# Triangle calculator: solve SSS, SAS, ASA, AAS and SSA

> Missing sides, angles, area, perimeter, heights, inradius and circumradius of any triangle from three measurements, with both SSA answers.

Interactive version: https://www.calcopenly.com/geometry/triangle-solver
Subject: Geometry calculators

Three measurements fix a triangle as long as at least one of them is a side. The law of cosines, c² = a² + b² − 2ab cos C, solves three sides (SSS) and two sides with the angle between them (SAS). The law of sines, a/sin A = b/sin B = c/sin C, solves two angles and a side (ASA, AAS). The area comes from ½ab sin C, and the results add the perimeter, all three heights, the inradius and the circumradius.

The default sides of 7, 8 and 9 cm give angles of 48.19°, 58.41° and 73.40° and an area of 26.8328 cm². Surveying and roof framing use the same two laws to find a length that cannot be measured directly from ones that can.

Two sides and an angle that is not between them (SSA) can fit two triangles, one or none. With a = 6, b = 8 and A = 35°, both B = 49.89° and B = 130.11° work, so both triangles are drawn and compared in a table.

## Inputs

- **I know** (options: SSS, SAS, ASA, AAS, SSA)
- **Side a (opposite A)**
- **Side b (opposite B)**
- **Side c (opposite C)**
- **Angle A**
- **Angle B**
- **Angle C (between a and b)**
- **Show results in** (options: Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km), Inches (in), Feet (ft), Yards (yd), Miles (mi))

## Results

- Area — main result
- Side a
- Side b
- Side c
- Angle A (°)
- Angle B (°)
- Angle C (°)
- Perimeter
- Inradius
- Circumradius
- Height onto a
- Height onto b
- Height onto c
- Type
- Number of triangles
- Second triangle: side c
- Second triangle: angle B (°)
- Second triangle: angle C (°)
- Second triangle: area

## Formula

$$
\begin{gathered} \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,\qquad c^2 = a^2 + b^2 - 2ab\cos C \\ \text{Area} = \tfrac{1}{2}ab\sin C,\qquad r = \frac{\text{Area}}{s} \end{gathered}
$$

## Worked examples

### SSS 3-4-5

- I know: SSS
- Side a (opposite A): 3 cm
- Side b (opposite B): 4 cm
- Side c (opposite C): 5 cm
- Show results in: Centimetres (cm)
- **Area: 6 cm²**
- **Angle C: 90 °**
- **Angle A: 36.869898 °**
- **Angle B: 53.130102 °**
- **Inradius: 1 cm**
- **Circumradius: 2.5 cm**
- **Type: Scalene, right**
- **Number of triangles: 1**
- Checked against: Right triangle: area ½·3·4, r = (3 + 4 − 5)/2, R = hypotenuse/2; Python 3.8 math: degrees(acos(0.8)), degrees(acos(0.6))

### Equilateral, side 5

- I know: SSS
- Side a (opposite A): 5 cm
- Side b (opposite B): 5 cm
- Side c (opposite C): 5 cm
- Show results in: Centimetres (cm)
- **Area: 10.825318 cm²**
- **Angle A: 60 °**
- **Angle B: 60 °**
- **Angle C: 60 °**
- **Type: Equilateral**
- **Circumradius: 2.886751 cm**
- Checked against: Python 3.8 math: sqrt(3)/4*25, 5/sqrt(3)

### SAS a = 5, b = 7, C = 49°

- I know: SAS
- Side a (opposite A): 5 cm
- Side b (opposite B): 7 cm
- Angle C (between a and b): 49 °
- Show results in: Centimetres (cm)
- **Side c: 5.298667 cm**
- **Angle A: 45.411694 °**
- **Angle B: 85.588306 °**
- **Area: 13.207418 cm²**
- Checked against: Python 3.8 math: law of cosines and ½ab·sin C

### ASA A = 40°, c = 10, B = 60°

- I know: ASA
- Side c (opposite C): 10 cm
- Angle A: 40 °
- Angle B: 60 °
- Show results in: Centimetres (cm)
- **Angle C: 80 °**
- **Side a: 6.527036 cm**
- **Side b: 8.793852 cm**
- **Area: 28.262897 cm²**
- Checked against: Python 3.8 math: C = 80°, law of sines a = 10·sin40/sin80

