Isosceles Triangle Calculator

An isosceles triangle has at least two sides of equal length. In the usual orientation, the unequal side is called the base, the angle opposite the base is the apex angle, and the two angles touching the base are equal. This symmetry makes an isosceles triangle especially convenient to solve. Once the equal side length and base are known, the height, area, perimeter, and all three angles follow from right-triangle geometry. Drawing the altitude from the apex to the base divides the triangle into two congruent right triangles. The altitude bisects both the base and the apex angle. Each right triangle has the equal side as its hypotenuse and half the base as one leg. The Pythagorean theorem therefore gives height as the square root of the equal side squared minus one quarter of the base squared. The area is one half times the full base times that height, and the perimeter is twice the equal side plus the base. The apex angle can be obtained with the law of cosines. If a is an equal side and b is the base, the cosine of the apex angle is the quantity two a squared minus b squared, divided by two a squared. Once the apex angle is known, each base angle is one half of 180 degrees minus the apex angle. The two base angles must match, and all three angles must sum to 180 degrees, providing useful checks on the result. Not every pair of positive lengths can form an isosceles triangle. The triangle inequality requires the base to be shorter than the sum of the two equal sides, so b must be less than 2a. Equality would flatten the shape into a line with zero area, while a larger base cannot connect the two side segments at all. The calculator enforces this strict inequality and rejects zero or negative lengths. Isosceles triangles appear in roof trusses, bridge supports, symmetric facades, surveying layouts, optics, and many classroom geometry problems. All input lengths must use the same unit. Height and perimeter retain that unit, while area uses its square. The displayed angles are in degrees. Results are rounded for readability, so use unrounded values or suitable engineering tolerances when a later calculation depends on high precision.