Triangle Calculator
Calculate triangle area with Heron formula. Identifies equilateral, isosceles, scalene, right.
Free · Runs in your browser · No signup
Examples to try
Heron’s idea
Perimeter = a+b+c. Area can use Heron’s formula with semi-perimeter s when all three sides are known.
Common mistakes
- Entering sides that violate the triangle inequality.
- Mixing units (cm with inches).
- Assuming every three numbers form a right triangle.
Scope: Side-side-side geometry helper. Related: circle calculator.
How to Use
Enter your values in the fields above and click Calculate to get instant results. All computations run locally in your browser. No data is ever uploaded or stored.
Related Calculators
Triangle Area & Classification
CalcSolver uses Heron's formula to compute area from three side lengths, then checks side equality and the Pythagorean relationship to classify the triangle.
Heron's formula: Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2
Perimeter: a + b + c
Right-triangle test: a² + b² = c² (within 10⁻⁶ tolerance)
Triangle Type Classification
By sides: Equilateral (3 equal) · Isosceles (2 equal) · Scalene (all different)
By angles: Right (one 90°) · Acute (all < 90°) · Obtuse (one > 90°)
The triangle inequality requires that the sum of any two sides exceeds the third. Interior angles always sum to exactly 180°. The most common Pythagorean triples are (3,4,5) and (5,12,13).
Triangle Geometry Calculations
CalcSolver's triangle calculator determines area, perimeter, and all three angles from known side lengths. It uses the Law of Cosines to find angles and Heron's formula for area: Area = √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter (a+b+c)/2. This elegant formula works for any triangle as long as you know all three side lengths, making it one of the most versatile tools in geometry.
Heron's Formula Explained
Heron's formula, attributed to Hero of Alexandria (10–70 AD), calculates a triangle's area without needing the height. First compute the semi-perimeter s = (a + b + c) / 2, then apply Area = √(s(s-a)(s-b)(s-c)). For example, a triangle with sides 7, 8, and 9 has s = 12, so Area = √(12 × 5 × 4 × 3) = √720 ≈ 26.83 square units. This is especially useful in surveying and land measurement where direct height measurement may be impractical.
The Pythagorean Theorem
For right triangles, the Pythagorean theorem states a² + b² = c², where c is the hypotenuse. This 2,500-year-old relationship is the foundation of Euclidean geometry. A 3-4-5 triangle is the smallest integer-sided right triangle: 3² + 4² = 9 + 16 = 25 = 5². Other common Pythagorean triples include 5-12-13, 8-15-17, and 7-24-25. The converse is equally useful: if a² + b² = c² for three side lengths, the triangle must be a right triangle.
Types of Triangles
Triangles are classified by sides and angles. By sides: equilateral (all three sides equal, all angles 60°), isosceles (two sides equal, two base angles equal), and scalene (all sides and angles different). By angles: acute (all angles less than 90°), right (one angle exactly 90°), and obtuse (one angle greater than 90°). An equilateral triangle is always acute; a right triangle can be isosceles (45-45-90) or scalene (30-60-90). The sum of interior angles is always 180°.
Real-World Applications
Triangle calculations appear everywhere in practical work. Architecture and construction: roof pitch, stair stringers, and truss design all rely on triangle geometry. Navigation: triangulation uses angles from known points to determine position — the same principle behind GPS technology. Engineering: bridge design uses triangular shapes for maximum structural strength because triangles are inherently rigid. Art and design: the golden triangle appears in compositions to create visual harmony.
Common Calculation Mistakes
- Forgetting to check triangle validity: the sum of any two sides must exceed the third side (a + b > c). Inputs like 1, 2, 5 cannot form a triangle.
- Confusing base with height: in the formula A = ½ × base × height, the height must be perpendicular to the base, not a side length.
- Using degrees instead of radians: the Law of Cosines requires angle measures in radians for most programming calculators. CalcSolver handles this conversion automatically.
- Rounding too early: carry at least 4 decimal places through intermediate steps to avoid compounding errors in the final answer.
Use CalcSolver's circle calculator for circular geometry, the equation solver for algebraic triangle problems, and the scientific calculator for trigonometric functions like sine, cosine, and tangent.
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Frequently Asked Questions
How do I find the area of a triangle?
The most common formula is A = ½ × base × height. If you know all three sides, use Heron's formula: A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2.
How do I find an unknown side?
Using the Pythagorean theorem for right triangles: a² + b² = c². For non-right triangles, use the law of cosines: c² = a² + b² − 2ab×cos(C).
What are the types of triangles?
By sides: equilateral (all equal), isosceles (two equal), scalene (all different). By angles: acute (all < 90°), right (one = 90°), obtuse (one > 90°).
Can three side lengths always form a triangle?
No. The triangle inequality theorem states that the sum of any two sides must be greater than the third side. For example, sides of 1, 2, and 5 cannot form a triangle because 1 + 2 = 3, which is not greater than 5.
What is the difference between Heron's formula and the base-height formula?
The base-height formula (A = ½ × base × height) requires knowing the perpendicular height, which is not always available. Heron's formula only requires the three side lengths, making it more versatile when you cannot directly measure the height.
How do I find the perimeter of a triangle?
Simply add all three side lengths together: Perimeter = a + b + c. For an equilateral triangle with side length s, the perimeter is 3s.