The area of a triangle is one of the most used formulas in geometry, and it takes only a base and a height. The method changes slightly with the information you are given, but the core idea never changes. You can check your work with the free tools on Calc Solver.
The Basic Formula
The area of a triangle is one half times base times height. The base can be any side, but the height must be measured straight up from that base to the opposite corner, at a right angle. Using a slanted side as the height is the most common error with this formula.
Step 1: Identify the Base and Height
Pick one side as the base, then find the height that meets it at a right angle. In a right triangle, the two sides around the right angle can serve as base and height directly. In other triangles, the height may fall inside the triangle or outside it, but the formula works the same way.
Step 2: Multiply and Halve
Multiply the base by the height, then take half. For a triangle with base 10 and height 6, the area is one half times 10 times 6, which is 30 square units. Always include square units in the final answer, because area measures surface, not length.
When You Only Know the Three Sides
If you know all three sides but no height, use Heron’s formula. First find the semi-perimeter s, which is half the sum of the three sides. Then the area is the square root of s times (s – a) times (s – b) times (s – c), where a, b, and c are the side lengths. For sides 3, 4, and 5, s is 6, and the area works out to 6 square units, matching the right-triangle result.
Common Mistakes to Avoid
First, using a slanted side as the height. Second, forgetting to halve the product, which doubles the answer. Third, mixing units, such as a base in centimeters and a height in meters. Convert to one unit before computing.
Practice Problems to Try
Find the areas of triangles with base and height (8, 5), (12, 9), and (7, 4). Then use Heron’s formula for sides 5, 5, and 6, and check whether your answer makes sense.