Isosceles Triangle Base (A) Calculator
Enter two known values to find the base of an isosceles triangle. Choose your solve mode: legs and height, vertex angle and legs, or area and height. The calculator shows the base length, full perimeter, area, altitudes, inradius, circumradius, and all three angles with step-by-step working.
Formula
Worked example
Given legs B = 10 cm and height H = 8 cm: half-base = sqrt(10^2 - 8^2) = sqrt(36) = 6 cm, so base A = 12 cm. Area = 0.5 x 12 x 8 = 48 cm^2. Base angle = arcsin(6/10) = 36.87 deg, vertex angle = 180 - 2 x 36.87 = 106.26 deg.
What is the base of an isosceles triangle?
An isosceles triangle has two equal sides called the legs (B) and one distinct side called the base (A). The base is the side that connects the two base angles, which are always equal to each other. The apex, or vertex, is the point where the two equal legs meet, and the angle there is called the vertex angle. Because the triangle is symmetric, the perpendicular height from the apex to the base bisects the base exactly in half, creating two congruent right triangles. This symmetry is the key to all three solve modes this calculator supports.
Three ways to find the base
This calculator solves for the base A in three different situations. Mode 1 - Legs and Height: If you know each leg length B and the perpendicular height H, the Pythagorean theorem gives half the base as sqrt(B^2 - H^2), so A = 2 x sqrt(B^2 - H^2). The height must be strictly shorter than the leg, otherwise no valid triangle exists. Mode 2 - Legs and Vertex Angle: If you know the legs B and the vertex angle beta, the law of cosines gives A = sqrt(2 x B^2 x (1 - cos(beta))). For beta = 60 degrees you get an equilateral triangle (A = B), and for beta = 90 degrees you get a right isosceles triangle (A = B x sqrt(2)). Mode 3 - Area and Height: If you know the area K and height H, rearranging the standard triangle area formula (K = 0.5 x A x H) gives A = 2K / H. The leg length follows as B = sqrt((A/2)^2 + H^2).
Derived properties: perimeter, angles, inradius and circumradius
Once the base A is known (and the leg B and height H), several other properties follow directly. Perimeter: P = 2B + A. Base angle: alpha = arcsin((A/2) / B). Because angles in a triangle sum to 180 degrees, the vertex angle is beta = 180 - 2 x alpha. Area: K = 0.5 x A x H. Inradius (inscribed circle): r = Area / semiperimeter = K / (P/2). Circumradius (circumscribed circle): R = B / (2 x sin(alpha)). All of these appear in the results panel so you have a complete picture of the triangle from just two input values.
Special cases and practical uses
Several special isosceles triangles appear across architecture, design, and mathematics. The right isosceles triangle (vertex angle 90 degrees, base angles 45 degrees each) is common in roof design and diagonal cuts. The golden gnomon and golden triangle are tied to the golden ratio and appear in pentagonal geometry and certain Islamic tile patterns. The equilateral triangle (vertex angle 60 degrees) is technically a special case of an isosceles triangle where all three sides and angles are equal. Practical applications include calculating roof pitch and rafter lengths, estimating the base of symmetric land plots from a measured height, working out equal-leg brace lengths in structural engineering, and finding the opening width of a compass set to a specific angle with a given arm length.
Common isosceles triangle configurations
| Name | Vertex angle | Base angles | Base / Leg ratio | Notes |
|---|---|---|---|---|
| Right isosceles | 90 deg | 45 deg each | 1.414 | Base = leg x sqrt(2) |
| Equilateral | 60 deg | 60 deg each | 1.000 | All sides equal |
| Golden gnomon | 36 deg | 72 deg each | 0.618 | Related to golden ratio |
| Golden triangle | 108 deg | 36 deg each | 1.618 | Leg / base = golden ratio |
| Half-equilateral | 120 deg | 30 deg each | 1.732 | Base = leg x sqrt(3) |
Notable special cases with their vertex angles and base-to-leg ratios.
Frequently asked questions
What is the formula for the base of an isosceles triangle?
It depends on what you know. If you have the leg length B and the height H, use A = 2 x sqrt(B^2 - H^2). If you have the leg B and the vertex angle beta (in degrees), use A = sqrt(2 x B^2 x (1 - cos(beta))). If you have the area K and height H, use A = 2K / H. All three give the same answer when applied to the same triangle.
Can the base be longer than the legs?
Yes. When the vertex angle is greater than 60 degrees the base is longer than each leg, giving a wide, flat triangle. When the vertex angle is exactly 60 degrees all sides are equal (equilateral). When the vertex angle is less than 60 degrees the base is shorter than the legs and the triangle is tall and narrow. The only constraints are that the height must be strictly less than the leg length and the vertex angle must be strictly between 0 and 180 degrees.
What is the height of an isosceles triangle?
The height (also called the altitude) is the perpendicular distance from the apex to the base. Because an isosceles triangle is symmetric, this height bisects the base exactly in half, creating two right triangles each with hypotenuse B (the leg), one leg equal to H (the height), and the other leg equal to A/2 (half the base). This is why the Pythagorean theorem is so useful for solving isosceles triangle problems.
What are the base angles of an isosceles triangle?
The base angles are the two angles at either end of the base, and they are always equal to each other. Once the base A and the leg B are known, each base angle equals arcsin((A/2) / B). The vertex angle (at the apex) is then 180 - 2 x base angle. If the vertex angle is 90 degrees, the base angles are each 45 degrees; if the vertex angle is 60 degrees, all three angles are 60 degrees and the triangle is equilateral.
What is the inradius of an isosceles triangle?
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 = Area / (P/2). For a triangle with base A = 12, legs B = 10, and height H = 8, the area is 48, the perimeter is 32, and the inradius is 48 / 16 = 3 length units.
How is the circumradius of an isosceles triangle calculated?
The circumradius R is the radius of the circle that passes through all three vertices. By the law of sines, R = B / (2 x sin(alpha)), where alpha is the base angle. Alternatively, R = (A x B x B) / (4 x Area). The circumscribed circle is always larger than the inscribed circle for any non-degenerate triangle.
What makes a right isosceles triangle special?
A right isosceles triangle has a vertex angle of exactly 90 degrees and base angles of 45 degrees each. The base A equals the leg B multiplied by sqrt(2), approximately 1.414. This makes it easy to work with because tan(45 deg) = 1, so the height equals exactly half the base: H = A/2. These triangles appear very often in carpentry, architecture, and geometry proofs.