Triangles Calculators

About Triangle Geometry

The triangle is the simplest polygon and one of the most important shapes in mathematics and engineering. Every triangle has three sides, three vertices, and three interior angles that always sum to exactly 180°. This constant angle sum makes triangles uniquely rigid — unlike rectangles or other polygons, a triangle cannot be deformed without changing its side lengths. This rigidity is why triangles are the structural backbone of bridges, roof trusses, and geodesic domes.

Types of Triangles

Triangles are classified in two ways — by their side lengths and by their angles:

By side lengths:

• Equilateral — all three sides equal; all three angles are 60°.
• Isosceles — two sides equal; the two base angles (opposite the equal sides) are equal.
• Scalene — all three sides different; all three angles different.

By angles:

• Acute — all three angles less than 90°.
• Right — one angle exactly 90°; the side opposite the right angle is the hypotenuse.
• Obtuse — one angle greater than 90°.

The Pythagorean Theorem

The Pythagorean theorem is the most famous result in triangle geometry: for a right triangle with legs a and b and hypotenuse c, a² + b² = c². It is used to find the length of any side of a right triangle when two sides are known. The theorem has hundreds of proofs — including Euclid's original proof from around 300 BCE and a proof attributed to US President James Garfield. The converse is also true: if a² + b² = c², the triangle is a right triangle.

Common Pythagorean triples (integer solutions): 3-4-5, 5-12-13, 8-15-17, 7-24-25. These are useful for quickly recognising right triangles in practical problems.

The Law of Sines and Law of Cosines

For non-right triangles, two powerful laws allow you to find missing sides and angles:

The Law of Sines states that in any triangle, the ratio of each side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This is most useful when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA, the ambiguous case).

The Law of Cosines generalises the Pythagorean theorem to non-right triangles: c² = a² + b² − 2ab·cos(C). When C = 90°, cos(90°) = 0 and the formula reduces to the Pythagorean theorem. The Law of Cosines is used when you know all three sides (SSS) or two sides and the included angle (SAS).

Triangle Area Formulas

There are several ways to calculate the area of a triangle depending on what information is available:

• Base and height: A = ½ × b × h (most common; height must be perpendicular to the base).
• Heron's formula (three sides): s = (a+b+c)/2; A = √(s(s−a)(s−b)(s−c)). Useful when no height is given.
• Two sides and included angle (SAS): A = ½ × a × b × sin(C).
• Coordinate formula (Shoelace): A = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| for vertices at known coordinates.

Triangle Centers

Every triangle has four classical centers, each defined by a different geometric construction:

• Centroid — intersection of the three medians (lines from each vertex to the midpoint of the opposite side). The centroid is the triangle's center of mass. It divides each median in the ratio 2:1.
• Circumcenter — intersection of the perpendicular bisectors of the three sides. The circumcenter is equidistant from all three vertices and is the center of the circumscribed circle.
• Incenter — intersection of the three angle bisectors. The incenter is equidistant from all three sides and is the center of the inscribed circle.
• Orthocenter — intersection of the three altitudes (perpendicular lines from each vertex to the opposite side).

Real-World Applications of Triangle Geometry

Triangles appear throughout engineering, surveying, navigation, and architecture. Structural engineers use triangulated frameworks (trusses) because triangles are the only rigid polygon shape — adding a diagonal to a rectangular frame creates two triangles and eliminates racking. Surveyors use triangulation to measure distances and map terrain: by measuring one baseline precisely and the angles to a distant point, the Pythagorean theorem and Law of Sines give the exact distance. GPS systems use trilateration (a 3D variant of triangulation) to determine position. Architects use the golden triangle (isosceles triangle with specific angle proportions) in classical design.

Frequently Asked Questions

What is the triangle inequality theorem?

The triangle inequality states that the sum of any two sides of a triangle must be greater than the third side: a + b > c, a + c > b, and b + c > a. If any of these conditions fails, the three lengths cannot form a valid triangle.

What is the difference between similar and congruent triangles?

Similar triangles have the same shape but different sizes — corresponding angles are equal and corresponding sides are proportional (related by a scale factor). Congruent triangles have the same shape and the same size — all corresponding sides and angles are equal.

Can a triangle have two right angles?

No. Since all interior angles must sum to 180°, two right angles would already total 180°, leaving 0° for the third angle — which is impossible. A triangle can have at most one right angle and at most one obtuse angle.