Isosceles Triangle Angles Calculator

Find the apex angle and base angles of an isosceles triangle from sides or other known values using the law of cosines.

Select Input Method

Angles

Apex Angle
--
degrees
Base Angle (each) --
Apex Angle --
Sum of Angles 180 degrees
Triangle Type --
Equal Side (a) --
Base (b) --

Step-by-Step Solution

Law of Cosines: cos(C) = (a^2 + b^2 - c^2) / (2ab)

How to Find the Angles of an Isosceles Triangle

In an isosceles triangle, two sides are equal (the "legs," labeled a) and the third side is the base (b). Because the legs are equal, the two base angles are also equal. The angle between the two legs is called the apex angle (or vertex angle). Since the sum of all angles in any triangle is 180 degrees, knowing any one angle lets you find the others.

Methods for Finding Angles

Law of Cosines (Sides Known)

When both the equal side and base are known.

cos(apex) = (2a2 - b2) / (2a2)

Using Inverse Cosine

The base angle from the right triangle formed by the altitude.

base angle = arccos(b / (2a))

From Height

Use arctan or arcsin when the height is known.

base angle = arctan(h / (b/2))

From One Known Angle

If the apex is known, each base angle is half the remainder.

base angle = (180 - apex) / 2

The Law of Cosines Method

The law of cosines states: c2 = a2 + b2 - 2ab cos(C), where C is the angle opposite side c. For an isosceles triangle with equal sides a and base b:

  1. Apex angle (between the two equal sides): The side opposite the apex angle is the base b. So: b2 = a2 + a2 - 2a2 cos(apex), which gives cos(apex) = (2a2 - b2) / (2a2).
  2. Base angle (opposite each equal side): base angle = (180 - apex) / 2.

Triangle Classification by Apex Angle

  • Acute isosceles: Apex angle < 90 degrees. All angles are less than 90 degrees.
  • Right isosceles: Apex angle = 90 degrees. Each base angle = 45 degrees.
  • Obtuse isosceles: Apex angle > 90 degrees. The base angles are each less than 45 degrees.
  • Equilateral: Apex angle = 60 degrees. All three angles are 60 degrees.

Common Isosceles Triangle Angles

Some frequently encountered isosceles triangles include the 45-45-90 right isosceles triangle (used in geometry and construction), the 72-72-36 "golden gnomon" (related to the golden ratio), and the 36-36-108 "golden triangle" (also related to the golden ratio and pentagons).

Practical Applications

Finding triangle angles is essential in surveying, navigation, roof pitch calculations, structural engineering, and computer graphics. The law of cosines is particularly useful when you have physical measurements of side lengths but need to determine angles for cutting, joining, or positioning.