Triangle Calculator
Calculate a triangle's area from base × height, or from its three sides using Heron's formula. Validates the triangle inequality and flags impossible sides.
Easiest when you can measure perpendicular height
Perpendicular to the base — not a slanted side
Area
30 square units
Area
30
Perimeter
—
Switch to Three Sides mode to compute
Method
½ × b × h
The calculation
Area = ½ × base × height
½ × 10 × 6 = 30
For a right triangle where you know two sides and want the third, use the Pythagorean Theorem Calculator. Angle-based triangle solving (SAS, ASA, AAS) lives with the Law of Sines and Law of Cosines calculators in the trig section.
How to use this calculator
- Pick a mode — “Base & Height” if you can measure the perpendicular height; “Three Sides” if you only have the side lengths.
- Enter the measurements. In Three Sides mode, the calculator checks that the sides can form a real triangle.
- Read the area in square units of whatever you measured in.
How it works
Base & Height mode uses the classroom formula:
Area = ½ × base × height
The height must be perpendicular to the base — not a slanted side. For a right triangle, the two legs are perpendicular to each other, so either can serve as the base.
Three Sides mode uses Heron’s formula:
s = (a + b + c) / 2 (the semi-perimeter)
Area = √(s × (s−a) × (s−b) × (s−c))
This works for any triangle, including obtuse and scalene shapes, as long as the three sides actually form a triangle (the triangle inequality must hold). The calculator flags impossible sides instead of returning a NaN.
For right triangles specifically, the Pythagorean Theorem Calculator is the right tool when you need to find a missing side from the other two.
Frequently Asked Questions
What's the area formula for a triangle? ▾
There are two common ones depending on what you can measure. If you know the base and its perpendicular height: Area = ½ × base × height. If you only know the three side lengths: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter — that's Heron's formula. This calculator handles both, in the two modes at the top.
What is Heron's formula? ▾
Heron's formula calculates a triangle's area from its three side lengths alone — no perpendicular height needed. It's named after Hero of Alexandria, who proved it in the first century AD. It's particularly useful for surveying, where you can measure side lengths with a tape measure but can't easily drop a perpendicular height across the middle of the triangle.
What is the triangle inequality? ▾
Every side of a triangle must be strictly shorter than the sum of the other two. If side c is greater than or equal to a + b, the 'triangle' is degenerate — it would have to fold flat. The calculator checks this in Three Sides mode and flags impossible inputs rather than returning a nonsense area.
What about right triangles? ▾
A right triangle is just a triangle with a 90° angle. Both modes here work for right triangles, but if you specifically need to find the hypotenuse or a missing leg given the other two, the Pythagorean Theorem Calculator is the right tool.
Can this calculator handle angles? ▾
Not in this version. The two modes here use sides only. For triangles where you know two sides and the angle between them (SAS), two angles and a side (ASA, AAS), or other angle-based combinations, the upcoming Law of Sines and Law of Cosines calculators in the trigonometry section will solve those.