Isosceles triangle angles calculator

What is an isosceles triangle

An isosceles triangle is defined as a triangle with two equal sides. These equal sides are referred to as the legs (denoted as $a$), while the third side is called the base (denoted as $b$). In an isosceles triangle, the angles adjacent to the base are also equal (denoted as $α$), and the angle between the legs is called the vertex angle (denoted as $β$).

Properties of an isosceles triangle

The isosceles triangle boasts several key properties:

1. Two sides of the triangle are equal ($a_1 = a_2 = a$).
2. The base angles are equal ($α_1 = α_2 = α$).
3. The altitude drawn to the base ($h_1$) is a median as well as an angle bisector.
4. The altitude $h_1$ bisects the base into two equal parts.
5. The sum of all angles in a triangle equals 180°.
6. In an isosceles triangle, the vertex angle and base angles are related by: $β + 2α = 180°$.

Calculating angles of an isosceles triangle

There are several methods to determine the angles of an isosceles triangle depending on the known elements:

Given the legs and the base

When you know the legs $(a)$ and the base $(b)$, you can find the angles using the following formulas:

Angle at the base $(α)$:

$$\alpha = \arccos\left(\frac{b}{2a}\right)$$

Vertex angle $(β)$:

$$β = 180° - 2α$$

Given one known angle

When one of the angles is known, the other angle is found using the formulas:

1. If the base angle $(α)$ is known:

$$β = 180° - 2α$$

2. If the vertex angle $(β)$ is known:

$$α = \frac{180° - β}{2}$$

Examples

Example 1

Given leg lengths $a = 10 \ \text{cm}$ and base $b = 12 \ \text{cm}$. Find the triangle’s angles.

Solution:

1. Calculate the base angle:

$$α = \arccos\left(\frac{12}{2 \cdot 10}\right) = \arccos(0.6) ≈ 53.13°$$

2. Calculate the vertex angle:

$$β = 180° - 2 \cdot 53.13° = 73.74°$$

Example 2

Given a vertex angle $β = 120°$. Find the base angles.

Solution:

$$α = \frac{180° - 120°}{2} = 30°$$

Practical application

Knowing the angles of an isosceles triangle has practical applications in various fields:

1. Architecture - especially in designing roof structures.
2. Construction - for constructing stable structures.
3. Surveying - for land measurement and mapping.
4. Navigation - for determining distances and directions.
5. Design - creating symmetric patterns and decorations.

Notes

1. Always remember that the sum of all angles in a triangle is 180°.
2. In an isosceles triangle, the altitude $h_1$ divides the triangle into two congruent right triangles.
3. Use a calculator to accurately determine values of trigonometric functions during computations.

Frequently asked questions

How to find the angles of an isosceles triangle if one leg is a = 15 cm, and the base is b = 14 cm?

Calculate the base angle:

$$\alpha = \arccos\left(\frac{14}{2 \cdot 15}\right) = \arccos(0.467) ≈ 62.16°$$

Calculate the vertex angle:

$$β = 180° - 2 \cdot 62.16° = 55.68°$$

Can an isosceles triangle have a right angle?

Yes, if the vertex angle is 90°, the base angles will each be 45°. Such a triangle is also known as an isosceles right triangle.

What are the angles of an isosceles triangle if it is also an equilateral triangle?

In an equilateral triangle, all sides and angles are equal. Each angle is 60°.

How can you determine if a triangle is isosceles, knowing only its angles?

If two angles in a triangle are equal, the triangle is isosceles.

What is the maximum possible vertex angle for an isosceles triangle?

Theoretically, the vertex angle can approach 180°, but it cannot exactly reach it. Practically, this means the legs are almost parallel, and the base is very small relative to the legs.