Triangle Third Side: worked example
| Side a | 7 |
|---|---|
| Side b | 9 |
| Angle between them (degrees) | 60 |
| Third side (c) | 8.2 |
| Triangle area | 27.3 |
| Perimeter | 24.2 |
Triangle Third Side: formula
Enter two sides and the angle between them, in degrees. This is the side-angle-side case: the law of cosines gives the third side as the square root of a squared plus b squared minus two a b cosine C, and the area is half a b sine C. The angle must be between the two known sides, or the formula does not apply.
Use one unit for the sides.
Triangle Third Side: changing Side a
| Side a | Third side (c) | Triangle area | Perimeter |
|---|---|---|---|
| 5.6 | 7.9 | 21.8 | 22.5 |
| 6.3 | 8 | 24.6 | 23.3 |
| 7 (example) | 8.2 | 27.3 | 24.2 |
| 7.7 | 8.4 | 30 | 25.1 |
| 8.4 | 8.7 | 32.7 | 26.1 |
Triangle Third Side: compare input ranges
| Input | Tested from | To | Third side (c): low input | Third side (c): high input | Swing |
|---|---|---|---|---|---|
| Angle between them (degrees) | 54 | 66 | 7.5 | 8.9 | 1.4 (17%) |
| Side b | 8 | 10 | 7.5 | 8.9 | 1.3 (16%) |
| Side a | 6.3 | 7.7 | 8 | 8.4 | 0.426 (5.2%) |
Triangle Third Side: individual input comparisons
| Input | Value | Third side (c) | Triangle area | Perimeter |
|---|---|---|---|---|
| Side a | 5.6 | 7.9 | 21.8 | 22.5 |
| 6.3 | 8 | 24.6 | 23.3 | |
| 7 | 8.2 | 27.3 | 24.2 | |
| 7.7 | 8.4 | 30 | 25.1 | |
| 8.4 | 8.7 | 32.7 | 26.1 | |
| Side b | 7 | 7 | 21.2 | 21 |
| 8 | 7.5 | 24.2 | 22.5 | |
| 9 | 8.2 | 27.3 | 24.2 | |
| 10 | 8.9 | 30.3 | 25.9 | |
| 11 | 9.6 | 33.3 | 27.6 | |
| Angle between them (degrees) | 48 | 6.8 | 23.4 | 22.8 |
| 54 | 7.5 | 25.5 | 23.5 | |
| 60 | 8.2 | 27.3 | 24.2 | |
| 66 | 8.9 | 28.8 | 24.9 | |
| 72 | 9.5 | 30 | 25.5 |
Triangle Third Side: combined input changes
| Side a / Side b | 7 | 9 | 11 |
|---|---|---|---|
| 5.6 | 6.4 | 7.9 | 9.5 |
| 6.3 | 6.7 | 8 | 9.6 |
| 7 | 7 | 8.2 | 9.6 |
| 7.7 | 7.4 | 8.4 | 9.8 |
| 8.4 | 7.8 | 8.7 | 10 |
Triangle Third Side: which assumptions move the result?
Triangle Third Side: values and definitions
| Quantity and definition | Value used |
|---|---|
| Side a — One side of the triangle. | 7 |
| Side b — A second side of the triangle. | 9 |
| Angle between them (degrees) — The angle formed where side a meets side b, from 0 to 180 degrees. | 60 |
| Third side (c) | 8.2 |
| Triangle area | 27.3 |
| Perimeter | 24.2 |
Triangle Third Side: half and double Angle between them (degrees)
| Angle between them (degrees) | Third side (c) |
|---|---|
| 30 | 4.6 |
| 60 | 8.2 |
| 120 | 13.9 |
Common error: Entering the angle in radians or mixing it with degrees. Sixty in the wrong mode gives a nonsense side length, so confirm the field is in degrees.
Angles must be entered in degrees, matching the calculator's mode. A small error in the angle changes the side noticeably for long sides. If the angle is not between the sides, use the law of sines instead.
Angles of 0 or 180 degrees give degenerate triangles.
Geometry calculators: Arc Length and Sector Area Calculator, Ellipse Area Calculator, Pyramid Volume Calculator, Right Triangle & Pythagorean Calculator, Aquarium Volume Calculator, Annulus Area Calculator.
Two sides and the angle between them lock a triangle in place, so the third side is fixed and only needs the law of cosines to find it. When the angle is 90 degrees the cosine term vanishes and you get the Pythagorean theorem. Any other angle bends that result up or down.
Sides of 7 and 9 meeting at 60 degrees close at about 8.19, shorter than the 11.4 a right angle would give.
The angle must sit between the two sides you entered. An angle elsewhere in the triangle needs the law of sines, not this calculation.
See the law of sines calculator for other cases, the right triangle and Pythagorean theorem calculators for right angles, and Heron's formula on the triangle area page. The trigonometry calculator gives sines and cosines, and the arc length page handles circles.
What if I know two angles and a side?
Use the law of sines, which relates each side to the sine of its opposite angle, after finding the third angle by subtracting from 180 degrees.
What is the law of cosines?
A formula for the third side of a triangle from two sides and the angle between them: c squared equals a squared plus b squared minus two a b cosine C.
How is it related to the Pythagorean theorem?
At a right angle the cosine term is zero, so the law of cosines reduces to the Pythagorean theorem.