Boneyard Tools

Three ways to find a triangle's area

Base and height, Heron's formula, and SAS each solve for area from different clues. Learn which to use and why they agree.

Open the Triangle Area Calculator →

Base times height, the everyday formula

The formula most people remember is one half times base times height. The catch is that the height must be perpendicular to the chosen base, measured as the straight-line distance from that base to the opposite corner. Any of the three sides can serve as the base as long as you pair it with the matching perpendicular height. A base of 10 and a height of 4 give an area of 20, no matter how slanted the triangle looks.

Heron's formula when you only have sides

Sometimes you know all three side lengths but no height, which is common for a plot of land or a fixed frame. Heron's formula handles exactly this case. Compute the semi-perimeter s as half the sum of the sides, then take the square root of s times each of the three (s minus side) terms. The famous 3-4-5 right triangle has a semi-perimeter of 6, and the formula returns an area of 6 and a perimeter of 12.

SAS for two sides and the angle between them

When you know two sides and the angle wedged between them, the area is one half times the two sides times the sine of that included angle. Sides of 5 and 7 with a 30-degree angle give one half times 35 times 0.5, which is 8.75. To report the perimeter, the tool finds the missing third side with the law of cosines, giving a perimeter of about 15.657625 for that triangle. The angle has to sit strictly between 0 and 180 degrees to describe a real corner.

Why all three methods agree

These are not three different areas but three routes to the same number, chosen by what information you have. Base and height come straight from the definition of area. Heron's formula is derived from the same idea using only the sides, and the SAS formula reconstructs an effective height as one side times the sine of the angle. Give any method a valid description of the same triangle and it lands on the identical area, which is a useful way to cross-check a measurement.

Frequently asked questions

Which method should I use?

Match the method to your data: use base and height if you have a height, three sides for Heron's formula, and two sides with the included angle for SAS. If you have more than enough information, any applicable method works.

What if I know two angles and a side instead?

This tool does not cover the angle-side-angle case directly. You would first solve for a second side using the law of sines, then feed two sides and their included angle into the SAS method.