Area of a triangle
Explain the half-base-times-height formula using a matching pair of triangles.
Before you begin
- Rectangle area
- Perpendicular lines
Relate a triangle to a parallelogram
Two identical triangles can be joined to form a parallelogram with the same base and perpendicular height. The triangle occupies half the parallelogram's area, giving A = bh/2. This works for right, acute and obtuse triangles. You may choose any side as the base, but must use the perpendicular distance from that side's line to the opposite vertex.
Find the correct height
The height meets the base, or its extension, at a right angle. It is not generally a sloping side. In an obtuse triangle, the height may lie outside the shape, yet it measures the same perpendicular separation. Mark a right-angle symbol before substituting values. Changing a triangle's slant while preserving its base and perpendicular height does not change its area.
Worked example
A triangle has base 9 cm and perpendicular height 4 cm. Find its area.
Show the worked solution
- Confirm that the 4 cm measurement is perpendicular to the 9 cm base.
- The matching parallelogram has area 9 × 4 = 36 cm².
- The triangle is half that area: 36 ÷ 2 = 18 cm².
18 cm².
Try it yourself
A triangle has base 12 m and area 30 m². What is its perpendicular height?
Common mistakes
- Use a perpendicular height, not an unlabelled sloping side.
- Do not forget to halve the base-height product.
What can you explain now?
Two triangles each have base 8 cm and perpendicular height 3.5 cm, but different sloping sides. Compare their areas.
Compare with the explanation
Both have area 14 cm².
Each area is ½ × 8 × 3.5 = 14. Base and perpendicular height determine the area, so the different slants do not change it.
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Common Core mathematics standards ↗