}$. Now, calculate the volume: $Volume = Area \times length = 24 \times 10 = 240$ cm$^{3}$.