When you know all three sides of a triangle you can find everything else. Enter the sides to get the three angles, the area using Heron's formula, the perimeter and whether the triangle is acute, right-angled or obtuse.
Enter the lengths of all three sides in the same unit.
Read the area, the angles and the triangle type.
Follow the working for the cosine rule and Heron's formula.
Semi-perimeter: s = (a + b + c) ÷ 2
Heron's area: Area = √(s(s − a)(s − b)(s − c))
Cosine rule: cos A = (b² + c² − a²) ÷ 2bc
If you know all three sides of a triangle, everything else about it is fixed. This triangle calculator takes sides a, b and c and returns all three angles using the cosine rule, the area using Heron's formula, the perimeter, and the type of triangle: equilateral, isosceles or scalene, and acute, right-angled or obtuse.
Surveyors, land measurers and anyone dividing an irregular plot into triangles often know side lengths but cannot easily measure a perpendicular height or an angle on the ground. Heron's formula is taught in Class 9 and the cosine rule in Class 11, and both are common in competitive exams. The calculator first checks the triangle inequality, so it tells you at once if three lengths cannot form a triangle.
1. Check the triangle inequality: each side must be less than the sum of the other two.
2. Find angle A with the cosine rule: cos A = (b² + c² − a²) ÷ 2bc, then A = cos⁻¹ of that value.
3. Find angle B the same way: cos B = (a² + c² − b²) ÷ 2ac.
4. Find angle C = 180° − A − B.
5. Compute the semi-perimeter s = (a + b + c) ÷ 2.
6. Find the area with Heron's formula: Area = √(s(s − a)(s − b)(s − c)).
7. Classify the triangle by comparing sides and by checking whether the largest angle is less than, equal to or more than 90°.
Drop a perpendicular from one vertex and apply Pythagoras to the two right triangles formed. Combining the results gives a² = b² + c² − 2bc cos A. When A = 90°, cos A = 0 and this reduces to Pythagoras' theorem. When A is obtuse, cos A is negative, so a² is larger than b² + c². This is why comparing the square of the longest side with the sum of the other two squares tells you whether a triangle is acute, right or obtuse.
Start from Area = ½bc sin A and use sin²A = 1 − cos²A with the cosine rule for cos A. After factorising the difference of squares twice, the expression becomes (a + b + c)(−a + b + c)(a − b + c)(a + b − c) ÷ 16. Writing s for half the perimeter turns each bracket into 2s, 2(s − a), 2(s − b) and 2(s − c), giving Area² = s(s − a)(s − b)(s − c). No height or angle is needed.
If one side equals the sum of the other two, the triangle collapses into a straight line with zero area, and if it is longer, the sides cannot meet at all. In both cases a factor in Heron's formula becomes zero or negative. The calculator rejects these inputs. It finds A and B by inverse cosine and C by subtraction, so the three angles always add to exactly 180°. The type test treats an angle within a tiny tolerance of 90° as right-angled.
Mohan owns a triangular corner plot whose sides measure 13 m, 14 m and 15 m, and he wants its area and corner angles before building a boundary wall.
Cosine rule for angle A: cos A = (b² + c² − a²) ÷ 2bc cos A = (196 + 225 − 169) ÷ 420 → A = 53.1301°
Angle B: cos B = (a² + c² − b²) ÷ 2ac → B = 59.4898°
Angle C: C = 180° − A − B = 67.3801°
Heron's formula: s = (13 + 14 + 15) ÷ 2 = 21 Area = √(s(s−a)(s−b)(s−c)) = √(21 × 8 × 7 × 6) = 84
Answer: Area 84; Angles A, B, C 53.13°, 59.49°, 67.38°; Perimeter 42
Using the full perimeter instead of the semi-perimeter in Heron's formula.
Pairing an angle with the wrong side. Angle A is opposite side a.
Leaving the calculator in radian mode when finding cos⁻¹ by hand.
Entering sides in mixed units, such as feet and metres.
Rounding intermediate values heavily, which can shift the angles noticeably in thin triangles.
Measuring the area of triangular or irregular land plots split into triangles.
Finding roof, truss and bracket angles from member lengths.
Checking whether a frame is right-angled from its three sides.
Solving Class 9 Heron's formula and Class 11 cosine rule problems.
Triangulation in surveying and navigation.
Why does it say not a valid triangle?
Any two sides must add up to more than the third side; otherwise the sides cannot meet.
Which unit is the area in?
The square of the unit you used for the sides, for example cm² for sides in cm.