Triangle Calculator
Solve a triangle from three sides, two sides and the angle between them, or two angles and a side.
Result
Area: 26.8328 ft²
A measurement, not a material estimate. Nothing is added for waste.
- Side a
- 7 ft (84 in, 2.1336 m)
- Side b
- 8 ft (96 in, 2.4384 m)
- Side c
- 9 ft (108 in, 2.7432 m)
- Angle A
- 48.1897 °
- Angle B
- 58.4119 °
- Angle C
- 73.3985 °
- Perimeter
- 24 ft (288 in, 7.3152 m)
https://calcgallery.com/geometry/triangle-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Triangle Calculator
- Choose what you know. Three sides, two sides and the angle between them, or two angles and one side.
- Enter those three parts. Side a is opposite angle A, side b opposite B, and side c opposite C.
- Read the area, then every side and angle below it, with the perimeter.
The four cases
| You know | Case | How the rest is found |
|---|---|---|
| Three sides | SSS | Each angle from the law of cosines |
| Two sides and the angle between them | SAS | The third side from the law of cosines, then as SSS |
| Two angles and the side between them | ASA | The third angle is what is left of 180°, then the law of sines |
| Two angles and a side not between them | AAS | As ASA |
Every case is turned into three sides first and finished the same way. The three angles therefore always add up to exactly 180 degrees, and the area always comes from the same formula.
The law of cosines
For any triangle, with C the angle between sides a and b:
c² = a² + b² − 2ab × cos CWhen C is 90 degrees, cos C is zero and this is Pythagoras’ theorem. The law is used in both directions. From two sides and the angle between them it gives the third side. From three sides it gives each angle, as cos A = (b² + c² − a²) ÷ 2bc.
The law of sines
Each side divided by the sine of its opposite angle gives the same number:
a ÷ sin A = b ÷ sin B = c ÷ sin COnce two angles are known, the third is 180 degrees minus both. One known side then sets that shared ratio, and the other two sides follow from it.
The area
The area is found from the three sides by Heron’s formula:
p = (a + b + c) ÷ 2area = √(p (p − a)(p − b)(p − c))It needs no height, so it works the same way for every case.
When there is no triangle
Three lengths make a triangle only if each is shorter than the other two combined, so sides of 1, 2 and 3 lie flat. Two angles leave room for a third only if they add up to less than 180 degrees. The page checks both and gives no result when either fails, rather than passing impossible values to the formulas.
With your numbers
- Law of cosines: c² = a² + b² − 2ab cos C, then each angle from the three sides.
- Sides: a = 5 ft, b = 7 ft, c = 6.245 ft
- Angles: A = 43.8979 °, B = 76.1021 °, C = 60 °
- Area = √(p(p − a)(p − b)(p − c)) = 15.16 ft²
Quick reference
A measurement, not a material estimate. Nothing is added for waste.
| Example | Area |
|---|---|
| Sides 3, 4, 5 m | 6 m² |
| Sides 7, 8, 9 m | 26.8328 m² |
| Equilateral, 6 m sides | 15.5885 m² |
| 5 and 7 ft with 60° between | 15.1554 ft² |
| 30° and 45° with 10 m between | 18.3013 m² |
| 30° and 45°, 10 m opposite the 30° | 68.3013 m² |
| 30° and 60° with 10 ft between | 21.6506 ft² |
Common questions about the Triangle Calculator
- Which case do I have?
- Count what you know. Three sides is SSS. Two sides and the angle where they meet is SAS. Two angles and the side joining them is ASA. Two angles and any other side is AAS. If you know two sides and an angle that is not between them, that is SSA, which this page does not solve; see below.
- Why can I not enter two sides and an angle that is not between them?
- That combination, SSA, is the ambiguous case. The same two sides and angle can describe no triangle, exactly one, or two different triangles, depending on the lengths. Picking one of two answers without telling you which would be wrong, so the page leaves the case out rather than guess. If you can measure the third side or the included angle instead, use SSS or SAS.
- Why can I not solve from three angles?
- Three angles fix the shape but not the size. A triangle with angles of 30, 60 and 90 degrees can have a hypotenuse of 1 cm or 1 km. At least one side is needed to set the scale.
- Why was my triangle refused?
- Either the sides cannot meet, because one is as long as the other two together or longer, or the angles leave no room for a third, because the two you entered add up to 180 degrees or more. In either case the page shows no result rather than a meaningless number; check the three values against those two rules.
- How accurate are the angles?
- As accurate as the sides you enter. The formulas are exact; the page rounds only for display, to four decimal places of a degree. A tape measured to the nearest quarter inch is the limit in practice, not the arithmetic.