Isosceles Triangle Height

Use this isosceles triangle height calculator to quickly find the altitude (height) of any isosceles triangle from its base and equal side lengths. An isosceles triangle has two sides of equal length (called legs) and a third side called the base. The height is the perpendicular distance from the apex (where the two equal sides meet) to the base. This calculator is ideal for geometry students, teachers, architects, engineers, carpenters, and anyone needing to compute triangular dimensions without manual calculations.

The height is calculated using the Pythagorean theorem applied to one half of the isosceles triangle. When you draw an altitude from the apex to the midpoint of the base, it splits the isosceles triangle into two congruent right triangles. Each right triangle has the equal side as its hypotenuse (s), the height as one leg (h), and half the base as the other leg (b/2). Applying Pythagoras: h = sqrt(s^2 - (b/2)^2) = sqrt(s^2 - b^2/4). For example, a triangle with base 10 units and equal sides 13 units has a height of sqrt(169 - 25) = sqrt144 = 12 units -- a 5-12-13 Pythagorean triple.

The calculator also validates the triangle inequality condition: each equal side must be longer than half the base length for a valid isosceles triangle. If you enter values that cannot form a valid triangle, the calculator silently returns no result, preventing mathematical errors in your work. Negative values and zero are also rejected as invalid inputs.

Who should use this calculator: Students learning geometry and trigonometry can verify homework problems and check their manual calculations involving isosceles triangle heights. Teachers can use it to generate examples and verify student work. Architects and construction professionals can measure triangular dimensions for roof pitch calculations, gable end framing, truss design, and decorative triangular elements. Carpenters and DIY enthusiasts working on projects with triangular components -- such as rafters, awnings, or shelving brackets -- will get instant, accurate height measurements without solving square roots by hand.

Regional Notes: Triangle geometry is universal -- this calculator works with any unit system (metric or imperial). Enter base and equal side lengths in meters, centimeters, inches, feet, or yards, and the result will be in the same unit. The formula and validation logic apply equally worldwide. Shareable URLs preserve your inputs so you can bookmark or share specific calculations with colleagues, classmates, or clients across any region.