How it works
The area of any triangle is
Area = ½ × base × height
The height must be perpendicular to the base you choose, and any side can be the base. In a right triangle the two legs are already perpendicular, so the area is just half their product.
When you know all three sides but no height, use Heron's formula. Let s be half the perimeter; then Area = √(s(s − a)(s − b)(s − c)). The 13-14-15 triangle (area 84) is a contest favourite.
You can also run the formula backwards: if you know the area and a base, the height to that base is 2 × Area ÷ base. Computing the same area with two different bases is a classic way to find an unknown height.
Triangle Area practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
A triangle has base 10 and height 6. Find its area.
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Answer: 30
Area = ½ × 10 × 6 = 30.
A right triangle has hypotenuse 13 and one leg 5. Find its area.
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Answer: 30
The other leg is √(13² − 5²) = √144 = 12. The legs are perpendicular, so the area is 5 × 12 ÷ 2 = 30.
A triangle has sides 13, 14, and 15. Find its area.
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Answer: 84
Half the perimeter is s = 21. Heron's formula gives √(21 × 8 × 7 × 6) = √7056 = 84.
A triangle has area 88.0 and base 16. What is its height?
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Answer: 11
From Area = ½ × base × height, the height is 2 × 88 ÷ 16 = 176 ÷ 16 = 11.
Keep going
Related topics
- Pythagorean theorem
- Special right triangles: 45-45-90 and 30-60-90
- Similar triangles and scale factor
- Polygon interior and exterior angle
Related practice sets in the app
All Geometry topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice