Area of a triangle
Also known as: triangle area formula
The area of a triangle is half the product of its base and height: A = ½bh. The height must be measured perpendicular to the chosen base, and any of the three sides can serve as the base.
Every triangle is exactly half of a parallelogram with the same base and height, which is where the formula A = ½bh comes from. The base b is any one side, and the height h — also called the altitude — is the perpendicular distance from that side to the opposite vertex. A triangle with base 10 and height 6 has area ½ × 10 × 6 = 30 square units.
The most common mistake is using a slanted side as the height. The altitude must meet the base at a right angle, and in an obtuse triangle it may fall outside the triangle entirely, requiring you to extend the base. In a right triangle the two legs are already perpendicular, so they serve directly as base and height: a right triangle with legs 3 and 4 has area ½ × 3 × 4 = 6.
Several alternate formulas handle cases where the height is not given. Heron's formula uses the three side lengths: with s = (a + b + c) / 2, the area is √(s(s − a)(s − b)(s − c)). The trigonometric form A = ½ab·sin(C) works when you know two sides and the angle between them. For an equilateral triangle with side s, the area is (√3 / 4)s². On the coordinate plane, you can take a horizontal or vertical side as the base and read the height off the axes.
Triangle area appears constantly on the SAT, ACT, CLT, and AMC 8. Test questions rarely hand you a clean base and height — they embed the triangle in a rectangle, split a larger figure into triangles, give coordinates instead of lengths, or supply side lengths that require the Pythagorean theorem to recover the missing altitude first.
Key takeaways
- The area of a triangle is A = ½bh, where h is perpendicular to the chosen base.
- Any side can be the base, as long as you pair it with the matching altitude.
- In a right triangle, the two legs act as base and height.
- Heron's formula, √(s(s − a)(s − b)(s − c)), gives the area from the three side lengths alone.
- A = ½ab·sin(C) gives the area from two sides and their included angle.