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How to Find the Area of a Triangle
Finding the area of a triangle is a core math concept taught in geometry and trigonometry. Whether you are cutting a piece of fabric, building a roof truss, or solving a homework problem, calculating the space inside a triangle is straightforward when you know the base and the height.
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Triangle Dimensions: Base and Perpendicular Height
The Standard Area Formula
The most common way to calculate the area of a triangle is by using its base (b) and its vertical, perpendicular height (h). The formula is:
In plain terms, you multiply the base length by the height, and then divide the result by 2 (or multiply by 0.5).
Why does this formula work? Any triangle can be thought of as exactly half of a corresponding parallelogram. Since the area of a parallelogram is simply base × height, the area of a triangle must be exactly half of that.
Step-by-Step Example Calculations
Example 1: Standard Calculation
Find the area of a triangle with a base of 10 centimeters and a height of 7 centimeters.
- Identify inputs: b = 10 cm, h = 7 cm
- Write the formula: Area = ½ × b × h
- Calculate: Area = 0.5 × 10 × 7
- Multiply base and height: 10 × 7 = 70
- Divide by 2: 70 ÷ 2 = 35
- Final Area: 35 cm²
Example 2: A Right Triangle
Find the area of a right-angled triangle whose perpendicular legs measure 6 inches and 8 inches.
- Identify legs: leg 1 = 6 in (acts as base b), leg 2 = 8 in (acts as height h)
- Write the formula: Area = ½ × b × h
- Calculate: Area = 0.5 × 6 × 8
- Multiply legs: 6 × 8 = 48
- Divide by 2: 48 ÷ 2 = 24
- Final Area: 24 sq in
Alternative Formulas (No Height Known)
Sometimes, you don't know the perpendicular height of the triangle. Depending on the information you have, you can use other formulas:
1. Side-Angle-Side (SAS) Formula
If you know two side lengths (a and b) and the interior angle (C) between them, you can find the area using trigonometry:
Area = ½ × a × b × sin(C)
2. Heron's Formula (SSS)
When you know all three side lengths (a, b, and c), first calculate the semi-perimeter (s):
s = (a + b + c) ÷ 2
Then, apply Heron's equation to find the area:
Area = √(s × (s − a) × (s − b) × (s − c))
Solve Triangles Instantly
Use our interactive geometry solvers to compute triangle properties automatically:
Frequently asked questions
The base can be any of the three sides of the triangle. The height (or altitude) is the perpendicular distance from the chosen base to the opposite corner (vertex). The key is that the height must meet the base at a 90-degree angle. If you choose a different side as the base, the corresponding height will also change.
Yes. If you know all three side lengths but not the height, you can use Heron's Formula. First, calculate the semi-perimeter (s = (a + b + c) ÷ 2). Then compute the area: Area = √(s × (s − a) × (s − b) × (s − c)), where a, b, and c are the side lengths.
A right triangle is the easiest case because its two legs are already perpendicular to each other. One leg acts as the base, and the other leg acts as the height. The area is simply: Area = ½ × leg 1 × leg 2.
In an obtuse triangle (where one angle is greater than 90 degrees), the perpendicular height drawn from the top vertex to the base will fall outside the body of the triangle. To find the height, you must imagine extending the baseline outward. The formula remains exactly the same: Area = ½ × base × height.