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Triangle Calculator

This triangle calculator needs exactly three known values, at least one of them a side – leave the rest blank. It fills in every side, angle and measurement.

Source: Wolfram MathWorld – Law of cosines. Updated: .

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Result

Sides and angles
a = 3 · b = 4 · c = 5 · α = 36.8699° · β = 53.1301° · γ = 90°
Area
6
Perimeter
12
Heights (altitudes)
hₐ = 4 · h_b = 3 · h_c = 2.4
Incircle and circumcircle
inradius r = 1, circumradius R = 2.5
Type of triangle
right, scalene
Working
  • Law of cosines: α = arccos((b² + c² − a²) ÷ (2·b·c)) = arccos((16 + 25 − 9) ÷ 40) = 36.8699°
  • Law of cosines: β = arccos((a² + c² − b²) ÷ (2·a·c)) = arccos((9 + 25 − 16) ÷ 30) = 53.1301°
  • Angle sum: γ = 180° − α − β = 90°
  • Heron’s formula: s = 6; A = √(s(s − a)(s − b)(s − c)) = 6
  • r = A ÷ s = 1; R = abc ÷ (4A) = 2.5
Solved as case SSS (three sides).

How it is calculated

A triangle is fixed by three values as long as at least one is a side: three sides (SSS), two sides and the included angle (SAS), or one side and two angles (ASA, AAS). Only two sides with a non-included angle (SSA) can give two different triangles – the “ambiguous case”.

Formulas

Labels

Sides are a, b, c and angles A, B, C, each angle opposite the side with the same letter. The height hₐ is perpendicular to side a through the opposite vertex. Enter angles in degrees.

Type of triangle

The calculator reports whether the triangle is acute, right or obtuse, and whether it is equilateral, isosceles or scalene. For right triangles with two known sides, the Pythagorean theorem calculator is quicker; sines and cosines of single angles come from the scientific calculator, and you can measure angles on screen with the online protractor.

Frequently asked questions

How do you find the angles of a triangle with three sides?

Use the law of cosines: cos A = (b² + c² − a²) ÷ (2bc). For a 3-4-5 triangle that gives A ≈ 36.87°, B ≈ 53.13° and C = 90°.

How do you find the area of a triangle?

Half base times height, or Heron’s formula from the three sides. From two sides and the included angle: area = ½ab·sin C.

How do you find a missing angle?

If two angles are known, subtract from 180°: C = 180° − A − B. Otherwise use the law of cosines or sines.

What is the ambiguous case?

With two sides and an angle not between them (SSA), the second angle can be acute or obtuse, so two triangles may fit. If the side opposite the given angle is the longer one, there is only one.

When is a triangle impossible?

When two sides together are not longer than the third (triangle inequality) or when the given angles already add up to 180° or more.

Sources and legal basis

As of: · Editorial policy

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