Triangle Calculator
Comprehensive triangle calculator using the Side-Side-Side (SSS) method to determine triangle properties. Enter three side lengths to calculate area using Heron's formula, perimeter, angles using the law of cosines, and height. The calculator identifies triangle types (equilateral, isosceles, scalene) and detects right triangles. Perfect for geometry students, engineers, architects, or anyone working with triangular measurements. All calculations are validated to ensure the sides can form a valid triangle before computing results. All calculations happen instantly in your browser with complete privacy—no data is stored or transmitted.
Triangle Calculator
How it works: This calculator uses the Side-Side-Side (SSS) method to calculate triangle properties. Enter three side lengths, and the calculator will determine if they form a valid triangle, then calculate the area using Heron's formula, perimeter, angles using the law of cosines, and height. The calculator also identifies the triangle type (equilateral, isosceles, or scalene) and checks if it's a right triangle.
What Is a Triangle Calculator?
A triangle calculator computes the properties of a triangle — area, perimeter, all three angles, and classification — from any valid combination of known sides and angles. Triangles are the fundamental building blocks of geometry: every polygon can be decomposed into triangles, and they appear throughout architecture, engineering, surveying, physics, and computer graphics.
This tool uses Heron's formula, the Law of Cosines, and the Law of Sines to solve triangles from different input combinations: three sides (SSS), two sides and an included angle (SAS), two sides and an opposite angle (SSA), two angles and a side (AAS/ASA), or a right triangle with two known values. All calculations happen instantly in your browser with no data sent to any server.
How to Use the Triangle Calculator
- Select the input type: sides only (SSS), sides and angle (SAS/SSA), or angles and side (AAS/ASA).
- Enter your known values in the input fields. Sides can be in any unit (cm, m, inches — results use the same unit).
- Angles are entered in degrees. If you know two angles, the third is computed automatically (angles sum to 180°).
- The calculator instantly outputs: all three sides, all three angles, area, perimeter, and triangle type.
- For right triangles, you can use the simpler input: two legs, or one leg and the hypotenuse.
Worked Example: The 3-4-5 Right Triangle
A triangle with sides a = 3, b = 4, c = 5 (SSS input):
Semi-perimeter (s): (3 + 4 + 5) ÷ 2 = 6
Area (Heron's formula): √(6 × 3 × 2 × 1) = √36 = 6
Perimeter: 3 + 4 + 5 = 12
Angle A (opp. side 3): arcsin(3/5) = 36.87°
Angle B (opp. side 4): arcsin(4/5) = 53.13°
Angle C (opp. side 5): 90° (right angle)
Type: Right Scalene
The 3-4-5 is a Pythagorean triple: 3² + 4² = 9 + 16 = 25 = 5². It satisfies the Pythagorean theorem exactly.
Triangle Types Reference
| By Angles | Definition | By Sides | Definition |
|---|---|---|---|
| Acute | All angles < 90° | Equilateral | All 3 sides equal; all angles = 60° |
| Right | One angle = exactly 90° | Isosceles | 2 sides equal; 2 base angles equal |
| Obtuse | One angle > 90° | Scalene | All 3 sides different lengths |
A triangle can combine both classifications, e.g., “Right Scalene” (3-4-5) or “Obtuse Isosceles.” An equilateral triangle is always acute.
Key Concepts: Triangle Formulas
Heron's formula calculates area from three side lengths without needing height: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. This is the most general area formula and works for any triangle where all three sides are known.
Law of Cosines relates all three sides and one angle: c² = a² + b² − 2ab·cos(C). It generalises the Pythagorean theorem (which is the special case where C = 90°, making the last term zero). It is used to find the third side when two sides and the included angle are known (SAS), or to find angles when all three sides are known (SSS).
Law of Sines relates sides to their opposite angles: a/sin(A) = b/sin(B) = c/sin(C). It is used for AAS and ASA cases. The SSA case (two sides and an angle not between them) can produce two, one, or zero valid triangles — this is the “ambiguous case” that the calculator handles automatically.
Tips and Common Mistakes
Check the triangle inequality. For a valid triangle, the sum of any two sides must be greater than the third side. If a + b ≤ c, no triangle exists with those measurements. This is the most common input error — for example, sides 1, 2, 10 cannot form a triangle (1 + 2 = 3 < 10).
Angles must sum to 180°. If you input two angles, the third is 180° minus their sum. If all three angles are specified and don't sum to 180°, the input is invalid. A common mistake is entering 90°, 45°, 46° (total = 181°) instead of 90°, 45°, 45°.
The SSA ambiguous case. When two sides and a non-included angle are given, there may be two valid triangles. For example, sides a = 7, b = 10, angle A = 30° can yield two different triangles. The calculator will show both solutions when they exist. In real-world applications (surveying, construction), always verify which solution applies to your specific situation.
Frequently Asked Questions
How do I find the area of a triangle without the height?
Use Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter and a, b, c are the three side lengths. This works for any triangle. For example, a triangle with sides 5, 6, 7 has s = 9, and Area = √(9×4×3×2) = √216 ≈ 14.70 square units.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c², forming a right triangle with integer sides. The most common is 3-4-5. Others include 5-12-13, 8-15-17, 7-24-25, and 9-40-41. Multiples also work: 6-8-10, 9-12-15. Pythagorean triples are used in construction for squaring corners.
Can a triangle have two right angles?
No. A triangle's angles must sum to exactly 180°. If one angle is 90°, the remaining two angles must sum to 90°, so neither can also be 90°. A triangle with two right angles would require 180° just for those two angles, leaving 0° for the third — which is impossible in Euclidean geometry.
What is the relationship between a triangle's sides and its angles?
The longest side is always opposite the largest angle, and the shortest side opposite the smallest angle. In an equilateral triangle, all sides and all angles are equal. This relationship is quantified by the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C), where lowercase letters are sides and uppercase are their opposite angles.
How do I find a missing angle if I know two angles?
All three interior angles of a triangle sum to 180°. So the missing angle = 180° − angle1 − angle2. For example, if two angles are 55° and 75°, the third angle is 180° − 55° − 75° = 50°. This works for any triangle, regardless of type.
What does the triangle calculator output?
The calculator outputs: all three side lengths, all three angles (in degrees), the area (in square units of the input unit), the perimeter, and the triangle type (e.g., Acute Isosceles, Right Scalene, Obtuse Scalene). For the SSA ambiguous case, it may output two valid triangle solutions.
What is the largest possible area for a triangle with a fixed perimeter?
For a fixed perimeter, the equilateral triangle has the maximum area. This is a consequence of the isoperimetric inequality applied to triangles. Among all triangles with perimeter P, the equilateral triangle with side P/3 encloses the most area. Maximising area is relevant in land surveying and layout problems.
How does this calculator handle the SSA ambiguous case?
SSA (two sides and an angle not between them) can yield 0, 1, or 2 valid triangles depending on the values. The calculator checks all possibilities: if the given side opposite the angle is shorter than the other given side, two solutions may exist. Both valid triangles are presented. If no valid triangle exists (the given side is too short to reach the opposite side), an error is shown.