Triangle Calculator
Calculate all sides, angles, and area of a triangle using SAS, ASA, SSS, or AAS modes.
Input Information
Two sides + included angle
Result
Enter values to calculate
Triangle Calculator is a geometry tool that calculates all missing sides, angles, and area of a triangle based on known side and angle information.
Four calculation modes are provided based on the minimum conditions needed to define a triangle:
- SAS - Two sides and included angle
- ASA - Two angles and included side
- SSS - Three sides
- AAS - Two angles and a non-included side
The sum of the three angles of a triangle is always 180°, and the triangle inequality (the sum of any two sides must be greater than the third side) must be satisfied.
Term Glossary
- SSS
- When a triangle is defined by the lengths of three sides only.
- SAS
- When two sides and the included angle are given.
- Law of Cosines
- c² = a² + b² - 2ab·cos(C): Formula expressing the relationship between sides and angles in any triangle.
- 1
Choose a calculation mode
Choose the mode that matches your known data: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side).
- 2
Enter your values
Enter the side lengths and angles required by the selected mode. Angles are entered in degrees (°).
- 3
Calculate
Press the calculate button to solve the triangle. The calculator derives all remaining sides and angles from the known data.
- 4
Read the results
Review the computed angles (the three should sum to 180°) and the area. For a right triangle, the hypotenuse and each angle are also shown.
Example 1 — 3-4-5 right triangle
Entering SSS mode with sides 3, 4, 5 gives: the two acute angles are about 36.87° and 53.13°, and the remaining angle is 90°. The area is ½ × 3 × 4 = 6.
Example 2 — Equilateral triangle (side 6)
Entering an equilateral triangle with sides 6, 6, 6 gives three 60° angles and an area of (√3/4) × 6² = 9√3 ≈ 15.588.
Law of Cosines
Used to find the third side.
Law of Sines
Used to find the remaining parts using a known side-angle pair.
Area Formulas
(Heron's formula)
Triangle Properties
- Sum of three angles = 180°
- Triangle inequality: sum of any two sides > remaining side
- A larger side is opposite a larger angle
- Isosceles triangle: two equal sides means two equal base angles
Special Triangles
- Right triangle: One angle is 90°, Pythagorean theorem (a² + b² = c²)
- Equilateral triangle: All three sides and angles are equal (60° each)
- 30-60-90 triangle: Side ratio 1 : √3 : 2
- 45-45-90 triangle: Side ratio 1 : 1 : √2
QDo the three angles of a triangle always add up to 180°?
Yes, in Euclidean geometry the sum of the three interior angles of a triangle is always 180°. So given two angles, the third is 180° minus their sum. Example: if two angles are 36.87° and 53.13°, the third is 180 − 36.87 − 53.13 = 90°.
QWhat is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse (longest side, c) equals the sum of the squares of the other two sides: a² + b² = c². Example: in a 3-4-5 right triangle, 3² + 4² = 9 + 16 = 25 = 5². It is used to find side lengths in right triangles.
QWhat is Heron's formula?
Heron's formula gives the area of a triangle from its three side lengths alone. With semiperimeter s = (a + b + c) / 2, area = √(s(s−a)(s−b)(s−c)). Example: sides 3, 4, 5 give s = 6 and area = √(6×3×2×1) = √36 = 6.
QWhat is the triangle inequality?
For a triangle to exist, the sum of any two side lengths must be greater than the third. Example: sides 2, 3, and 6 cannot form a triangle because 2 + 3 = 5 < 6. This calculator shows an error for inputs that violate this condition.
QHow do you calculate the area of a triangle?
The most basic formula is area = ½ × base × height. If base and height are unknown, use the sine rule: area = ½ × a × b × sin(C) (two sides and the included angle), or Heron's formula when only the three sides are known.