Deriving the Area of a Triangle

1. Area of a rectangle and a parallelogram

The area of a shape is how much flat surface it covers. We measure it by counting unit squares, such as squares of side 1 cm, which have an area of 1 square centimetre (1 cm²). A rectangle 6 units long and 4 units wide holds 4 rows of 6 squares, so its area is $6\times 4=24$ square units. In general, $$\text{Area of rectangle}=\text{length}\times \text{width}$$

A parallelogram looks like a rectangle that has been pushed over to one side. Cut a right-angled triangle off one end, slide it to the other end, and you have a rectangle with the same base and the same height as the parallelogram. Nothing was added or lost, so the areas are equal: $$\text{Area of parallelogram}=\text{base}\times \text{perpendicular height}$$

The word perpendicular is important. The height is measured straight up at 90° to the base, not along the slanting side. If the base of a parallelogram-shaped plot is 12 m and its perpendicular height is 5 m, its area is $12\times 5=60$ m², even if the slanting sides are 6 m or 8 m long.

2. A triangle is half of a rectangle

Now look at a triangle on a grid. Take a triangle with a base of 6 units and a perpendicular height of 4 units, and draw the rectangle round it with the same base and height. The rectangle has area $6\times 4=24$ square units. When you count the squares inside the triangle, you find 12, exactly half. The animation below shows this for the triangle.

▶ Count the squares in the triangle and in its rectangle. The triangle is half of the rectangle.

This is easy to see for a right-angled triangle, because the rectangle is cut along its diagonal into two identical triangles. But is it true for every triangle, even one with its top corner in the middle? Yes, and we can show why with the next picture.

▶ Check that the red outside pieces are the same size as the green inside pieces.

The height line from the top corner splits the big rectangle into two smaller rectangles. In each small rectangle, the slanting side of the triangle cuts it into two identical triangles, one inside the triangle and one outside. So the inside triangle takes up exactly half of the rectangle, whatever the position of the top corner.

3. A triangle is half of a parallelogram

There is a second way to see the same thing, which works even if you cannot draw the rectangle neatly. Take any triangle, trace a second identical copy of it and turn the copy through half a turn (180°) so that it fits against the first along one side. The two triangles together form a parallelogram.

▶ The parallelogram is made of two identical triangles, so each triangle is half of it.

The parallelogram has the same base $b$ and the same perpendicular height $h$ as the triangle. We know that the area of a parallelogram is $b\times h$. Because the parallelogram is made of two equal triangles, one triangle is half of it. So we have found the formula:

$$\text{Area of triangle}=\frac{1}{2}\times b\times h\text{or}A=\frac{bh}{2}$$

Both pictures give the same result. Half of a rectangle and half of a parallelogram with the same base and height are the same thing, because the rectangle and the parallelogram have equal areas.

4. The perpendicular height and common mistakes

The height used in the formula is always the perpendicular height. In a right-angled triangle, the two sides that make the right angle can serve as the base and the height. In other triangles, you draw a dashed line from the top corner straight down to the base (or to the base extended, if the triangle is wide, like an open book) so that it meets the base at 90°.

Whichever side you choose as the base, you must use the height that matches it. A triangle has three possible bases and three matching heights, but they all give the same area. We will see this in the next lesson.

Worked examples

Try each one yourself first, then open the solution and compare every step.