## What Is an Isosceles Triangle?

A triangle with two sides of equal length is an isosceles triangle.

### Examples of Isosceles Triangle:

### Not an Isosceles Triangle:

### Examples of Isosceles Triangles in Real Life:

Many things in the world have the shape of an isosceles triangle. Some popular examples of these triangles in real life are:

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Parts of an Isosceles Triangle

Parts of an isosceles triangle

**1. Legs:** The two equal sides of an isosceles triangle are known as ‘legs’. In the triangle ABC (given above), AB and AC are the two legs of the isosceles triangle.

**2. Base:** The ‘base’ of an isosceles triangle is the third and unequal side. In the triangle ABC, BC is the base of the isosceles triangle.

**3. Vertex angle:** The ‘vertex angle’ is the angle formed by two equal sides of an isosceles triangle. ∠BAC is a vertex angle of the isosceles triangle.

**4. Base angles:** The ‘base angles’ are the angles that involve the base of an isosceles triangle. ∠ABC and ∠ACB are the two base angles of the isosceles triangle.

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## Properties of an Isosceles Triangle

Here is a list of some properties of isosceles triangles:

- In an isosceles triangle, if two sides are equal, then the angles opposite to the two sides correspond to each other and are also always equal.

In the isosceles triangle given above, the two angles ∠B and ∠C, opposite to the equal sides AB and AC are equal to each other.

- The isosceles triangle has three acute angles, meaning that the angles are less than 90°.
- The sum of three angles of an isosceles triangle is always 180°.

## Types of Isosceles Triangles

Generally, isosceles triangles are classified into three different types:

**Isosceles acute triangle:**An isosceles acute triangle is a triangle in which all three angles are less than 90°, and at least two of its angles are equal in measurement. One example of the angles of an isosceles acute triangle is 50°, 50°, and 80°.

**Isosceles right triangle:**The following is an example of a right triangle with two legs (and their corresponding angles) of equal measure.

**Isosceles obtuse triangle:**An isosceles obtuse triangle is a triangle in which one of the three angles is obtuse (lies between 90° and 180°), and the other two acute angles are equal in measurement. One example of isosceles obtuse triangle angles is 30°, 30°, and 120°.

## Area and Perimeter of Isosceles Triangle

- The area of an isosceles triangle is given by the following formula:

Area (A) = ½ × base (b) × height (h)

- The perimeter of the isosceles triangle is given by the formula:

Perimeter (P) = 2a + base (b)

Here, ‘a’ refers to the length of the equal sides of the isosceles triangle and ‘b’ refers to the length of the third unequal side.

## Solved Examples

### Example 1

What is the height of an isosceles triangle with an area of 12 sq. cm and a base of 6 cm?

**Solution:**

Area of isosceles triangle = ½ x base x height

i.e. 12 = ½ x 6 x height

i.e. 12 = 3 x height

i.e. height = 4 cm

### Example 2

What is the perimeter of an isosceles triangle, if equal sides are ‘a’ cm each and the unequal side is ‘b’ cm?

**Solution:**

Perimeter of an isosceles triangle = sum of its sides

Perimeter of an isosceles triangle = (a + a + b) cm, i.e., (2a + b) cm

### Example 3

Find the perimeter of an isosceles triangle if the base is 16 cm and the equal sides are 24 cm each.

**Solution: **

Formula of the perimeter of an isosceles triangle, P = 2a + b

Here, a (sides) = 24 cm and b (base) = 16 cm

Therefore, perimeter of an isosceles triangle, P = 2(24) + 16 = 64 cm.

Hence, the perimeter is 64 cm.

## Triangle Games

With **SplashLearn**, there are several games about triangles for children to try. Let us look at a few of them:

**Identify Types of Triangles****:**In this game, your child will identify various types of triangles. They will use the attributes given to determine the correct triangle and learn the characteristics of a triangle. Students will choose the correct answer from the options.**Classify Triangles****:**The game challenges your child to sharpen their skills by solving a series of problems involving two-dimensional shapes to identify different types of triangles. Students will analyze a triangle’s sides and angle measurements and classify each into the appropriate category.

### Other Games

**Classify Triangles and Rectangles as Closed Shapes****:**This game will help children classify different types of shapes and help memorize them. It will help your child recognize various shapes quicker and easier.**Sort the Shapes by Name****:**This will turn out to be a fun game, with your child learning all about different shapes at the end! The game involves sorting shapes based on their names, and by doing so, your young mathematician will gain more practice with 2D shape concepts. This game will push your child towards mastery while developing overall mathematical capabilities.

Students may also find it a bit overwhelming to remember the properties of isosceles triangles. But that’s precisely where you will require a great deal of patience while teaching your kid. Allow your child to shine bright with **SplashLearn**.

## Practice Problems on Isosceles Triangles

## Isosceles Triangles

### What is the height of an isosceles triangle with an area of 10 sq. cm and a base of 5 cm?

Area of isosceles triangle = ½ x base x height i.e. 10 cm

^{2}= ½ x 5 cm x height... i.e. height = 4 cm

### In ΔABC, if ∠A = ∠B then

Sides opposite to equal angles are also equal. ∠A = ∠B, BC is opposite to ∠A and AC is opposite to angle B. Therefore, AC = BC in ΔABC.

### What is the area of the isosceles triangle given below?

^{2}

^{2}

^{2}

^{2}

^{2}

Area of isosceles triangle = ½ x base x height = ½ x 15 cm x 6 cm = 45 cm

^{2}

## Frequently Asked Questions

**How do we know if a triangle is isosceles?**

A triangle is said to be an isosceles triangle if any of its two sides are equal. Let’s take a triangle that has AB, BC, and CA as its three sides. If any of these are true—AB = BC, BC = CA or CA = AB—then the triangle is isosceles.

**Can a right triangle also be an isosceles triangle?**

Yes, a right triangle or right-angle triangle can be an isosceles triangle. An isosceles right triangle will have 1 right angle and 2 other angles as equal angles.

**Can you find all the angles of an isosceles triangle if you know one of the equal angles?**

Yes, if we know the two equal angles, then we can easily subtract the sum of it from 180°, since the sum of all angles of a triangle is equal to 180°.

**What are some of the properties of an isosceles triangle?**

- It has two sides of equal length.
- The angles opposite to the equal sides are also equal in measure.