Area of Triangles, Parallelograms and Trapezia

What you will learn

• Find the area of a parallelogram, a triangle and a trapezium.
• Use the perpendicular height correctly.
• Explain where each formula comes from.

Before you start: you should know how to find the area of a rectangle and what perpendicular means. See Area and Perimeter and Lines, Segments, Parallel and Perpendicular.

The idea

Every formula here starts from the rectangle, $A = \text{base} \times \text{height}$. The height is the perpendicular distance from the base to the opposite side or corner. It is shown with a dashed line and a small square, and it is not the length of a slanted side.

Parallelogram

Cut a right-angled triangle off one end and slide it to the other end. The parallelogram turns into a rectangle with the same base and height, so $A = b \times h$.

Triangle

Two copies of a triangle, one turned upside down, fit together to make a parallelogram. So one triangle is half of it: $A = \tfrac{1}{2} b h$.

Trapezium

A trapezium has one pair of parallel sides, $a$ and $b$. Two copies, one flipped, make a parallelogram with base $a + b$. One trapezium is half of that: $A = \tfrac{1}{2}(a + b)h$. You can also say: average the parallel sides, then multiply by the height.

Worked examples

Example 1: Triangle

A triangle has a base of $14$ cm and a perpendicular height of $9$ cm. Find its area.

1. $A = \tfrac{1}{2} \times 14 \times 9$.
2. $14 \times 9 = 126$, and half of $126$ is $63$.

Answer: $63 \text{ cm}^2$

Example 2: Parallelogram with decimals

A parallelogram has a base of $12.5$ m and a perpendicular height of $6$ m. Find its area.

1. $A = 12.5 \times 6 = 75$.

Answer: $75 \text{ m}^2$

Example 3: Trapezium

A garden bed is a trapezium. Its parallel sides are $8$ m and $14$ m and it is $7$ m wide between them. Find its area.

1. Add the parallel sides: $8 + 14 = 22$.
2. $A = \tfrac{1}{2} \times 22 \times 7 = 11 \times 7 = 77$.

Answer: $77 \text{ m}^2$

Watch out: the slanted side of a triangle or parallelogram is almost never the height. Look for the right-angle mark.

Why it matters

Land is rarely a perfect rectangle. Surveyors split fields into triangles and trapezia to find their area, roofers work out the area of sloping roof panels, and sail makers cut triangular sails. Any straight-sided shape can be cut into triangles, which makes these formulas the key to finding almost any area.