Area of a triangle

What happens when you cut a rectangle along a diagonal?

Take a rectangular sheet of paper and draw a line from one corner to the opposite corner. This line is a diagonal. Cut along it and you get two triangles.

Now place one triangle on top of the other. They cover each other exactly, so they have the same area. Since the two triangles together make the whole rectangle, each triangle has half the area of the rectangle. Try it with long rectangles, short rectangles and squares: it always works.

Is every triangle in a rectangle half of it?

A triangle covers half of a rectangle when:

• one side of the triangle is a full side of the rectangle (its base), and
• the third corner of the triangle lies anywhere on the opposite side.

Why? Draw a line straight down from the top corner to the base. It splits the rectangle into two smaller rectangles, and the triangle into two parts. Each part is half of one smaller rectangle. Half plus half gives half of the whole rectangle.

A worked example on grid paper

A rectangle on grid paper is 6 units long and 4 units high, so its area is 6 × 4 = 24 sq units.

1. Triangle along the diagonal: half of 24 = 12 sq units.
2. Triangle with its top corner 2 units from the left on the top side: the dashed line splits the rectangle into a 2 × 4 part and a 4 × 4 part. Half of 8 + half of 16 = 4 + 8 = 12 sq units.

Both triangles look very different, but each covers exactly half of the rectangle: 12 sq units.

Two more quick examples

• A square cut along its diagonal: a square of side 8 cm has an area of 8 × 8 = 64 sq cm, so each triangle is half of 64 = 32 sq cm.
• A triangle in a 10 cm × 6 cm rectangle with its top corner 3 cm from the left: the line down from the top corner makes a 3 × 6 part and a 7 × 6 part. Half of 18 + half of 42 = 9 + 21 = 30 sq cm, which is half of 60, as expected.

How do you find the area of shapes made of rectangles and triangles?

Split the shape into pieces whose areas you can find, then add them.

Example: a shape is a 4 cm × 3 cm rectangle with a triangle on top. The triangle’s base is the 4 cm top side of the rectangle, and its peak is 2 cm above it.

• Rectangle: 4 × 3 = 12 sq cm.
• Triangle: half of a 4 cm × 2 cm rectangle = half of 8 = 4 sq cm.
• Total area = 12 + 4 = 16 sq cm.

On grid paper, you can also count: full squares count as 1, and squares cut exactly in half by a slanting line count as ½.

Why do triangles matter?

Many shapes around us, like roofs, kites, flags and plots of land, can be split into rectangles and triangles. Once you can find the area of these two shapes, you can find the area of almost any figure with straight sides. Before moving on, revise the area of a rectangle and square, because every triangle’s area here starts with a rectangle.