Isosceles Triangle Calculator
Solve an isosceles triangle from any two of: the equal sides, the base, or the height — pick your combo in the mode menu. Returns area, perimeter, all three sides, and every angle, in whatever unit you enter.
Example: with What do you know? Equal sides (a) + base (b) · Equal side a (used in modes with a) 5 · Base b (used in modes with b) 6 · Height h to the base (used in modes with h) 4 → Area: 12 square units.
- All three sides (a, a, b)5, 5, 6 units
- Height h4 units
- Perimeter16 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
Every isosceles triangle hides two mirror-image right triangles; the Pythagorean theorem does all the solving.
Split it down the middle
Drop the altitude from the apex to the base and an isosceles triangle becomes two mirror-image right triangles, each with hypotenuse a (an equal side) and legs h and b/2. Every formula on this page falls out of that picture: h = √(a² − b²/4), a = √(h² + b²/4), and b = 2√(a² − h²). Give it any two of the three lengths and the third is one Pythagorean step away.
The default example is the classic 5-5-6: half-base 3, height 4, a perfect 3-4-5 right triangle on each side, area 12.
The angles come along free
Half the base over the equal side is the sine of half the apex angle, so apex = 2·arcsin((b/2)/a), and the two base angles split what remains: (180° − apex)/2 each. Those base angles are always equal — that is the defining symmetry of an isosceles triangle. If the apex angle passes 90°, the triangle is obtuse but every formula here still holds.
How it’s calculated
The altitude from the apex splits the triangle into two right triangles with hypotenuse a and legs h and b/2, giving h = √(a² − b²/4), a = √(h² + b²/4), and b = 2√(a² − h²) depending on the mode. Area = b·h/2; perimeter = 2a + b; apex angle = 2·arcsin((b/2)/a); base angles = (180° − apex)/2. Rounded to 3 decimals.
The height is measured to the base (the unequal side); a height drawn to one of the equal sides is a different length and needs a general triangle solver.
Which formula each mode uses
| You know | Missing piece | Formula |
|---|---|---|
| Equal sides a + base b | height | h = √(a² − b²/4) |
| Base b + height h | equal sides | a = √(h² + b²/4) |
| Equal side a + height h | base | b = 2√(a² − h²) |
All three follow from the Pythagorean theorem on the two mirror right triangles inside every isosceles triangle.
Common mistakes
- Entering an equal side no longer than half the base (a ≤ b/2) — the two sides cannot reach each other, so no triangle exists.
- Treating the equal side a as the height; the height is the perpendicular from the apex to the base.
- Using the full base b instead of b/2 inside the Pythagorean step.
- Assuming the apex angle matches the base angles; only the two base angles are equal.
Frequently asked questions
What are the isosceles triangle formulas?
Height h = √(a² − b²/4); equal side a = √(h² + b²/4); base b = 2√(a² − h²); area = b·h/2; perimeter = 2a + b; apex angle = 2·arcsin((b/2)/a). All come from splitting the triangle into two right triangles.
Why does the calculator show a dash for my inputs?
The two values cannot form a triangle. In sides + base mode, the equal side must exceed half the base (a > b/2); in side + height mode the equal side must exceed the height (a > h). Check for swapped entries.
Which angles of an isosceles triangle are equal?
The two base angles — the ones touching the unequal side b. The apex angle between the two equal sides is generally different, and the three always sum to 180°.
Is an equilateral triangle isosceles?
Yes — it is the special case a = b, where the apex angle and base angles all become 60°. Enter equal values for a and b here and you will see exactly that.