Triangle calculator
Any three values in, the whole triangle out.
Your files stay on your device
This tool runs entirely in your browser. What you paste is processed on your device and is never sent to a server, so it is safe to use with tokens, keys and unpublished text.
Example
Example: the angles and area of a triangular lot of 60, 85 and 110 ft
A corner lot is a triangle whose sides were measured with a tape: 60 ft, 85 ft and 110 ft. Type them as sides a, b and c and leave the angles empty; with three sides, the law of cosines gives every angle.
- cos α = (85² + 110² − 60²) ÷ (2 × 85 × 110), so α ≈ 32.76°; the same way β ≈ 50.06°.
- γ = 180° − α − β ≈ 97.18°: more than 90°, so the lot is an obtuse triangle.
- Area = ½ × 60 × 85 × sin 97.18° ≈ 2530 square feet; perimeter 255 ft; height on c ≈ 46 ft.
| You know | You type | The calculator finds |
|---|---|---|
| Three sides | a = 60, b = 85, c = 110 | α ≈ 32.76°, β ≈ 50.06°, γ ≈ 97.18°, area ≈ 2530 |
| Two sides and the angle between them | a = 12, b = 15, γ = 60 | c ≈ 13.748, α ≈ 49.11°, β ≈ 70.89°, area ≈ 77.942 |
| A side and two angles | c = 100, α = 35, β = 65 | γ = 80°, a ≈ 58.242, b ≈ 92.029 |
| Two sides and an angle not between them | a = 7, b = 9, α = 40 | two triangles: c ≈ 10.836 or c ≈ 2.953 |
| Three sides that don’t close | a = 3, b = 4, c = 8 | No triangle has these values. |
The lot is about 2530 square feet, with its widest corner, about 97°, opposite the 110 ft side. The units are whatever you measured in: feet in, square feet out.
How it works
- Type three values of the triangle — sides a, b, c and angles α, β, γ, each angle opposite the side of the same letter. At least one must be a side.
- The missing sides and angles appear first; where two triangles fit (two sides and an angle not between them) both are shown.
- Below: the law used at each step, the area, the perimeter and the height on c, and what kind of triangle it is.
Frequently asked questions
- What is the law of cosines?
- c² = a² + b² − 2ab·cos γ: with two sides and the angle between them it gives the third side, and with three sides it gives any angle. When γ = 90° it is the Pythagorean theorem.
- What is the law of sines?
- a / sin α = b / sin β = c / sin γ: with a side and the angle opposite it, the other sides follow from their angles.
- Why can there be two answers?
- Two sides and an angle that is not between them can close in two ways — the ambiguous case. The calculator shows both triangles, or says that none exists.
- How is the area worked out?
- As half of two sides times the sine of the angle between them: A = ½ · a · b · sin γ.
- Why does it say no triangle has these values?
- Each side must be shorter than the other two together: 3, 4 and 8 can’t close, since 3 + 4 is less than 8. Two angles that add up to 180° or more leave no room for a third either.
- Which units should I use?
- Any, as long as all the sides are in the same one: the missing sides come out in that unit and the area in its square, so feet give square feet and centimetres square centimetres. Angles are always in degrees.
- Is anything uploaded?
- No. It is worked out in your browser, and nothing you type leaves your computer.
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