Triangle Calculator
Enter Side Lengths
About This Triangle Calculator
This triangle calculator is a free tool that finds all properties of a triangle from three side lengths. Enter the three sides, and it computes area, perimeter, all three interior angles, heights, circumradius, inradius, and median — with a visual diagram that updates in real time.
It uses Heron's formula for area and the law of cosines for angles. All calculations run in your browser — nothing is sent to a server.
How It Works
- Enter the lengths of all three sides (a, b, c).
- The calculator checks the triangle inequality (a + b > c for all sides).
- If valid, it computes area, perimeter, angles, heights, circumradius, inradius, median, and triangle type.
Triangle Types
Triangles are classified by both their sides and their angles:
- Equilateral: All three sides are equal. All angles = 60°.
- Isosceles: Exactly two sides are equal. Two angles are equal.
- Scalene: All three sides have different lengths. All angles are different.
- Right: One angle is exactly 90°. Satisfies the Pythagorean theorem (a² + b² = c²).
- Acute: All three angles are less than 90°.
- Obtuse: One angle is greater than 90°.
A triangle has both a side type and an angle type. For example, a 3-4-5 triangle is scalene (all sides different) and right (one 90° angle).
Triangle Inequality Theorem
Three lengths can form a triangle only if the sum of any two sides is strictly greater than the third side.
For sides a, b, c: a + b > c, a + c > b, and b + c > a.
If any condition fails, those lengths form a line segment, not a triangle.
Formulas Used
- Heron's formula (area): Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) / 2
- Law of cosines (angles): cos(A) = (b² + c² − a²) / (2bc)
- Heights: hₐ = 2 × Area / a
- Circumradius: R = (a × b × c) / (4 × Area)
- Inradius: r = Area / s
- Median to side a: mₐ = ½ √(2b² + 2c² − a²)
Common Pythagorean Triples
These integer side lengths always produce right triangles:
- 3-4-5 (and multiples: 6-8-10, 9-12-15, …)
- 5-12-13
- 8-15-17
- 7-24-25
Limitations
- This calculator requires all three side lengths. It does not solve triangles from angles alone or from two sides and an angle (use the law of sines or trigonometric formulas for those cases).
- For extremely flat triangles (nearly degenerate), floating-point precision may affect the last decimal places.
- Inputs must be positive numbers. Zero or negative side lengths are not valid.
Frequently Asked Questions
How do I calculate the area of a triangle from three sides?
Use Heron's formula. First find the semi-perimeter: s = (a + b + c) / 2. Then Area = √[s(s − a)(s − b)(s − c)]. For example, a triangle with sides 5, 6, 7 has s = 9 and area = √(9 × 4 × 3 × 2) ≈ 14.70 square units.
How do I find the angles of a triangle from its sides?
Use the law of cosines: cos(A) = (b² + c² − a²) / (2bc), where A is the angle opposite side a. Calculate two angles this way, then subtract their sum from 180° to find the third.
What is the triangle inequality theorem?
It states that the sum of any two sides must be greater than the third side. For sides a, b, c: a + b > c, a + c > b, and b + c > a. If any condition fails, those lengths cannot form a valid triangle.
How do I know if a triangle is right, acute, or obtuse?
Compare the square of the longest side to the sum of the squares of the other two. If a² + b² = c² (longest side), it's right. If a² + b² > c², it's acute. If a² + b² < c², it's obtuse.
What is the circumradius of a triangle?
The circumradius (R) is the radius of the circle that passes through all three vertices. It is calculated as R = (a × b × c) / (4 × Area).
What is the inradius of a triangle?
The inradius (r) is the radius of the largest circle that fits inside the triangle, tangent to all three sides. It is calculated as r = Area / s, where s is the semi-perimeter.
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Privacy & Limitations
- All calculations run entirely in your browser -- nothing is sent to any server.
- Results are computed using standard formulas and should be verified for critical applications.
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Triangle Calculator FAQ
How do I calculate the area of a triangle from three sides?
Use Heron's formula. First find the semi-perimeter s = (a + b + c) / 2. Then Area = √[s(s − a)(s − b)(s − c)]. For example, a triangle with sides 5, 6, 7 has s = 9 and area = √(9 × 4 × 3 × 2) ≈ 14.70 square units.
How do I find the angles of a triangle from its sides?
Use the law of cosines: cos(A) = (b² + c² − a²) / (2bc), where A is the angle opposite side a. Calculate two angles this way, then subtract their sum from 180° to find the third.
What is the triangle inequality theorem?
The triangle inequality theorem states that the sum of any two sides must be greater than the third side. For sides a, b, c: a + b > c, a + c > b, and b + c > a. If any condition fails, those lengths cannot form a valid triangle.
How do I know if a triangle is right, acute, or obtuse?
Compare the square of the longest side to the sum of the squares of the other two. If a² + b² = c² (longest side), it's a right triangle. If a² + b² > c², it's acute. If a² + b² < c², it's obtuse.
What is the circumradius of a triangle?
The circumradius (R) is the radius of the circle that passes through all three vertices. It is calculated as R = (a × b × c) / (4 × Area). A larger circumradius means the vertices are more spread out.
What is the inradius of a triangle?
The inradius (r) is the radius of the largest circle that fits inside the triangle, tangent to all three sides. It is calculated as r = Area / s, where s is the semi-perimeter. The inradius represents the largest inscribed circle.