Triangle Calculator | Solve Triangles & Find Properties
Calculate triangle area, perimeter, angles, and sides. Solve SSS, SAS, ASA triangles with detailed visual diagrams and step-by-step solutions.
The Triangle Calculator is a comprehensive geometry tool that helps you calculate all properties of a triangle given different combinations of known values. Whether you know side lengths, angles, or a combination of both, this calculator provides accurate calculations including area, perimeter, height, and more.
Types of Triangles
By Sides
- Equilateral: All three sides equal
- Isosceles: Two sides equal
- Scalene: All sides different
By Angles
- Acute: All angles < 90°
- Right: One angle = 90°
- Obtuse: One angle > 90°
Special Triangles
- 30-60-90: Special right triangle
- 45-45-90: Isosceles right triangle
- 3-4-5: Pythagorean triple
Triangle Formulas
Area Formulas
Area = ½ × base × height
Using base and height
Heron's Formula: √[s(s-a)(s-b)(s-c)]
Where s = (a+b+c)/2
Area = ½ × a × b × sin(C)
Using SAS (Side-Angle-Side)
Perimeter & Angles
P = a + b + c
Perimeter (sum of sides)
A + B + C = 180°
Sum of interior angles
a² = b² + c² - 2bc·cos(A)
Law of Cosines
Triangle Properties
| Property | Formula | Description |
|---|---|---|
| Perimeter | P = a + b + c | Sum of all three sides |
| Semi-perimeter | s = (a+b+c)/2 | Half of perimeter |
| Area (Heron's) | √[s(s-a)(s-b)(s-c)] | Area using all sides |
| Height | h = 2A/base | Altitude to base |
| Inradius | r = A/s | Radius of incircle |
| Circumradius | R = abc/4A | Radius of circumcircle |
Input Combinations
SSS (Side-Side-Side)
When you know all three side lengths. Use Heron's formula for area, Law of Cosines for angles.
SAS (Side-Angle-Side)
When you know two sides and the included angle. Use Law of Cosines to find third side.
ASA (Angle-Side-Angle)
When you know two angles and the side between them. Use Law of Sines to find other sides.
AAS (Angle-Angle-Side)
When you know two angles and a side not between them. Find third angle, then use Law of Sines.
Special Right Triangles
30-60-90 Triangle
Side ratios: 1 : √3 : 2
- Shortest side opposite 30° angle
- Long leg = short leg × √3
- Hypotenuse = short leg × 2
45-45-90 Triangle
Side ratios: 1 : 1 : √2
- Isosceles right triangle
- Two equal legs
- Hypotenuse = leg × √2
Triangle Theorems
Pythagorean Theorem
For right triangles: a² + b² = c², where c is the hypotenuse.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C) = 2R (circumdiameter)
Law of Cosines
a² = b² + c² - 2bc·cos(A) (generalization of Pythagorean theorem)
Triangle Inequality
The sum of any two sides must be greater than the third side: a + b > c
Tips for Accurate Calculations
- Angles must sum to 180° in any triangle
- Ensure triangle inequality holds for side lengths
- Angles are typically in degrees (0-180)
- For right triangles, ensure one angle equals 90°
- Use consistent units for all measurements
- Round results appropriately based on input precision
Frequently Asked Questions
What is the minimum information needed to solve a triangle?
You need at least three pieces of information, with at least one being a side length. Common combinations are SSS, SAS, ASA, AAS, or SSA (with caution).
How do I know if my triangle is valid?
Check triangle inequality: sum of any two sides > third side. Also ensure angles sum to 180°. The calculator validates inputs automatically.
What is Heron's formula?
Heron's formula calculates area from side lengths: Area = √[s(s-a)(s-b)(s-c)], where s = semi-perimeter = (a+b+c)/2.
Can I calculate triangle height without area?
Yes, height to side 'a' can be calculated as: h = b × sin(C) = c × sin(B). The calculator automatically computes all heights.
This triangle calculator uses mathematical formulas to compute triangle properties. Results are rounded for readability. For critical applications, verify calculations with professional tools. All angles are in degrees.