Isosceles Triangle Height Calculator
Calculate the perpendicular height of an isosceles triangle — supply the equal sides and base, the area and base, or the equal side and apex angle.
Known inputs
Perpendicular distance from apex to base
- 1
Half base
6 ÷ 2 = 3 - 2
Height (Pythagorean theorem)
√(5² − 3²) = √(16) = 4
How does this calculator work?
The height of an isosceles triangle from apex to base is h = √(a² − (b/2)²). You can also find it from area and base (h = 2A/b) or from the equal side and apex angle (h = a cos(α/2)). All three input modes produce the full triangle: area, perimeter, and all angles.
Formula
How this is calculated
The height of an isosceles triangle is the perpendicular line from the apex to the base. Because an isosceles triangle is bilaterally symmetric, this altitude bisects both the apex angle and the base, creating two congruent right triangles. In each right triangle the hypotenuse is the equal side a, the base leg is b/2, and the vertical leg is h. The Pythagorean theorem gives h = √(a² − (b/2)²) directly from the sides.
If instead you know the area and the base, the triangle area formula Area = ½ × base × height rearranges to h = 2 × Area / b. This is handy when you have measured or are given the area (for example from a grid or coordinate calculation) but not the side lengths. The equal side a is then recovered from h and b using a = √(h² + (b/2)²).
The third mode uses the apex angle α. The altitude bisects the apex angle into two equal halves of α/2, and in the right triangle cos(α/2) = h / a, so h = a × cos(α/2). The base is then b = 2a × sin(α/2). All three input modes yield identical results when the same triangle is described — they are just different ways of specifying it.
Frequently asked questions
Yes — for an isosceles triangle, the altitude from the apex, the median from the apex to the base, and the angle bisector of the apex angle all coincide. This is a consequence of the bilateral symmetry of the triangle.
The altitude from one of the two base vertices is a different line. Its length equals 2 × Area / a (using the equal side as the base of that height). The calculator reports the altitude to the unequal base; for the altitude to an equal side, compute 2 × Area / a from the reported values.
Knowing the equal side a and the apex angle α fully determines the triangle — there is no remaining degree of freedom. The calculator derives b = 2a sin(α/2) and h = a cos(α/2) to complete the picture.
TG we-Calculate Editorial Team. (2026). Isosceles Triangle Height Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/isosceles-triangle-height-calculator
TG we-Calculate Editorial Team. "Isosceles Triangle Height Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/isosceles-triangle-height-calculator.
TG we-Calculate Editorial Team, "Isosceles Triangle Height Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/isosceles-triangle-height-calculator
@misc{wecalculate_isosceles_triangle_height_calculator, title = {Isosceles Triangle Height Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/isosceles-triangle-height-calculator}}, year = {2026}, note = {TG we-Calculate} }
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