Math
Isosceles Triangle Calculator
An isosceles triangle has two equal legs, a base, an apex angle between the legs, and two equal base angles.
Any two of its measurements fix the rest. Choose the pair you know and the calculator finds the legs, base, both angles, height, area, perimeter, inradius, and circumradius.
The unequal side.
Try an example
Result
Legs, base
—
- Apex and base angles
- —
- Height to base
- —
- Area
- —
- Perimeter
- —
- Inradius, circumradius
- —
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Study path
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What the isosceles triangle calculator solves
The height from the apex splits an isosceles triangle into two identical right triangles with legs h and base/2 and The longest side of a right triangle, always opposite the right angle. equal to the leg. Every formula below comes from that right triangle, which is why any two measurements are enough.
| Known pair | First find | Formula |
|---|---|---|
| Legs, base | height | √(leg² − (base/2)²) |
| Legs, apex angle | base | 2·leg·sin(apex/2) |
| Legs, base angle | base | 2·leg·cos(base angle) |
| Base, height | legs | √(h² + (base/2)²) |
| Base, area | height | 2·area / base |
| Height, apex angle | base | 2·h·tan(apex/2) |
| Legs, perimeter | base | perimeter − 2·legs |
How to use it
- Pick the pair of measurements you know from the list.
- Enter the two values (angles in degrees).
- Read the legs and base, then the apex and base angles, height, area, perimeter, and both radii.
- Open the steps to see the right-triangle relationship used.
How to read the answer
The two base angles are always equal and always acute; the apex can be acute, right, or obtuse. If the apex is exactly 60° the triangle is equilateral, and if it is 90° the triangle is also a 45-45-90 triangle. The height listed is the one to the base; heights to the legs equal 2·area / leg.
Common mistakes and edge cases
- The two legs must add to more than the base, so legs of 3 cannot span a base of 6 or more.
- The height must be shorter than the legs — it is one leg of the inner right triangle.
- The base angle must be under 90°; two base angles of 90° or more leave nothing for the apex.
- The apex angle is opposite the base. If you were given an angle at the base, use the base-angle option.
- The perimeter must exceed twice the legs (or twice the base), or no triangle closes.
Worked examples
Legs 5, base 6
Height 4, area 12
Legs, base
legs = 5, base = 6
Legs 10, apex 40°
Base = 20·sin 20°
Legs, base
legs = 10, base = 6.8404
Base 8, base angle 60°
Equilateral
Legs, base
legs = 8, base = 8
Base 6, height 4
Legs 5
Legs, base
legs = 5, base = 6
Legs 5, area 12
Two possible triangles; acute one shown
Legs, base
legs = 5, base = 6
Height 4, apex 90°
45-45-90 triangle, base 8
Legs, base
legs = 5.6569, base = 8
Legs 5, perimeter 16
Legs, base
legs = 5, base = 6
Legs 3, base 7
Legs too short
Legs, base
—
Decimal legs 2.5, base 3
Legs, base
legs = 2.5, base = 3
Frequently asked questions
How do I find the height of an isosceles triangle?+
Split it down the middle. The height is h = √(leg² − (base/2)²). With legs 5 and base 6, h = √(25 − 9) = 4.
How do I find the base angles?+
cos(base angle) = (base/2) / leg. Both base angles are equal, and the apex angle is 180° minus twice the base angle.
How do I find the area of an isosceles triangle?+
Area = ½ · base · height, or with legs and apex angle, Area = ½ · leg² · sin(apex).
Can an isosceles triangle be a right triangle?+
Yes. When the apex angle is 90° it is the 45-45-90 triangle: base = leg·√2.
Is an equilateral triangle isosceles?+
Yes. Equilateral is the special case where the base equals the legs and all angles are 60°.
Why does legs + area give a warning?+
Because two apex angles share the same sine. A triangle with legs 5 and area 12 can have apex 73.7° (base 6) or apex 106.3° (base 8).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026