Triangle Calculator

What Is a Triangle Solver?

A triangle solver calculates all unknown sides, angles, and derived properties of a triangle from a minimum set of known values. Any triangle is uniquely determined (up to congruence) by three independent measurements, provided at least one is a side. The five classic input combinations are:

• SSS — three sides
• SAS — two sides and the included angle
• ASA — two angles and the included side
• AAS — two angles and a non-included side
• SSA — two sides and a non-included angle (ambiguous case)

From the solved triangle, this calculator also derives area, perimeter, semiperimeter, inradius, circumradius, altitudes, and medians — all the properties that appear in geometry courses, surveying, navigation, and engineering.

How to Use This Calculator

1. Choose a solve mode (SSS, SAS, ASA, AAS, or SSA) depending on which values you know.
2. Select the angle unit — degrees or radians.
3. Enter the known values in the input fields that appear for your chosen mode.
4. View the results: all six elements (three sides and three angles), area, perimeter, inradius, circumradius, altitudes, medians, and step-by-step solution formulas.

For the SSA (ambiguous) case, both valid triangles are displayed when they exist.

Key Formulas

Law of Cosines

Generalises the Pythagorean theorem to all triangles. Used in SSS mode (to find angles from three sides) and SAS mode (to find the third side from two sides and the included angle). When $C = 90^\circ$, the formula reduces to $c^2 = a^2 + b^2$.

Law of Sines

Relates each side to the sine of its opposite angle. The common ratio equals the diameter of the circumscribed circle ($2R$). Used in ASA, AAS, and SSA modes.

Heron's Formula (Area from Three Sides)

Where $s$ is the semiperimeter. This formula computes the area directly from the three side lengths without needing any angle or height.

Area (Two Sides and Included Angle)

Where $a$ and $b$ are two sides and $C$ is the angle between them.

Inradius and Circumradius

The inradius $r$ is the radius of the inscribed circle (tangent to all three sides). The circumradius $R$ is the radius of the circumscribed circle (passing through all three vertices).

Altitudes

The altitude from vertex A is the perpendicular distance from A to side $a$. Each altitude equals twice the area divided by its corresponding side.

Medians

A median connects a vertex to the midpoint of the opposite side. The formula above gives the median from vertex A; analogous formulas apply for $m_b$ and $m_c$.

Worked Examples

Example 1: Fencing a Triangular Plot (SSS)

A surveyor measures a triangular lot with sides a = 85 m, b = 120 m, and c = 95 m. Find all angles and the area.

1. Angle A (Law of Cosines): $\cos A = \frac{b^2+c^2-a^2}{2bc} = \frac{14400+9025-7225}{2 \times 120 \times 95} = \frac{16200}{22800} = 0.7105$
2. $A = \arccos(0.7105) \approx 44.73°$
3. Angle B: $\cos B = \frac{a^2+c^2-b^2}{2ac} = \frac{7225+9025-14400}{2 \times 85 \times 95} = \frac{1850}{16150} = 0.1146$
4. $B = \arccos(0.1146) \approx 83.42°$
5. $C = 180° - 44.73° - 83.42° = 51.85°$
6. Area (Heron's formula): $s = 150$, $A = \sqrt{150 \times 65 \times 30 \times 55} \approx 4{,}010.9 \text{ m}^2$

Practical note: The perimeter is 300 m of fencing, and the area is about 0.99 acres — useful for cost estimation and land valuation.

Example 2: Roof Truss Design (SAS)

An architect designs a roof truss with rafters a = 7 m and b = 10 m meeting at an apex angle of C = 45°. Find the span (third side) and the roof area.

1. Span c (Law of Cosines): $c = \sqrt{7^2 + 10^2 - 2(7)(10)\cos 45°} = \sqrt{49 + 100 - 98.99} = \sqrt{50.01} \approx 7.07 \text{ m}$
2. Area: $A = \frac{1}{2}(7)(10)\sin 45° = 35 \times 0.7071 \approx 24.75 \text{ m}^2$
3. Angle A: $A = \arcsin\!\left(\frac{7 \sin 45°}{7.07}\right) \approx 44.5°$
4. $B = 180° - 44.5° - 45° = 90.5°$

Practical note: This is close to a right triangle at vertex B. The span of 7.07 m determines the wall-to-wall support distance.

