A triangle's area measures how much flat space is enclosed by its three sides. The most familiar formula is A = 1/2 bh, but it only works directly when you know a base and the perpendicular height to that base. In many problems, the height is not given, so other methods are needed.
Learning three area methods helps you choose the fastest path from the information you have to the answer you need.
The formula A = 1/2 ab sin C uses two side lengths and the included angle between them, making it powerful for trigonometry and coordinate geometry problems. Heron's formula uses only the three side lengths, so it is useful when no height or angle is known. All three methods find the same area when applied correctly.
The key skill is matching the formula to the given measurements and checking that units are squared.
Key Facts
- Base-height method: A = 1/2 bh, where h is perpendicular to the chosen base.
- Sine method: A = 1/2 ab sin C, where C is the included angle between sides a and b.
- Heron's formula: A = sqrt(s(s - a)(s - b)(s - c)), where s = (a + b + c)/2.
- Any side of a triangle can be chosen as the base, but the height must meet that base at a right angle.
- For a right triangle with legs x and y, A = 1/2 xy because the legs are perpendicular.
- Area units are always square units, such as cm^2, m^2, or in^2.
Vocabulary
- Base
- The side of a triangle chosen as the reference side for measuring height.
- Height
- The perpendicular distance from a vertex to the line containing the chosen base.
- Included angle
- The angle formed between two given sides of a triangle.
- Semiperimeter
- Half the perimeter of a triangle, written as s = (a + b + c)/2.
- Heron's formula
- A formula that finds the area of a triangle using only its three side lengths.
Common Mistakes to Avoid
- Using a slanted side as the height is wrong because height must be perpendicular to the chosen base.
- Using A = 1/2 ab sin C with a non-included angle is wrong because C must be the angle between sides a and b.
- Forgetting to divide by 2 in A = 1/2 bh or A = 1/2 ab sin C gives an area twice as large as the correct value.
- Using Heron's formula without checking the triangle inequality is wrong because side lengths must be able to form a real triangle.
Practice Questions
- 1 A triangle has base 14 cm and perpendicular height 9 cm. Find its area.
- 2 A triangle has sides 8 m and 11 m with an included angle of 35 degrees. Use A = 1/2 ab sin C to find its area to the nearest tenth.
- 3 A triangle has side lengths 6, 7, and 10, but no height or angle is given. Which area method should you use, and why?