The Area of a Triangle

A sail on a small boat is a triangle of strong cloth. So is a scarf folded corner to corner, and the end wall of a house under its roof. If you had to buy the cloth for a sail, how much would you need? The sail is not a rectangle, so length × width does not work. How do we find the area of a triangle?

Half a rectangle

Start with the easiest triangle: one with a right angle. Draw a rectangle and cut it from one corner to the opposite corner, along its diagonal. You get two right-angled triangles, exactly the same size.

A rectangle 6 cm long and 4 cm high, cut along its diagonal into two equal right-angled triangles, one blue and one amber

The rectangle's area is 6 × 4 = 24 cm². Each triangle is half of it: 24 ÷ 2 = 12 cm².

Two copies make a parallelogram

What about a triangle with no right angle? Cut out two copies of it. Turn the second copy upside down and put it next to the first, so that one side of each touches. The two triangles fit together into a parallelogram.

A blue triangle and an amber copy of it turned upside down, fitted together along one side to make a parallelogram with the same base and height as the triangle

In The Area of a Parallelogram we found that a parallelogram's area is base × height. The parallelogram here has the same base and the same height as the triangle. And it is made of exactly two triangles. So one triangle is half of it:

area of a triangle = half of base × height

This works for every triangle, because any triangle and its upside-down copy always fit together into a parallelogram.

The sail

The sail has a base of 8 m along the bottom, and its top is 5 m straight above the base. Base × height is 8 × 5 = 40. Half of 40 is 20. So the sail needs 20 m² of cloth.

Just as with the parallelogram, the height is measured straight up from the base at a right angle, not along a slanted side.

Careful

In a triangle with an obtuse angle at the bottom, the top leans out past the end of the base. The height is still the distance straight down from the top to the line of the base, even if it lands outside the triangle.

Try it at home

Fold a sheet of paper from one corner to the opposite corner and cut along the fold. Lay one triangle on the other. Are they the same?
Find the area of a triangle with a base of 10 cm and a height of 6 cm.
A triangle and a rectangle have the same base and the same height. Which has the bigger area, and by how much?

Answers: 1. Yes: each is exactly half the sheet. 2. 10 × 6 = 60, and half of 60 is 30 cm². 3. The rectangle: it has twice the area of the triangle.

Words to know

area of a triangle - half of base × height, because two copies of a triangle make a parallelogram

Big idea

Every triangle is half of a parallelogram with the same base and height. So the area of a triangle is half of base × height.