How Heron's formula works
Heron's formula finds a triangle's area directly from its three side lengths, without needing an angle or height. It first computes the semi-perimeter (half the perimeter), then combines it with each side.
- a, b, c — the triangle's three side lengths
- s — semi-perimeter = (a + b + c) / 2
More detail
Why some side combinations don't work
Three lengths only form a real triangle if any two of them add up to more than the third — the triangle inequality. If a side is as long as (or longer than) the sum of the other two, the sides can't meet to close a shape, so the calculator reports no triangle and the area shows as "—".
Quick check. The classic 3-4-5 right triangle gives area 6 and perimeter 12 — a fast way to sanity-check the calculator or your own hand math.
Frequently asked questions
What is the area of a 3-4-5 triangle?
Using Heron's formula: s = (3+4+5)/2 = 6, so Area = √(6·(6−3)·(6−4)·(6−5)) = √(6·3·2·1) = √36 = 6. The perimeter is simply 3+4+5 = 12.
Why does the calculator show "—" for my three side lengths?
That happens when the three lengths can't form a real triangle — either a side is 0 or negative, or the triangle inequality is violated (one side is greater than or equal to the sum of the other two, e.g. sides 1, 1, 10).
Do I need to know an angle or height to use this?
No. Heron's formula only needs the three side lengths — no angles, no height. That makes it useful when you can measure the sides directly but don't know the triangle's angles.
What's the difference between the area and perimeter results?
The perimeter is just the sum of the three sides (a + b + c) — the distance around the triangle. The area is the space enclosed inside, found with Heron's formula from the same three sides.
Does the formula work for any type of triangle?
Yes — Heron's formula works for any valid triangle: right, acute, obtuse, scalene, isosceles, or equilateral, as long as the three lengths satisfy the triangle inequality.