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## Isosceles Trapezoid

An isosceles trapezoid is a trapezoid with congruent base angles and congruent non-parallel sides. A trapezoid is a quadrilateral with only one of its sides parallel. An isosceles trapezoid has many interesting properties that make it unique and help us differentiate it from the other quadrilaterals. Let us discuss them in detail.

## Isosceles Trapezoid Definition

An isosceles trapezoid can be defined as a trapezoid in which non-parallel sides and base angles are of the same measure. In other words, if two opposite sides (bases) of the trapezoid are parallel, and the two non-parallel sides are of equal lengths, then it is an isosceles trapezoid. Look at the image below-showing sides c and d are equal in lengths and opposite sides a and b (bases of the trapezoid) are parallel to each other.

## Properties of Isosceles Trapezoid

Following are the properties of an isosceles trapezoid according to the figure given below.

- It has an axis of symmetry. It has no rotational symmetry and one line of symmetry joining the midpoint of the parallel sides.
- One pair of sides is parallel and they are the base sides. (AB II DC in the given image)
- The remaining sides other than the base are non-parallel and are equal in length. (c = d in the given image)
- The diagonals are the same in length. (AC = BD)
- The base angles are the same. (∠D = ∠C, ∠A=∠B)
- The sum of opposite angles is 180° or supplementary. (∠A + ∠C = 180° and ∠B + ∠D = 180°)
- The line segment joining the midpoints of the parallel sides is perpendicular to the bases. (PQ ⊥ DC)

## Isosceles Trapezoid Formula

Following are the formulas to calculate the area and perimeter of the isosceles trapezoid.

### Area of isosceles trapezoid

To find the area of the isosceles trapezoid we have to add the base sides or parallel sides and divide it by 2 and then multiply the result with height.**Area of Isosceles Trapezoid = (sum of parallel sides ÷ 2) × h**

### Perimeter of isosceles trapezoid

To find the perimeter of the isosceles trapezoid we have to add all the sides of the isosceles trapezoid.**Perimeter of Isosceles Trapezoid = sum of all sides**

## Examples on Isosceles Trapezoid

**Example 1:**Find the height of the isosceles trapezoid if the area is 128 inches

^{2 }and the lengths of the bases are 12 inches and 20 inches.

**Solution:**Given area = 128 inches

^{2}, Bases = 12 inches and 20 inches

we know that area of isosceles trapezoid = (sum of parallel sides ÷ 2) × height

therefore, 128 = [(12 + 20) ÷ 2] × height

Height = 128/16 = 8 inches

**Example 2:**Find the area of an isosceles trapezoid if its bases are 3 inches and 5 inches and its height is 4 inches.

**Solution:**Area of isosceles trapezoid = (sum of parallel sides ÷ 2) × height

given, bases = 3 inches and 5 inches, height = 4 inches

Area = [(3 + 5) ÷ 2] × 4

Area = 16 inches

^{2}

**Example 3:**Find the perimeter of an isosceles trapezoid if its bases are 20 inches and 25 inches and non-parallel sides are 30 inches each.

**Solution:**Perimeter of an isosceles trapezoid = sum of all sides of isosceles trapezoid

Perimeter of an isosceles trapezoid = 20 + 25 + 30 + 30 = 105 inches

## FAQs on Isosceles Trapezoid

### What is an Isosceles Trapezoid?

An isosceles trapezoid is a type of trapezoid that has nonparallel sides equal to each other. An isosceles trapezoid is a type of quadrilateral where the line of symmetry bisects one pair of the opposite sides. The bases of an isosceles trapezoid are parallel to each other along with the legs being equal in measure.

### What are the Properties of an Isosceles Trapezoid?

In an isosceles trapezoid, the number of sides is four. The two opposite sides (bases) are parallel to each other and the other two sides are equal in lengths but non-parallel to each other.

