Triangle Calculator
Solve a triangle from three known parts: SSS, SAS, ASA, AAS, or SSA. The result lists all three sides, all three angles in degrees, perimeter, and area. SSA reports a second triangle when the ambiguous case exists.
Triangle Calculator tool
About the Triangle Calculator
Side a is opposite angle A, side b opposite B, and side c opposite C. SAS uses sides a and b with included angle C. ASA uses angles A and B with included side c. AAS uses angles A and B with side a. SSA uses sides a and b with angle A opposite a.
This is plane trigonometry, not spherical navigation. Sides must share a unit. Angles are degrees, not radians — use the scientific calculator if you need a radian conversion.
How the Triangle Calculator works
SSS uses the law of cosines for the angles. SAS uses the law of cosines for the third side. ASA and AAS use the law of sines after the third angle is 180° minus the other two. SSA uses the law of sines and may yield zero, one, or two triangles.
c² = a² + b² − 2ab cos C
a / sin A = b / sin B = c / sin C
Area = √[s(s − a)(s − b)(s − c)], s = (a + b + c) ÷ 2
Example
Sides 3, 4, 5 form a right triangle: C = 90°, area = 6, perimeter = 12.
How to Use the Triangle Calculator
- Choose the known-parts mode.
- Enter the three required values.
- Read sides, angles, area, and perimeter.
- If SSA shows two solutions, both are listed.
- Reject any set that violates the triangle inequality or angle sum.
Triangle Calculator Examples
SSS 3-4-5
The right angle sits opposite the side of 5. Area is 6.
SAS
a = 3, b = 4, included C = 90° recovers c = 5.
SSA ambiguous
a = 7, b = 8, A = 30° has two geometric completions. The calculator reports both.
How to Solve a Triangle
Identify which three parts you know. Use cosines when the included angle or all sides are known. Use sines when you have an angle opposite a known side. Always compute the third angle as 180° minus the other two when you already have two angles.
Mode cheat sheet
Match the labels on the form.
| Mode | Enter |
|---|---|
| SSS | a, b, c |
| SAS | a, b, included C |
| ASA | A, B, included c |
| AAS | A, B, side a |
| SSA | a, b, angle A |
Impossible inputs
Sides 1, 2, 5 cannot meet (1 + 2 < 5). An SSA pair with a too short opposite side has no triangle. The calculator shows an error instead of imaginary sides.
Frequently Asked Questions
What does SSS mean?
Three sides. The angles follow from the law of cosines.
When is SSA ambiguous?
When the known angle is acute and the side opposite it is shorter than the other known side but long enough to meet the opposite ray twice.
Do angles have to sum to 180°?
Yes. The calculator checks that after it solves.
Can I enter radians?
No. Enter degrees.
Is this the area calculator’s triangle?
The area calculator only needs base and height. This page solves the full triangle.
Guides that use this calculator
- How to Find Slope From Two Points
m = rise/run from two points. Vertical lines are undefined. Here is y = mx + b, a table of cases and the free slope calculator.
- How to Calculate Area of Shapes (With Formulas)
Area is the space inside a 2-D shape. Here are the formulas our area calculator uses, including parallelograms and circular sectors.