An isosceles triangle is a triangle consisting of two equal sides. Since these triangles have two equal sides, it also means that they have two equal angles. Similar to other triangles, isosceles triangles have three vertices, three edges, and their interior angles add up to 180°.

Here, we will learn about the fundamental properties of isosceles triangles. Also, we will look at its most important formulas and use them to solve some exercises.

##### GEOMETRY

**Relevant for**…

Learning about the characteristics of isosceles triangles.

##### GEOMETRY

**Relevant for**…

Learning about the characteristics of isosceles triangles.

## Fundamental characteristics of isosceles triangles

An isosceles triangle has the following characteristics:

- Two sides are congruent with each other, that is, two sides have the same length.
- The third side of an isosceles triangle, which is uneven to the other two sides, is called the base of the isosceles triangle.
- The two angles opposite the equal sides are congruent with each other. This means that it has two congruent base angles.
- The angle that is not congruent with the other angles is called the apex angle.
- The height from the apex angle of an isosceles triangle bisects the base into two equal parts and also bisects the apex angle into two equal angles.
- The height from the apex angle divides the triangle into two right triangles.

## Important isosceles triangle formulas

The most important formulas for isosceles triangles are the perimeter, height, and area formulas.

### Isosceles triangle perimeter formula

The perimeter of isosceles triangles is calculated by adding the lengths of all the sides of the triangle. In this case, two of the lengths are equal, so we can use the following formula:

$latex p=b+2a$ |

where *b* is the length of the base and *a* is the length of the congruent sides.

### Isosceles triangle area formula

The area of any triangle can be calculated by multiplying the length of the base by the length of the height and dividing by 2. Therefore, we have:

$latex A= \frac{1}{2} \times b \times h$ |

where *b* is the length of the base and *h* is the length of the height.

### Formula for the height of isosceles triangles

We can calculate the height of isosceles triangles using the lengths of the triangle’s sides. Therefore, we have:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$ |

where *a* is the length of the congruent sides and *b* is the length of the base.

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## Examples of isosceles triangle problems

### EXAMPLE 1

- What is the perimeter of an isosceles triangle with a base of length 12 m and congruent sides of length 8 m?

**Solution:** We have the following values:

- Base, $latex b=12$ m
- Sides, $latex a=8$ m

Using these values in the perimeter formula, we have:

$latex p=b+2a$

$latex p=12+2(8)$

$latex p=12+16$

$latex p=28$

The perimeter is 28 m.

### EXAMPLE 2

- An isosceles triangle has a base of 15 m and a height of 10 m. What is its area?

**Solution:** We recognize the following values:

- Base, $latex b=15$ m
- Height, $latex h=10$ m

We substitute these values in the formula for the area:

$latex A= \frac{1}{2}bh$

$latex A= \frac{1}{2}(15)(10)$

$latex A=75$

The area is 75 m².

### EXAMPLE 3

- What is the height of an isosceles triangle that has a base of length 8 m and congruent sides of length 12 m?

**Solution:** We have the following information:

- Base, $latex b=8$ m
- Sides, $latex a=12$ m

We substitute these values in the formula for the area:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$

$latex h= \sqrt{{{12}^2}- \frac{{{8}^2}}{4}}$

$latex h= \sqrt{144- \frac{64}{4}}$

$latex h= \sqrt{128}$

$latex h=11.3$

The height of the triangle is 11.3 m.

## Isosceles triangles – Practice problems

## See also

Interested in learning more about isosceles triangles? Take a look at these pages:

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