Triangle Calculator

A triangle has six basic measurements: three sides and three interior angles. Knowing any three of them, as long as at least one is a length, is usually enough to pin down the rest. Fill in the boxes you know and leave the others blank. Sides are labeled with lowercase letters and the angle across from each side carries the same letter in uppercase, so angle A always faces side a. Angles can be entered in degrees or radians, and any box accepts simple expressions such as 2√3, 3/4 or pi/6.

The calculator recognizes the classic cases from geometry class. Three sides (SSS) are solved with the law of cosines, after checking that each side is shorter than the other two together. Two sides and the angle between them (SAS) give the third side by the law of cosines. Two angles and any side (ASA or AAS) give the third angle by subtraction from 180° and the remaining sides by the law of sines.

Two sides and an angle that is not between them (SSA) is the tricky case. Depending on the numbers, there may be no triangle, exactly one, or two different triangles that both fit. When two exist, the main results show the one with the smaller angle opposite the second side, and a note lists the other. Three angles alone cannot be solved, because similar triangles share the same angles at every size.

The default example, two sides of length 1 with a 60° angle between them, produces an equilateral triangle: every side is 1, every angle is 60°, and the area is √3/4 ≈ 0.433013.

Triangle facts, theorems, and laws

The interior angles of any flat triangle add up to exactly 180°, or π radians. That single fact means two angles always determine the third, and it explains why a triangle can contain at most one right angle or one obtuse angle. An exterior angle, formed by extending one side, is equal to the two remote interior angles added together.

Triangles are classified by their sides and by their angles. An equilateral triangle has three equal sides and three 60° angles; an isosceles triangle has at least two equal sides, and the angles opposite those sides are equal too; a scalene triangle has no equal sides. By angles, a triangle is acute when all three angles are under 90°, right when one is exactly 90°, and obtuse when one is larger than 90°.

The triangle inequality states that the sum of any two sides must be greater than the third. Lengths 2, 3 and 6 cannot form a triangle because 2 + 3 is less than 6; lengths 2, 3 and 5 collapse into a straight line. A related rule says that the longest side always sits opposite the largest angle and the shortest side opposite the smallest angle.

Two laws connect sides and angles in every triangle, not just right triangles. The law of sines says a ÷ sin A = b ÷ sin B = c ÷ sin C, and each of these ratios equals twice the circumradius. It is the tool of choice when you know two angles and a side. The law of cosines, c² = a² + b² − 2ab·cos C, generalizes the Pythagorean theorem: when C is 90°, the cosine is zero and the familiar c² = a² + b² remains. It finds the third side from two sides and the included angle, or any angle from three sides.

For right triangles, the trigonometric ratios give shortcuts: the sine of an acute angle is the opposite side over the hypotenuse, the cosine is the adjacent side over the hypotenuse, and the tangent is the opposite side over the adjacent side.

Area of a Triangle

The most familiar area formula is half the base times the height, where the height is the perpendicular distance from the base to the opposite corner. Any side can serve as the base, so a triangle has three heights, and the product of each side and its own height is all equal to twice the area. For an obtuse triangle, the height to one of the shorter sides falls outside the triangle, on the extension of that side.

When the height is unknown, other formulas avoid it. Given two sides and the angle between them, the area is ½ab·sin C. Given all three sides, Heron's formula uses the semiperimeter s. For sides 3, 4 and 5, s = 6 and the area is √(6 × 3 × 2 × 1) = 6, which matches half of 3 × 4 because this is a right triangle. On a coordinate grid, the area of a triangle with corners (x₁, y₁), (x₂, y₂) and (x₃, y₃) is half the absolute value of x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂).

Area = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2

a, b, c

the three side lengths

s

the semiperimeter, half the perimeter

Heron's formula needs only the three sides. With two sides and the angle between them, Area = ½ab·sin C gives the same result.

Median, inradius, and circumradius

A median joins a corner to the midpoint of the opposite side. Every triangle has three medians, and they always meet at one point, the centroid, which is the triangle's balance point. The centroid divides each median in a 2 to 1 ratio, with the longer piece next to the corner. The length of the median to side a is mₐ = ½√(2b² + 2c² − a²), and the other two follow the same pattern.

The inradius is the radius of the incircle, the largest circle that fits inside the triangle and touches all three sides. Its center, the incenter, is where the three angle bisectors cross. The inradius equals the area divided by the semiperimeter, r = Area ÷ s. For a 3-4-5 right triangle, r = 6 ÷ 6 = 1.

The circumradius is the radius of the circumcircle, the circle that passes through all three corners. Its center, the circumcenter, lies where the perpendicular bisectors of the sides meet. It can be found from the law of sines as R = a ÷ (2 sin A), or from the sides and the area as R = abc ÷ (4 × Area). The circumcenter is inside an acute triangle, exactly at the midpoint of the hypotenuse of a right triangle, and outside an obtuse triangle.

For an equilateral triangle, the centroid, incenter and circumcenter are the same point, and the circumradius is exactly twice the inradius. Euler showed that in every triangle the distance d between these two centers satisfies d² = R(R − 2r), so R is never less than 2r.

Frequently asked questions