Classifying Triangles Calculator
Enter the three side lengths of any triangle and this calculator instantly tells you its type by sides and by angles. You also get all three interior angles, the perimeter, area, three altitudes, the inradius and the circumradius. A validity check confirms whether the three values can even form a triangle, and a step-by-step panel shows every formula used.
Formula
Worked example
For a triangle with sides a = 5, b = 5, c = 5: all sides equal so it is equilateral. cos A = (25 + 25 - 25)/(2 x 25) = 0.5, so A = 60 deg. Perimeter = 15, s = 7.5, Area = sqrt(7.5 x 2.5 x 2.5 x 2.5) = 10.825. Inradius = 10.825/7.5 = 1.443. Circumradius = 125/(4 x 10.825) = 2.887.
Classifying triangles by their sides
A triangle is classified as equilateral when all three sides are the same length, isosceles when exactly two sides are equal, and scalene when all three sides differ. Because side lengths determine the interior angles, the side classification automatically limits which angle classes are possible. An equilateral triangle always has three 60-degree angles and is always acute. An isosceles triangle can be acute, right, or obtuse depending on the size of the apex angle. A scalene triangle can also belong to any of the three angle classes.
Classifying triangles by their angles
Angle classification depends on the size of the largest interior angle. If every interior angle is less than 90 degrees the triangle is acute. If one angle equals exactly 90 degrees it is a right triangle, and the side opposite that angle is the hypotenuse, which satisfies the Pythagorean theorem: a squared plus b squared equals c squared. If one angle exceeds 90 degrees the triangle is obtuse. Because the angles of any triangle sum to exactly 180 degrees, a triangle can have at most one right or obtuse angle.
Key triangle properties: area, inradius and circumradius
When only the three side lengths are known, area is most conveniently found with Heron's formula. First compute the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s(s-a)(s-b)(s-c)). The inradius r is the radius of the largest circle that fits inside the triangle: r = Area / s. The circumradius R is the radius of the circle that passes through all three vertices: R = abc / (4 * Area). Both are useful in advanced geometry and engineering applications.
The triangle inequality and why it matters
Three positive numbers can form a triangle only if each one is strictly less than the sum of the other two. This is called the triangle inequality theorem. If even one of the three conditions a + b > c, a + c > b, b + c > a fails, the three segments cannot close into a triangle. This calculator checks the inequality before attempting any classification and displays a clear error if the sides are invalid, so you know immediately whether your measurements are feasible.
Triangle classification reference
| Name | Side class | Angle class | Key property |
|---|---|---|---|
| Equilateral | Equilateral | Acute | a = b = c; all angles 60 deg |
| Acute isosceles | Isosceles | Acute | a = b; all angles < 90 deg |
| Right isosceles | Isosceles | Right | a = b; angles 45-45-90 deg |
| Obtuse isosceles | Isosceles | Obtuse | a = b; one angle > 90 deg |
| Acute scalene | Scalene | Acute | No equal sides; all angles < 90 deg |
| Right scalene | Scalene | Right | No equal sides; one angle = 90 deg |
| Obtuse scalene | Scalene | Obtuse | No equal sides; one angle > 90 deg |
Any triangle belongs to exactly one side-class and one angle-class. Equilateral is always acute.
Frequently asked questions
How do I classify a triangle by its sides?
Count how many sides are equal. If all three have the same length the triangle is equilateral. If exactly two sides match it is isosceles. If no two sides are equal it is scalene. This calculator does the comparison automatically when you enter the three side lengths.
How do I classify a triangle by its angles?
Find the largest interior angle. If it is less than 90 degrees the triangle is acute. If it equals exactly 90 degrees the triangle is right. If it exceeds 90 degrees the triangle is obtuse. Interior angles are calculated here from the sides using the Law of Cosines.
Can a triangle be both equilateral and right?
No. An equilateral triangle always has three 60-degree angles, so none of them can equal 90 degrees. The only possible combination for equilateral is equilateral-acute.
Can a triangle have two right angles?
No. Interior angles always sum to exactly 180 degrees. Two right angles would already total 180, leaving zero degrees for the third angle, which is impossible. A valid triangle can have at most one right angle.
What is the difference between inradius and circumradius?
The inradius is the radius of the inscribed circle, the largest circle that fits entirely inside the triangle, touching all three sides. The circumradius is the radius of the circumscribed circle, the circle that passes through all three vertices. For an equilateral triangle the circumradius is exactly twice the inradius.
Does this calculator work if I only know the angles?
This tool classifies from side lengths. If you only have angles, you can still determine the angle-based type (acute, right, or obtuse) directly from the angle values without needing the sides. You cannot determine side lengths from angles alone, since infinitely many differently-sized triangles share the same angle set.