Why do three sides give the angles?

Because three sides determine a triangle uniquely, up to reflection. The cosine rule recovers each angle from the sides with no further measurement.

What is the triangle inequality?

Any two sides must sum to more than the third. Three lengths that fail it describe no triangle, which is why the page declines rather than returning a number.

Math & Statisticsgeometry

Triangle Calculator

Every property of a triangle from its three sides. Three sides fix a triangle completely, so the angles, heights and both circle radii follow without further measurement, and any two sides must exceed the third or no triangle exists.

Also called: heron formula calculator, triangle area from sides.

Area

26.832816

Area 26.832816 and perimeter 24. The angles are 48.19, 58.41 and 73.4 degrees, so it is scalene and acute. The heights to each side are 7.67, 6.71 and 5.96. Any two sides must sum to more than the third. These do, so the triangle exists.

Perimeter

Angle opposite side a

48.19

Angle opposite side b

58.41

Angle opposite side c

73.4

Height to side a

7.67

Height to side b

6.71

Height to side c

5.96

Inscribed circle radius

2.24

Circumscribed circle radius

4.7

Semi-perimeter

Classification

scalene and acute

On the triangle inequality

Any two sides must sum to more than the third. These do, so the triangle exists.

How this is calculated

Method and background

How this is calculated

Heron's formula gives the area from the sides alone, and the cosine rule gives each angle. Because three sides determine a triangle uniquely, everything else follows: the height to each side is twice the area over that side, the inscribed circle radius is the area over the semi-perimeter, and the circumscribed radius is the product of the sides over four times the area. The triangle inequality is checked first, since three lengths where one is at least the sum of the other two describe no triangle at all.

Heron's formula for the area and the cosine rule for the angles, both from the three sides alone

s: Semi-perimeter
a, b, c: The three sides

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a 7-8-9 triangle

Side a: 7
Side b: 8
Side c: 9

Area26.832816

Heron's formula on s = 12

Open this example

a 3-4-5 triangle is right angled

Side a: 3
Side b: 4
Side c: 5

Area6

boundary: the circumradius of a right triangle is half the hypotenuse

Open this example

Method and limits

What it assumes

A Euclidean triangle with all three sides given.

What it deliberately does not model

Very thin triangles lose precision in Heron's formula; a numerically stable variant matters at extremes.
Degenerate triangles, where one side equals the sum of the others, have zero area and are rejected here.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why do three sides give the angles?: Because three sides determine a triangle uniquely, up to reflection. The cosine rule recovers each angle from the sides with no further measurement.
What is the triangle inequality?: Any two sides must sum to more than the third. Three lengths that fail it describe no triangle, which is why the page declines rather than returning a number.