### AAS A = 30°, B = 45°, a = 10

- I know: AAS
- Side a (opposite A): 10 cm
- Angle A: 30 °
- Angle B: 45 °
- Show results in: Centimetres (cm)
- **Angle C: 105 °**
- **Side b: 14.142136 cm**
- **Side c: 19.318517 cm**
- **Area: 68.30127 cm²**
- Checked against: Python 3.8 math: b = 10·sin45/sin30 = 10√2, c = 10·sin105/sin30

### SSA ambiguous case, two triangles

- I know: SSA
- Side a (opposite A): 6 cm
- Side b (opposite B): 8 cm
- Angle A: 35 °
- Show results in: Centimetres (cm)
- **Number of triangles: 2**
- **Angle B: 49.886408 °**
- **Angle C: 95.113592 °**
- **Side c: 10.419047 cm**
- **Area: 23.904479 cm²**
- **Second triangle: angle B: 130.113592 °**
- **Second triangle: angle C: 14.886408 °**
- **Second triangle: side c: 2.687386 cm**
- **Second triangle: area: 6.165685 cm²**
- Checked against: Python 3.8 math: asin(8·sin35°/6) and its supplement, c = 6·sin C/sin35°

## Questions

### How do you find a missing side of a triangle that is not right-angled?

With two sides and the angle between them, use the law of cosines: c² = a² + b² − 2ab cos C. For a = 5, b = 7 and C = 49°, c = √(74 − 70 cos 49°) ≈ 5.2987. With two angles and any side, use the law of sines, a/sin A = b/sin B. When C = 90° the cosine term is zero and the law of cosines becomes Pythagoras' theorem.

### What is the ambiguous case of the law of sines?

It is the SSA case: two sides and an angle opposite one of them can fit two different triangles. For an acute angle A, find the height h = b sin A. If a < h there is no triangle, if a = h one right triangle, if h < a < b two triangles, and if a ≥ b one. For a = 6, b = 8 and A = 35°, h = 4.589, so B is either 49.89° or 130.11°.

### How do you find the angles of a triangle from its three sides?

Rearrange the law of cosines: cos A = (b² + c² − a²)/(2bc), and the same pattern for B and C. For sides 7, 8 and 9, cos A = (64 + 81 − 49)/144 = 0.6667, so A = 48.19°; B = 58.41° and C = 73.40° follow the same way, and the three add up to 180°. The largest angle is always opposite the longest side.

### How do you find the area of a triangle without the height?

Take half the product of two sides and the sine of the angle between them: Area = ½ab sin C. Sides 7 and 8 with C = 73.40° give ½ × 7 × 8 × 0.9583 ≈ 26.83. From three sides alone, Heron's formula gives the same answer: the semi-perimeter is 12, and √(12 × 5 × 4 × 3) = √720 ≈ 26.83.

### How can you tell if a triangle is acute, right or obtuse from its sides?

Compare the square of the longest side c with the sum of the squares of the other two. If c² < a² + b² the triangle is acute, if they are equal it has a right angle, and if c² is larger it is obtuse. Sides 7, 8 and 9 give 81 < 113, so acute; 3, 4 and 5 give 25 = 25, right; 4, 5 and 8 give 64 > 41, obtuse.

### How accurate is the triangle calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 8 worked examples whose answers come from independent sources; for example, “SSS 3-4-5” is checked against Right triangle: area ½·3·4, r = (3 + 4 − 5)/2, R = hypotenuse/2; Python 3.8 math: degrees(acos(0.8)), degrees(acos(0.6)).

### Where does the method come from?

Weisstein, E. W. “Law of Sines”, “Law of Cosines”, “Triangle” — MathWorld; OpenStax Precalculus 2e, §8.1 Non-right Triangles: Law of Sines (ambiguous case).

## Sources

- [Weisstein, E. W. “Law of Sines”, “Law of Cosines”, “Triangle” — MathWorld](https://mathworld.wolfram.com/LawofSines.html)
- [OpenStax Precalculus 2e, §8.1 Non-right Triangles: Law of Sines (ambiguous case)](https://openstax.org/books/precalculus-2e/pages/8-1-non-right-triangles-law-of-sines)