Triangle Calculator
Solve any triangle — find all sides, angles, area, perimeter, and type — using SSS, SAS, ASA, AAS, or SSA inputs.
Last updated: March 2026
How to Solve Any Triangle
A triangle has six elements: three sides (a, b, c) and three angles (A, B, C), where angle A is opposite side a, B opposite b, and C opposite c. Knowing any three of these (with at least one side) is enough to determine all six. The five standard input combinations are:
- SSS — Three sides known. Use the Law of Cosines to find angles.
- SAS — Two sides and the angle between them. Use the Law of Cosines for the third side, then the Law of Sines for the remaining angles.
- ASA — Two angles and the side between them. The third angle = 180° − A − B. Use the Law of Sines for the remaining sides.
- AAS — Two angles and a non-included side. Same approach as ASA once the third angle is known.
- SSA — Two sides and a non-included angle (the ambiguous case). May yield 0, 1, or 2 triangles.
Key Formulas
Law of Cosines
c² = a² + b² − 2ab·cos(C)
Used for SSS and SAS cases. When C = 90°, cos(C) = 0 and this reduces to the Pythagorean theorem: c² = a² + b².
Law of Sines
a / sin(A) = b / sin(B) = c / sin(C)
Used for ASA, AAS, and SSA cases. Each side-to-sine ratio equals 2R where R is the circumradius.
Heron's Formula (Area from Three Sides)
s = (a + b + c) / 2, Area = √(s(s−a)(s−b)(s−c))
No angles needed. The semi-perimeter s makes the arithmetic manageable. For a 3-4-5 triangle: s = 6, Area = √(6·3·2·1) = 6.
Special Triangles
30-60-90: sides in ratio 1 : √3 : 2. 45-45-90: sides in ratio 1 : 1 : √2. These arise from bisecting an equilateral triangle and from a unit square's diagonal, respectively.
Types of Triangles
By sides: Equilateral (all sides equal, all angles 60°), Isosceles (two equal sides, two equal base angles), Scalene (all sides and angles different).
By angles: Acute (all angles < 90°), Right (one angle = 90°), Obtuse (one angle > 90°).
A right isosceles triangle (45-45-90) combines both classifications. An equilateral triangle is always acute.
How to Calculate Triangle Area If...
How to find the area of a triangle with base 10 and height 6?
Area = (1/2) × base × height = (1/2) × 10 × 6 = 30 square units. The base-height formula is the simplest when you have a perpendicular height.
How to find the area of a triangle with sides 5, 6, and 7?
Using Heron's formula: s = (5+6+7)/2 = 9. Area = √(9·4·3·2) = √216 ≈ 14.70 square units.
How to find the area of a 30-60-90 triangle with hypotenuse 10?
The legs are 5 and 5√3 ≈ 8.66. Area = (1/2) × 5 × 5√3 = 12.5√3 ≈ 21.65 square units.
How to find the missing angle of a triangle if two angles are 45° and 75°?
The third angle = 180° − 45° − 75° = 60°. The angles of any triangle always sum to exactly 180°.
How to find the perimeter of an equilateral triangle with side 8?
Perimeter = 3 × 8 = 24 units. All three sides are equal in an equilateral triangle.
How to determine if three sides form a valid triangle?
Check the triangle inequality: every pair of sides must sum to more than the third. Sides 3, 4, 8 fail because 3 + 4 = 7 < 8. Sides 3, 4, 5 pass.
How to find the height of an equilateral triangle with side 6?
Height = (√3/2) × side = (√3/2) × 6 = 3√3 ≈ 5.196 units.
How to find the area of an isosceles triangle with equal sides 10 and base 12?
Height = √(10² − 6²) = √(100 − 36) = √64 = 8. Area = (1/2) × 12 × 8 = 48 square units.
People Also Search For
How to solve a triangle step by step?
Identify which values you know (sides and/or angles), then choose the appropriate law (Cosines for SSS/SAS, Sines for ASA/AAS/SSA). Find the third angle using 180° − A − B, compute all sides, then use Heron's formula or ½ab·sin(C) for the area.
What is the formula for the perimeter of a triangle?
Perimeter = a + b + c — simply add all three side lengths. For an equilateral triangle with side s, this simplifies to 3s.
How to check if a triangle is a right triangle given three sides?
If a² + b² = c² (where c is the longest side), it is a right triangle. Example: 3² + 4² = 9 + 16 = 25 = 5², confirming a right triangle. This is the converse of the Pythagorean theorem.
What are Pythagorean triples?
Pythagorean triples are integer side lengths that form right triangles. Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples also work: 6-8-10, 9-12-15.
Difference between SSS, SAS, ASA, AAS, and SSA
These are the five triangle congruence/solving cases. SSS and SAS use the Law of Cosines. ASA, AAS, and SSA use the Law of Sines. SSA is the "ambiguous case" that may yield two solutions.
How to find the circumradius of a triangle?
Circumradius R = abc / (4 × Area). This is the radius of the circle that passes through all three vertices of the triangle.
Frequently Asked Questions
What is Heron's formula?
Heron's formula calculates the area of a triangle from its three side lengths: s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). No height or angle measurement required.
What is the sum of angles in a triangle?
Always 180°. This holds for every triangle in Euclidean (flat) geometry, regardless of size or type.
What is the ambiguous SSA case?
When you know two sides and a non-included angle, there may be 0, 1, or 2 possible triangles. The calculator solves for the primary solution and alerts you if a second solution exists.
Can I use this for obtuse triangles?
Yes. The Law of Cosines and Sines work for all triangles: acute, right, and obtuse. The calculator handles all cases automatically.
What are the 30-60-90 triangle side ratios?
The ratio is 1 : √3 : 2. The short leg (opposite 30°) is the base unit, the long leg (opposite 60°) is √3 times the short leg, and the hypotenuse is twice the short leg.
What are the 45-45-90 triangle side ratios?
The ratio is 1 : 1 : √2. Both legs are equal, and the hypotenuse is leg × √2 ≈ leg × 1.414.
What is the formula for triangle area with two sides and an angle?
Area = (1/2) × a × b × sin(C), where C is the angle between sides a and b. This is useful when you know SAS.
How do I know which type of triangle I have?
By sides: equilateral (a=b=c), isosceles (two equal), scalene (none equal). By angles: right (one = 90°), obtuse (one > 90°), acute (all < 90°). The calculator identifies both classifications automatically.
What is the triangle inequality theorem?
The sum of any two sides must exceed the third side. If a + b ≤ c, the three values cannot form a triangle. The calculator warns you if inputs violate this rule.
Is my data private?
All calculations run in your browser. No data is transmitted to any server.