Isosceles Triangle Calculator
An isosceles triangle has two equal legs and one base. Enter any two known values, choose your solve mode, and get every dimension instantly, including the area, perimeter, all three angles, inradius, and circumradius.
Formula
Worked example
For base b = 6 and leg a = 5: half-base = 3, height h = sqrt(25 - 9) = 4, area = 0.5 * 6 * 4 = 12, perimeter = 16, semiperimeter = 8, apex angle = arccos(14/50) = 73.74°, base angles = 53.13°, inradius = 12/8 = 1.5, circumradius = 5*5*6/(4*12) = 3.125.
How the isosceles triangle formulas work
An isosceles triangle has two sides of equal length called the legs (a) and a third side called the base (b). Because the legs are equal the triangle is symmetric about the altitude drawn from the apex. That altitude hits the base exactly at its midpoint, splitting the triangle into two congruent right triangles. Each right triangle has a base of b/2 and a hypotenuse equal to the leg a, so the Pythagorean theorem gives the height as h = sqrt(a^2 - (b/2)^2). Area then follows from the standard formula A = (1/2) * b * h, and the perimeter is simply P = 2a + b. The semiperimeter s = P/2 shows up in Heron's formula and in the inradius formula.
Five solve modes: working from different known values
You rarely have both the base and leg available at the start. Sometimes you know the leg and the angle at the apex, or the base and a base angle. This calculator covers five common input combinations. Given leg a and apex angle A: the base follows from b = 2a * sin(A/2). Given base b and apex angle A: the leg is a = b / (2 * sin(A/2)). Given leg a and base angle B: the base is b = 2a * cos(B). Given base b and base angle B: the leg is a = b / (2 * cos(B)). Once any two of those four quantities are pinned down, the full triangle is determined and every other measurement can be derived without ambiguity.
Angles and the Isosceles Triangle Theorem
The two base angles of an isosceles triangle are always equal because they sit opposite the two equal legs. This is the Isosceles Triangle Theorem. The apex angle A is the one between the two legs. By the law of cosines, A = arccos((2a^2 - b^2) / (2a^2)). Because all interior angles sum to 180°, each base angle is (180 - A) / 2. A taller, narrower triangle has a small apex angle and large base angles. A wide, flat triangle has a large apex angle and small base angles approaching zero. The limiting cases (A = 0 or A = 180) are degenerate lines, not valid triangles.
Inradius, circumradius, and the inscribed/circumscribed circles
The inradius r is the radius of the largest circle that fits inside the triangle, touching all three sides. It equals the area divided by the semiperimeter: r = A / s. The circumradius R is the radius of the circle that passes through all three vertices. For an isosceles triangle it works out to R = (a^2 * b) / (4A). The ratio R/r is sometimes used as a measure of triangle "regularity": an equilateral triangle has R/r = 2 (the smallest possible), while very flat or very thin isosceles triangles push this ratio much higher.
Triangle validity and the golden triangle
Not every pair of values produces a real triangle. For the two legs to meet above the base, each leg must exceed half the base (a > b/2), which is equivalent to the triangle inequality 2a > b. When the leg-to-base ratio exactly equals the golden ratio (phi = (1 + sqrt(5)) / 2 = 1.618...), the result is a golden gnomon or golden triangle. These special isosceles triangles arise in the geometry of regular pentagons, Penrose tilings, and the five-pointed star. The calculator flags this case automatically in the insight panel.
Isosceles triangle examples
| Base (b) | Leg (a) | Height | Area | Perimeter | Apex angle | Inradius | Circumradius |
|---|---|---|---|---|---|---|---|
| 6 | 5 | 4.000 | 12.000 | 16 | 73.74° | 1.500 | 3.125 |
| 8 | 5 | 3.000 | 12.000 | 18 | 106.26° | 1.333 | 3.472 |
| 10 | 13 | 12.000 | 60.000 | 36 | 44.76° | 3.333 | 7.042 |
| 4 | 4 | 3.464 | 6.928 | 12 | 60.00° | 1.155 | 2.309 |
| 6.18... | 10 | 9.511 | 29.389 | 26.18 | 36.00° | 2.247 | 5.117 |
Height uses sqrt(a^2 - (b/2)^2), area uses (1/2)*b*h, inradius = Area/s, circumradius = a^2*b/(4*Area).
Frequently asked questions
How do you find the height of an isosceles triangle?
Drop a perpendicular from the apex to the base. It bisects the base into two halves of b/2. Each half forms a right triangle with the leg as the hypotenuse, so the height is h = sqrt(a^2 - (b/2)^2). For a base of 6 and legs of 5, the height is sqrt(25 - 9) = sqrt(16) = 4.
How do you calculate the area of an isosceles triangle?
First find the height h = sqrt(a^2 - (b/2)^2), then apply the standard formula A = (1/2) * b * h. Equivalently, A = (b/4) * sqrt(4a^2 - b^2). For a base of 6 and legs of 5, the area is (1/2) * 6 * 4 = 12 square units.
What is the inradius of an isosceles triangle?
The inradius r is the radius of the inscribed circle that touches all three sides. It equals the area divided by the semiperimeter: r = Area / s, where s = (2a + b) / 2. For a base of 6 and legs of 5, r = 12 / 8 = 1.5 units.
What is the circumradius of an isosceles triangle?
The circumradius R is the radius of the circle passing through all three vertices. It equals R = (a^2 * b) / (4 * Area). For a base of 6 and legs of 5, R = (25 * 6) / (4 * 12) = 150 / 48 = 3.125 units.
Why must the leg be longer than half the base?
The two equal legs only meet above the base if each one reaches past the base midpoint. If a leg equals half the base the triangle flattens to a line with zero height; if shorter the legs never meet. This is equivalent to the triangle inequality: 2a > b.
Can I solve an isosceles triangle if I only know an angle and one side?
Yes, as long as the side and angle together uniquely define the triangle. This calculator supports five modes: base + leg, leg + apex angle, base + apex angle, leg + base angle, and base + base angle. Each combination fully determines the isosceles triangle.
What is a golden triangle?
A golden triangle is an isosceles triangle where the ratio of leg to base equals the golden ratio phi = (1 + sqrt(5)) / 2 = 1.618... It has an apex angle of exactly 36° and base angles of 72°. It appears in the geometry of regular pentagons and Penrose tilings. This calculator detects and labels it automatically.