🔺 Triangle Calculator
Solve any triangle from three known values – sides, angles, area, perimeter and type – with a scale drawing and the law of sines and cosines worked out.
Quick answer: A triangle with sides 3, 4 and 5 has angles of 36.87°, 53.13° and 90°, an area of 6 and a perimeter of 12 – it is a right scalene triangle.
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Triangle Calculator inputs
Result
Solved triangle (right scalene)
A = 36.87°, B = 53.13°, C = 90°
Sides 3, 4, 5
- Area
- 6
- Perimeter
- 12
- Type
- right scalene
| Sides a, b, c | 3, 4, 5 |
| Angles A, B, C | 36.8699°, 53.1301°, 90° |
| Angles in radians | 0.6435, 0.9273, 1.5708 |
| Heights hₐ, h_b, h_c | 4, 3, 2.4 |
| Semi-perimeter s | 6 |
| Inradius r = Area ÷ s | 1 |
| Circumradius R = a ÷ (2 sin A) | 2.5 |
Step by step
- Law of cosines: A = arccos((b² + c² − a²) ÷ 2bc) = arccos((4² + 5² − 3²) ÷ (2 × 4 × 5)) = 36.8699°
- B = arccos((a² + c² − b²) ÷ 2ac) = 53.1301°
- C = 180° − A − B = 90°
- Area (Heron’s formula): √(s(s − a)(s − b)(s − c)) = √(6 × 3 × 2 × 1) = 6
- Perimeter: 3 + 4 + 5 = 12
How to solve a triangle
Label the sides a, b and c, and the angles opposite them A, B and C. Choose the case that matches what you know and enter three values; the calculator finds the rest.
Law of cosines: c² = a² + b² − 2ab·cos C · Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C
| You know | Method |
|---|---|
| SSS – three sides | Law of cosines for each angle |
| SAS – two sides + included angle | Law of cosines for the third side, then angles |
| ASA / AAS – two angles + a side | Third angle = 180° − A − B, then law of sines |
| SSA – two sides + non-included angle | Law of sines; may have 0, 1 or 2 solutions |
Worked example (SSS)
Sides 3, 4 and 5:
- A = arccos((4² + 5² − 3²) ÷ (2 × 4 × 5)) = arccos(0.8) = 36.87°.
- B = arccos((3² + 5² − 4²) ÷ (2 × 3 × 5)) = arccos(0.6) = 53.13°.
- C = 180° − 36.87° − 53.13° = 90°, so it is a right triangle.
- s = 6, so Area = √(6 × 3 × 2 × 1) = 6, and the perimeter is 12.
Classifying triangles
- By sides: equilateral (all equal), isosceles (two equal), scalene (none equal).
- By angles: acute (all under 90°), right (one exactly 90°), obtuse (one over 90°).
Checks
- The three angles always add up to 180°.
- Triangle inequality: any two sides must add up to more than the third, so 2, 3 and 6 is impossible.
- The longest side is always opposite the largest angle.
Frequently asked questions
What do SSS, SAS, ASA, AAS and SSA mean?
They describe which three measurements you know, in order around the triangle. S is a side, A is an angle. SAS means two sides with the angle between them; AAS means two angles and a side that is not between them. Any of these determines the triangle, except SSA, which can give zero, one or two triangles.
Why can SSA give two triangles?
Knowing two sides and an angle that is not between them, the third vertex can sometimes swing to two positions. The law of sines gives sin B, and both B and 180° − B may work. This is the “ambiguous case”; the calculator shows both triangles when they exist.
Why can’t I solve a triangle from three angles?
Three angles fix the shape but not the size – every scaled copy has the same angles. You need at least one side length.
How is the area calculated?
Once all three sides are known the calculator uses Heron’s formula, Area = √(s(s − a)(s − b)(s − c)) with s the semi-perimeter. Equivalent formulas are ½ab·sin C and ½ × base × height.