Triangle Calculator — Step-by-Step Solutions

The Triangle Calculator solves triangles given various combinations of sides and angles. It supports SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and AAS (two angles and a non-included side) configurations.

Enter the known values and the calculator will find all missing sides, angles, area, and perimeter using trigonometric formulas and Heron's formula.

c^2 = a^2 + b^2 - 2ab\cos(C)

Side a

Side b

Side c

Your calculations will appear here

How to Use the Triangle Calculator

Select the configuration you know: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), or AAS (two angles and a non-included side).
Enter the known values. Angles should be in degrees.
The calculator finds all missing sides and angles, plus area and perimeter.

Formula and Theory

Triangles are solved using trigonometric laws. The Law of Cosines (c^2 = a^2 + b^2 - 2ab*cos(C)) relates sides and angles, generalising the Pythagorean theorem. The Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) provides another relationship.

The area of a triangle can be calculated several ways: (1/2)*base*height, (1/2)*a*b*sin(C), or Heron's formula using the semi-perimeter s: area = sqrt(s(s-a)(s-b)(s-c)).

Worked Examples

Example 1: SSS triangle

Problem: Find the angles of a triangle with sides 3, 4, 5

Solution: cos(C) = (9 + 16 - 25) / (2*3*4) = 0, so C = 90 degrees. This is a right triangle.

Answer: Angles: 36.87 degrees, 53.13 degrees, 90 degrees

Example 2: Area using Heron's formula

Problem: Find the area of a triangle with sides 7, 8, 9

Solution: s = (7+8+9)/2 = 12. Area = sqrt(12*5*4*3) = sqrt(720)

Answer: Area = 26.83 square units

Frequently Asked Questions

What is the sum of angles in a triangle?

The interior angles of any triangle always sum to exactly 180 degrees.

What is Heron's formula?

Heron's formula calculates the area of a triangle from its three side lengths: Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2.

Can a triangle have two right angles?

No. Since the angles must sum to 180 degrees, having two 90-degree angles would leave 0 degrees for the third angle, which is impossible.

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