Isosceles Triangle Calculator

Complete isosceles triangle calculator. Enter base and equal sides or angles to find area, height, perimeter, and all angles.

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Triangle Properties

Area
--
square units
Perimeter --
Height (h) --
Base (b) --
Equal Side (a) --
Base Angles --
Apex Angle --

Step-by-Step Solution

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What Is an Isosceles Triangle?

An isosceles triangle is a triangle that has at least two sides of equal length. These equal sides are called the "legs" and the third side is called the "base." The angles opposite the equal sides (the base angles) are also equal. The angle between the two equal sides is called the apex angle or vertex angle.

Key Formulas for Isosceles Triangles

Height

The height from the apex to the base bisects the base.

h = sqrt(a2 - (b/2)2)

Area

Half the base multiplied by the height.

A = (1/2) b h

Perimeter

Sum of all three sides.

P = 2a + b

Base Angle

Using the law of cosines or inverse trig.

alpha = arccos(b / (2a))

Apex Angle

The angle at the top vertex between the two equal sides.

beta = 180 - 2 alpha

From Base Angle

Derive height when the base angle is known.

h = (b/2) tan(alpha)

Properties of Isosceles Triangles

  • Two sides are equal in length (the legs).
  • The base angles (angles opposite the equal sides) are equal.
  • The altitude from the apex to the base bisects both the base and the apex angle.
  • The altitude, median, angle bisector, and perpendicular bisector from the apex all coincide.
  • An equilateral triangle is a special case where all three sides are equal.

Types of Isosceles Triangles

Depending on the apex angle, an isosceles triangle can be acute (apex angle < 90 degrees), right (apex angle = 90 degrees), or obtuse (apex angle > 90 degrees). When the apex angle equals 60 degrees, it becomes an equilateral triangle.

Practical Applications

Isosceles triangles appear frequently in architecture, from roof trusses and gable ends to decorative arches. They are fundamental in engineering design, optics (prisms), and in the construction of trusses and bridges. Navigation and surveying also make use of isosceles triangle properties for triangulation.