The Isosceles Triangle Calculator finds the area, height, perimeter, and base angles of an isosceles triangle from its two equal legs and its base. Because an isosceles triangle has exactly two sides of equal length, the geometry simplifies enough that a height dropped to the base always splits the triangle into two identical right triangles.
An isosceles triangle sits between the fully symmetric equilateral triangle and the general scalene triangle, with just enough symmetry to make several formulas cleaner than the general case, while still needing two independent measurements rather than one.
Calculate the height of an isosceles triangle
The height dropped from the apex (the vertex between the two equal legs) to the midpoint of the base splits the triangle into two congruent right triangles, each with the height as one leg, half the base as the other leg, and the equal side as the hypotenuse.
h = √(leg² − (base/2)²)Worked example: an isosceles triangle with equal legs of 10 units and a base of 12 units.
1. Halve the base. 12 / 2 = 6.
2. Square the leg and half-base. 10² = 100, 6² = 36.
3. Subtract and take the square root. √(100 − 36) = √64 = 8 units.
Calculate the area of an isosceles triangle
With the height found, area follows the standard triangle formula.
A = ½ × base × hWorked example: the same triangle with base 12 and height 8.
1. Multiply base by height. 12 × 8 = 96.
2. Divide by 2. 96 / 2 = 48 square units.
Calculate the perimeter
Perimeter simply adds the base and twice the equal leg length.
P = base + 2 × legWorked example: base 12, legs 10.
12 + (2 × 10) = 12 + 20 = 32 units.
Find the base angles
The isosceles triangle theorem states that the two angles opposite the equal sides (the base angles) are themselves equal. Once the height is known, basic trigonometry finds those angles directly.
base angle = arccos((base/2) / leg)Worked example: base 12, leg 10 (half-base = 6).
1. Divide half-base by leg. 6 / 10 = 0.6.
2. Take the inverse cosine. arccos(0.6) ≈ 53.13°.
Both base angles measure 53.13°, and the apex angle is 180 − (2 × 53.13) = 73.74°, confirming the angle sum theorem still holds.
Understand the isosceles triangle theorem
The isosceles triangle theorem states that if two sides of a triangle are equal, the angles opposite those sides are also equal, and the converse holds too: if two angles are equal, the sides opposite them are equal.
This theorem is what guarantees the base angles calculated above will always match exactly, without needing a separate check, whenever the two legs are genuinely equal.
Recognize the special height property
Because the two legs are equal, the height dropped from the apex to the base always lands exactly at the base's midpoint, and that same height line bisects the apex angle into two equal halves.
Neither property holds for a general scalene triangle, where the height from a vertex does not generally bisect either the opposite side or the angle at the vertex it was dropped from.
Work through a second full example
An isosceles triangle with equal legs of 13 units and a base of 10 units shows every formula together. 1. Height. √(13² − 5²) = √(169 − 25) = √144 = 12 units. 2. Area. ½ × 10 × 12 = 60 square units.
3. Perimeter. 10 + (2 × 13) = 36 units.
4. Base angles. arccos(5/13) ≈ arccos(0.3846) ≈ 67.38° each.
5. Apex angle. 180 − (2 × 67.38) ≈ 45.24°.
The same Pythagorean and inverse-cosine steps used in the 10-and-12 example above apply here without any change in method, only in the substituted numbers.
Frequently asked questions
How do you find the height of an isosceles triangle?
Use h = √(leg² − (base/2)²), which comes from applying the Pythagorean theorem to one of the two right triangles formed by the height. For legs of 10 and a base of 12, the height is 8 units.
How do you find the area of an isosceles triangle?
Find the height first, then apply A = ½ × base × height. With base 12 and height 8, the area is 48 square units.
What is the isosceles triangle theorem?
If two sides of a triangle are equal, the angles opposite those sides are equal, and vice versa. This guarantees the two base angles of an isosceles triangle always match.
How do you find the base angles of an isosceles triangle?
Use base angle = arccos((base/2) / leg). For a base of 12 and legs of 10, each base angle is about 53.13°.
Does the height always bisect the base in an isosceles triangle?
Yes, specifically the height dropped from the apex (the vertex between the two equal sides) to the base always meets it at the exact midpoint, a property unique to this triangle type among general triangles.
Is an equilateral triangle a special case of isosceles?
Yes. An equilateral triangle satisfies the isosceles condition (at least two equal sides) with all three sides equal, making it a special, more symmetric case.
What if the two given legs can't form a valid triangle with the base?
If twice the leg length is less than or equal to the base, no valid triangle exists, since the two legs could not meet above the base at all; the calculator will flag this combination.
Summary
An isosceles triangle's height, area, perimeter and base angles all follow from its two equal legs and its base, since the height from the apex always bisects the base and splits the triangle into two congruent right triangles.
Legs of 10 units and a base of 12 units give a height of 8 units, an area of 48 square units, and base angles of about 53.13° each, consistent with the isosceles triangle theorem that guarantees equal angles opposite equal sides.