Solve a triangle from its side lengths or angles. Calculate missing sides, interior angles, area, perimeter, heights, medians, inradius, and circumradius with this free online triangle solver.
Choose the information you know, enter your measurements, and calculate the remaining properties. Lengths can be entered in metric or imperial units.
Select a method based on the measurements available to you.
Enter positive side lengths. Angles must be greater than 0° and their sum must be less than 180°. Results are rounded for display.
Standard notation: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The diagram is illustrative and not drawn to scale.
The correct formula depends on which measurements are known. These are the main formulas used by the triangle calculator.
Multiply the base by the perpendicular height and divide by two. The height must meet the base at a right angle; it is not necessarily the length of one of the sloping sides.
Use Heron's formula when all three side lengths are known but the height is not. Here, s is the semiperimeter.
Use this formula to find a missing side when two sides and their included angle are known. Similar formulas find the other sides or angles.
This relationship is useful when two angles and one side are known. First calculate the third angle, then use the sine ratios to find the missing side lengths.
The three interior angles of a Euclidean triangle always add up to 180 degrees. If two angles are known, subtract their sum from 180°.
In a right triangle, the hypotenuse c is opposite the 90° angle. The Pythagorean theorem calculates its length from the two perpendicular legs.
The perimeter is the total length around the triangle. The semiperimeter is half of the perimeter.
The formulas give the altitude to side a, the inradius r of the inscribed circle, and the circumradius R of the circumscribed circle.
Use these examples to check your calculations and understand how different triangle-solving methods work.
Given sides a = 3, b = 4, and c = 5.
The perimeter is 12 units. Its angles are approximately 36.87°, 53.13°, and 90°.
Given a = 5, b = 7, and included angle C = 60°.
The law of cosines gives the third side. The calculator can then determine the remaining angles, area, and perimeter.
Given A = 45°, B = 60°, and side c = 10.
The law of sines gives side a, and the same method finds side b. This method is useful when angles are measured directly.
Given perpendicular legs a = 6 and b = 8.
The hypotenuse is 10 units, and the perimeter is 24 units. The right angle is 90°.
Triangles can be classified by their side lengths or by their interior angles.
| Triangle type | Definition | Useful property |
|---|---|---|
| Equilateral | All three sides are equal. | Every angle is 60°. |
| Isosceles | At least two sides are equal. | Angles opposite equal sides are equal. |
| Scalene | All three sides have different lengths. | All three interior angles are different. |
| Right | One angle is exactly 90°. | Follows the Pythagorean theorem. |
| Acute | All three angles are less than 90°. | No interior angle is a right or obtuse angle. |
| Obtuse | One angle is greater than 90°. | Only one angle can be obtuse. |
Three positive lengths form a valid triangle only if the sum of any two sides is greater than the third side.
If these conditions fail, the three lengths cannot form a proper triangle. The calculator checks this condition before calculating a triangle from three sides.
The calculator uses the selected length unit consistently for side lengths, heights, medians, and radii. Areas are reported in the corresponding squared unit.
| Unit | Equivalent in meters | Typical use |
|---|---|---|
| Millimeter (mm) | 0.001 m | Small components and technical drawings |
| Centimeter (cm) | 0.01 m | School geometry and everyday measurements |
| Meter (m) | 1 m | Rooms, construction, and surveying |
| Kilometer (km) | 1,000 m | Large distances |
| Inch (in) | 0.0254 m | Imperial measurements |
| Foot (ft) | 0.3048 m | Building and property measurements |
| Yard (yd) | 0.9144 m | Longer imperial measurements |
Remember: converting a length to a different unit changes the numerical value of its area by the square of the length conversion factor. For example, 1 m² = 10,000 cm².
Multiply the base by the perpendicular height and divide by two. If all three side lengths are known, Heron's formula calculates the area without requiring a separate height measurement.
The law of cosines is useful when two sides and their included angle are known. The law of sines works when two angles and one side are known. For right triangles, use the Pythagorean theorem.
Yes, for a triangle in ordinary flat, Euclidean geometry, the three interior angles total 180 degrees.
Perimeter measures the total distance around the triangle and is expressed in length units. Area measures the two-dimensional space inside it and is expressed in squared units.
Three valid angles determine the triangle's shape, but not its size. You also need at least one side length to calculate the actual dimensions, area, and perimeter.
The inradius is the radius of the circle that fits inside the triangle and touches all three sides. It can be calculated as the area divided by the semiperimeter.
The circumradius is the radius of the circle passing through all three vertices. For a triangle with area K and side lengths a, b, and c, the formula is R = abc / (4K).
No. Two right angles already total 180 degrees, leaving no angle for the third vertex. A triangle can have only one right angle.