Example 3: Navigation Bearing (AAS)

A ship sails 10 km on bearing 040°, then turns. The angle at the start is A = 70° and at the waypoint is B = 70°, with the known side a = 10 km (opposite to angle A). Find the return distance.

1. $C = 180° - 70° - 70° = 40°$
2. Side c (Law of Sines): $c = \frac{a \sin C}{\sin A} = \frac{10 \times \sin 40°}{\sin 70°} = \frac{10 \times 0.6428}{0.9397} \approx 6.84 \text{ km}$
3. Side b: $b = \frac{a \sin B}{\sin A} = \frac{10 \times \sin 70°}{\sin 70°} = 10 \text{ km}$

Practical note: Since A = B, the triangle is isosceles and sides a and b are equal. The direct return distance is about 6.84 km.

Example 4: SSA Ambiguous Case

Given a = 7, b = 10, A = 30°. How many valid triangles exist?

1. $\sin B = \frac{b \sin A}{a} = \frac{10 \times 0.5}{7} = 0.7143$
2. $B_1 = \arcsin(0.7143) \approx 45.58°$ and $B_2 = 180° - 45.58° = 134.42°$
3. $C_1 = 180° - 30° - 45.58° = 104.42°$ (valid, > 0)
4. $C_2 = 180° - 30° - 134.42° = 15.58°$ (valid, > 0)
5. Two valid triangles exist. Triangle 1: B ≈ 45.58°, C ≈ 104.42°. Triangle 2: B ≈ 134.42°, C ≈ 15.58°.

Practical note: The SSA ambiguous case arises when angle A is acute and side a (opposite A) is shorter than side b. Always check the supplementary angle to determine if a second solution exists.

Solve Mode Reference

Common Mistakes to Avoid

Frequently Asked Questions

What is the difference between the Law of Sines and the Law of Cosines?

The Law of Cosines relates three sides to one angle: $c^2 = a^2 + b^2 - 2ab\cos C$. Use it when you know SSS (three sides) or SAS (two sides and the included angle). The Law of Sines relates each side to its opposite angle: $a/\sin A = b/\sin B$. Use it for ASA, AAS, and SSA problems. For SSS and SAS, the Law of Cosines is more numerically stable.

What is the SSA ambiguous case?

When given two sides and the angle opposite the first side (SSA), the Law of Sines may yield $\sin B \leq 1$ with two possible values of B. If both yield a positive third angle C, two distinct triangles satisfy the given data. If $\sin B > 1$, no triangle exists. If $\sin B = 1$, exactly one (right) triangle exists.

How do I find the area of a triangle without knowing the height?

Use Heron's formula if you know all three sides, or $\frac{1}{2}ab\sin C$ if you know two sides and the included angle. Both methods bypass the need for a perpendicular height measurement.

What are the inradius and circumradius?

The inradius ($r$) is the radius of the largest circle that fits inside the triangle, tangent to all three sides. The circumradius ($R$) is the radius of the smallest circle that passes through all three vertices. For an equilateral triangle with side $a$, $r = \frac{a}{2\sqrt{3}}$ and $R = \frac{a}{\sqrt{3}}$.

Can three angles alone determine a triangle?

No. Three angles define a family of similar triangles (same shape, any size). You need at least one side to pin down the scale. This is why all five solve modes require at least one side measurement.

How accurate is this calculator?

The calculator uses IEEE 754 double-precision floating-point arithmetic (about 15–16 significant digits). Results are accurate to at least 4 decimal places for typical inputs. Extreme cases (very flat or very large triangles) may have slightly reduced precision due to floating-point limitations.

Disclaimer

This calculator is provided for educational and convenience purposes only. While the formulas are based on standard mathematical definitions (sourced from Wolfram MathWorld and standard geometry textbooks), the results should not be used as the sole basis for critical engineering, construction, surveying, or navigation decisions. Always verify measurements independently and consult a qualified professional for applications where precision is essential.