Abc Triangle Calculator

Background.

The textbook area formula, half the base times the height, assumes you know the height. In practice you rarely do. Stand in a triangular garden bed, a sloping attic gable or a wedge-shaped plot of land with a tape measure and the three sides are easy to measure; the perpendicular height from one corner to the opposite side is not, and neither are the angles.

Three sides are enough. Any three lengths that can close into a triangle fix its shape completely — this is the side-side-side rule from congruence — so the area must follow from them alone. Heron of Alexandria, writing in the first century AD, gave the recipe: take half the perimeter, called the semiperimeter s, multiply it by how far it exceeds each side in turn, and take the square root. No trigonometry is involved, only arithmetic.

The same idea extends to awkward shapes. A four-sided field with no right angles can be split along a diagonal into two triangles; measure the four sides and the diagonal, find each triangle's area from its three sides, and add them. Surveyors broke land into triangles for centuries for exactly this reason.

The sides must actually make a triangle. Each one has to be shorter than the other two together: 3, 4 and 8 cannot meet at the corners, and 3, 4 and 7 collapse into a flat line with no area at all. Lengths that fail this test are refused rather than producing a meaningless number. Any unit works as long as all three sides use the same one; the perimeter comes back in that unit and the area in its square.

What is abc triangle calculator?

An SSS triangle is one specified by its three side lengths, conventionally labelled a, b and c. Because three valid sides determine a unique triangle, both its perimeter and its area follow directly: the perimeter is the sum of the sides, and the area comes from Heron's formula, which uses only those lengths. Angles and heights are not needed and are not reported here.

How to use this calculator.

1. 01Measure the three sides in the same unit — all metres, all feet or all centimetres — and enter them as side a, side b and side c. The order does not matter.
2. 02Check that the longest side is shorter than the other two added together; if it is equal or longer, the lengths cannot form a triangle and will be refused.
3. 03Read the area in square units of whatever you measured in, and the perimeter in the original unit.
4. 04For a four-sided shape, also measure one diagonal, run each of the two resulting triangles separately and add the areas.

The formula.

First the semiperimeter: s = (a + b + c) ÷ 2. Then Heron's formula, A = √[s(s − a)(s − b)(s − c)]. Each bracket, s minus a side, is how much half the perimeter exceeds that side, and the triangle inequality guarantees all three are positive for a real triangle. When the longest side approaches the sum of the other two, its bracket shrinks toward zero and so does the area, which is why a long, thin triangle can have a large perimeter and very little area. Scaling all three sides by a factor k multiplies the perimeter by k and the area by k², so doubling every side quadruples the area. The perimeter output is simply a + b + c, which equals 2s.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.