Enter any 3 values (sides or angles, with at least one side length) to solve for all remaining sides, angles, area, perimeter, inradius, circumradius, medians, and altitudes.
A triangle is a closed polygon consisting of three straight line segments (edges) joining three non-collinear points (vertices). The sum of its three interior angles is always exactly 180 degrees (or π radians).
Relates side lengths directly to the sines of their opposite angles and the circumradius R:
Generalization of the Pythagorean theorem for any triangle:
Computes exact area from all three sides using semiperimeter s = (a+b+c)/2:
Radii of inscribed and circumscribed circles:
Three known angles establish geometric similarity, fixing the proportional shape and ratios of the triangle, but not its physical scale. An infinite number of scaled triangles share identical angles. At least one side length is required to fix the unique dimension.
When two side lengths and a non-included angle are known (SSA), the opposite side can potentially swing to create two different valid triangles (one acute and one obtuse), one right triangle, or no valid triangle if the opposite side is too short to reach the base line.
The theorem states that for any geometrically valid triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining third side (a + b > c, a + c > b, b + c > a). If a + b ≤ c, the lines collapse into a flat segment.
An incircle is the largest circle that fits completely inside the triangle, touching all three sides tangentially with radius r = Area/s. A circumcircle is the circle passing through all three triangle vertices with radius R = (abc)/(4×Area).
A median is a line segment connecting a vertex to the midpoint of the opposite side. The three medians always intersect at a single point called the Centroid, which divides each median in a 2:1 ratio and serves as the physical center of mass.