The area of a triangle and its heights — Mathematics, 11–13
A triangle is half a parallelogram: area = base × height ÷ 2, with three heights and a position that depends on the type of triangle.
The three heights and the area of a triangle
In a triangle you can pick any side as the base. The matching height is the segment from the opposite vertex, at right angles to the base or to its extension. So a triangle has three heights. The area is base × height ÷ 2. In an acute-angled triangle all three heights lie inside it; in a right-angled one, two of them are the sides of the right angle; in an obtuse-angled one, two fall outside the triangle.
Why we divide by 2
Make two equal copies of a triangle and turn one of them a half turn: together they form a parallelogram with the same base and height. With base 10 cm and height 6 cm, the parallelogram has 10 × 6 = 60 cm² and each triangle is half of it: 30 cm². If you fix the base and drag the third vertex along a line parallel to the base, the height does not change: acute, right or obtuse, the area is always 30 cm².
The three heights of a right-angled triangle
A triangle has a right angle, the sides forming it measure 6 cm and 8 cm, and the longest side measures 10 cm. With base 8, the height is the other side, 6 cm: area = 8 × 6 ÷ 2 = 24 cm². With base 6, the height is 8 cm: 6 × 8 ÷ 2 = 24 cm². With base 10, the height h lies inside the triangle: 10 × h ÷ 2 = 24, so 10 × h = 48 and h = 4.8 cm. Three bases, three heights (6, 8 and 4.8 cm), the same area.
The trap: forgetting the half, or using a slanted side
Two very common mistakes. First: forgetting the ÷ 2 and ending up with the area of the parallelogram (10 × 6 = 60 instead of 30). Second: using a slanted side as the height. The height is measured at right angles to the base, even when it falls outside the triangle, on the extension of the base: that is what happens to two of the heights of an obtuse-angled triangle. It is not a drawing mistake, it is how a height works.
Where you see this outside school
A triangular sail, a pennant or the triangular wall at the top of a roof: to know how much cloth or paint you need, measure a base and its height and work out base × height ÷ 2. If two triangles have the same base and the same height, they cover the same area, even when their shapes look very different.
Keep exploring
Other languages
Loading MyLeoNes™…