Isosceles Triangle
Quickly compute isosceles triangle area, height, perimeter, base angles, and vertex angle from base and equal leg lengths. Free online geometry solver for students, architects, and DIY projects.
About This Calculator
An isosceles triangle is a triangle with two sides of equal length, called legs, and a third side called the base. The base angles (angles adjacent to the base) are also equal. This calculator computes five properties -- area, height, perimeter, base angles, and vertex angle -- from just the base and equal side lengths using fundamental geometry.
The height is found via the Pythagorean theorem applied to the right triangle formed by the height, half the base, and a leg: h = sqrt(s^2 - (b/2)^2). The area follows from the standard triangle area formula A = 1/2 x b x h, which simplifies to A = (b/4) x sqrt(4s^2 - b^2). The perimeter is simply b + 2s. To find the base angles, we use arcsin(h/s); since the legs are equal, both base angles are identical. The vertex angle is 180 deg - 2 x base angle. For example, an isosceles triangle with base 6 and equal sides 5 has height = 4, area = 12, base angles ≈ 53.13 deg, and vertex angle ≈ 73.74 deg.
Isosceles triangles appear throughout architecture (roof trusses, bridge supports, Greek pediments), engineering (gable frames, structural bracing), and design (logo shapes, packaging). This tool is ideal for students, teachers, carpenters, architects, and DIY enthusiasts who need quick, accurate triangle measurements without manual trigonometric calculations.
Regional Notes: Triangle geometry is universal and does not vary by region. The calculator works with any unit system -- enter base and sides in the same unit (cm, inches, feet, meters, etc.) and results are reported in the corresponding square units for area and linear units for height and perimeter. No currency or region-specific settings are needed.
Frequently Asked Questions
What is an isosceles triangle?
An isosceles triangle is a triangle with at least two equal sides. The two equal sides are called the legs, and the third side is called the base. The base angles (angles adjacent to the base) are also equal. If all three sides are equal, it is also an equilateral triangle.
How do you calculate the area of an isosceles triangle?
The area of an isosceles triangle is A = (1/2) x base x height. The height is found using the Pythagorean theorem: height = sqrt(s^2 - (b/2)^2), where s is the equal side length and b is the base. So the formula can also be written as A = (b/4) x sqrt(4s^2 - b^2).
How do you find the base angles of an isosceles triangle?
In an isosceles triangle, base angles are equal. They can be found using trigonometry: base angle = arcsin(height / s) or base angle = arccos(b / (2s)), where height = sqrt(s^2 - (b/2)^2). The vertex angle = 180 degrees - 2 x base angle.
What is the condition for a valid isosceles triangle?
For an isosceles triangle with equal sides s and base b, the triangle inequality requires that 2s > b (the sum of the two equal sides must exceed the base) and s > b/2 (each equal side must be longer than half the base). If equal sides are shorter than half the base, the triangle cannot exist.
What is the height of an isosceles triangle?
The height of an isosceles triangle from the vertex angle to the base is calculated using the Pythagorean theorem: height = sqrt(equalSide^2 - (base/2)^2). This height bisects both the base and the vertex angle. For example, an isosceles triangle with base 6 and equal sides 5 has height = sqrt(25 - 9) = sqrt(16) = 4 units.
Is the Isosceles Triangle Calculator free to use?
Yes, all calculators on Calculy are completely free to use. No registration, login, or payment is required. You can bookmark any calculation with shareable URLs that preserve your inputs.
What is the perimeter of an isosceles triangle?
The perimeter of an isosceles triangle is P = base + 2 x equalSide. For an isosceles triangle with base 6 and equal sides 5, the perimeter is 6 + 2 x 5 = 16 units.