# Isosceles Triangle Angles Calculator

Welcome to the **isosceles triangle angles calculator**, where we'll show you how to calculate the angles of an isosceles triangle. Along the way, we hope to teach you more about:

- What kinds of angles an isosceles triangle has;
- What the vertex angle of an isosceles triangle is; and
- What angles an isosceles right triangle has.

## How do I use the isosceles triangle angles calculator?

The isosceles triangle angles calculator gets to the point without cutting any triangles' corners. Here's how to use it:

**Enter the length**of your triangle's legs and the base length ($a$ and $b$, respectively).**See how the calculator**works out the vertex angle and base angles ($\beta$ and $\alpha$).- The calculator can work
**backward**, too! Try inputting your mystery triangle's angles together with one side's length (the isosceles triangle angles calculator needs some sense of proportion) and see how it works out the remaining side's length.

If you still want to learn what an isosceles triangle is and how to calculate its angles, read on!

## What is an isosceles triangle? What kind of angles does an isosceles triangle have?

An **isosceles triangle** is a triangle with two sides of equal length. We usually call these two sides the "legs" ($a$ below) and the remaining side the "base" ($b$).

Because the legs of an isosceles triangle are the same length, the **two angles** they form with the base **are also equal**. We call the angles adjacent to the base side the **base angles** ($\alpha$) and the remaining angle the **vertex angle** ($\beta$).

## How do I find the angles of an isosceles triangle?

**Finding the angles** $\alpha$ **and** $\beta$ **of an isosceles triangle** is easy — all you need is some **geometry tricks**!

**Find**$a$**and**$b$, the length of the isosceles triangle's**base and legs**.**Split the isosceles triangle**down its axis of symmetry (i.e. from its vertex angle straight down the middle of the base) to obtain**two mirrored right triangles**.**Use trigonometry**to work out the angles, which will be $90^\circ$, $\alpha$, and $\beta/2$.

## What is an isosceles right triangle?

An **isosceles right triangle** is the child of a **right triangle** and an **isosceles triangle**, and so it has all of its parents' attributes. It has two equal sides and one angle (in this case, the **vertex angle**) that is $90^\circ$. The two **base angles** are then $45^\circ$ each.

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## FAQ

### What is the vertex angle of an isosceles triangle?

The **vertex angle** of an isosceles triangle is the angle **formed by the triangle's two legs** (the two sides that are of equal length). It is unique in the triangle **unless** all three sides are equal and the triangle is **equilateral**.

### Can an isosceles triangle have a 90 degree angle?

**Yes** — an isosceles triangle can have a 90-degree angle. This angle would be its **vertex angle**. We can call an isosceles triangle with a 90-degree angle a **right isosceles triangle**. The base angles of an isosceles triangle can't be 90 degrees — a shape with two or more 90° angles can't even be a triangle at all!

### What are the angles of an isosceles triangle with a vertex angle of 90°?

If an isosceles triangle has a **vertex angle β = 90°**, we only need to calculate one more angle — the

**base angle,**, which features twice.

`α`

- The sum of a triangle's angles is
`180°`

, i.e.:

`2α + β = 180°`

. - Make
`α`

the subject of the equation:

`α = (180° − β) / 2`

- Substitute
`β = 90°`

:

`α = (180° − 90°) / 2`

- Work out the equation to obtain
`α = 45°`

.