### If One Base Angle of Isosceles Trapezoid is 30°. Find the Other Base Angle.

According to the property of an isosceles trapezoid, the base angles are equal, therefore if one base angle is 30°, then the other base angle will be equal to 30°.

### What is the Difference Between a Trapezoid and an Isosceles Trapezoid?

In a trapezoid, each side is of different lengths and the diagonals are not congruent, whereas, in an isosceles trapezoid the non-parallel sides are equal, the base angles are equal, the diagonals are congruent and the opposite angles are supplementary.

### What is the Formula for Area of an Isosceles Trapezoid?

The formula to calculate the area of an isosceles trapezoid is Area = (sum of parallel sides ÷ 2) × height.

### What is the Formula for Perimeter of an Isosceles Trapezoid?

The formula to calculate the perimeter of an isosceles trapezoid is Perimeter = sum of all sides of the isosceles trapezoid

## Isosceles Trapezoid

## Definition

An isosceles trapezoid is a trapezoid with oblique sides congruent.

## Properties

- The oblique sides are congruent
- The angles adjacent to their respective bases are congruent
- Diagonals are congruent
- All the Generic Trapezoid formulas are valid

## Isosceles Trapezoid Formula

As a quadrilateral, the trapezoid is a four-sided shape. The unique property about the trapezoid is that it has only one pair of parallel sides. Isosceles trapezoid is a type of trapezoid where the non-parallel sides are equal in length.

**There are two isosceles trapezoid formulas**

The Perimeter of isosceles trapezoid formula is

## Isosceles Trapezoid

An isosceles trapezoid (called an isosceles trapezium by the British; Bronshtein and Semendyayev 1997, p. 174) is trapezoid in which the base angles are equal and therefore the left and right side lengths are also equal.

From the Pythagorean theorem,

## Isosceles Trapezoid Formulas

**Trapezoid** is a quadrilateral in which a pair of opposite sides are parallel. It is also known as Trapezium. There are three types of Trapezoids and Isosceles Trapezoid is one of its type. The types of Trapezoids are:

- Right Trapezoid
- Isosceles Trapezoid
- Scalene Trapezoid

### Isosceles Trapezoid

Isosceles Trapezoid is a trapezium with congruent base angles and congruent non parallel sides. A trapezium is called an Isosceles trapezoid when two opposite sides (Bases) are parallel and the other two sides (legs) are of the same length.

Area and Perimeters are the formulas of Isosceles Trapezoid.

### Area of Isosceles Trapezoid

The area of the Isosceles Trapezoid can be calculated by adding the lengths of two parallel sides (bases) and dividing this by 2 and multiplying the result with the height of the trapezium to get the area. The area formula is given by-

Area = ((a+b)/2) × hWhere,

a, b are the length of parallel sides

and h is the height.

To get more understanding let’s solve a few examples

### Sample Problems on Area of Isosceles Trapezium

**Question 1: What is the area of isosceles trapezium if the length of parallel sides are 7cm, 5cm and height is 4cm.**

**Solution:**

Given

Length of parallel sides (a) = 7cm, b = 5cm

Height (h) = 4cm

Area = ((a + b)/2) × h

= ((7 + 5)/2) × 4

= (12/2) × 4

= 6 × 4

= 24 cm

^{2}

Area of given isosceles trapezoid is 24cm^{2}.

**Question 2: Find the** **height of isosceles trapezium if the length of parallel sides are 6cm, 4cm and area is 24cm ^{2}.**

**Solution:**

Given

Length of parallel sides (a) = 6cm, b = 4cm

Area = 24cm

^{2}Area = ((a+b)/2) × h

24 = ((6+4)/2) × h

24 = (10/2) × h

24 = 5 × h

h = 24/5

= 4.8cm

So from given area, base lengths the height of an isosceles trapezium is 4.8cm

### Perimeter of Isosceles Trapezoid

The perimeter of an isosceles trapezoid can be calculated by adding all sides of the trapezoid. The perimeter formula is given by-

Perimeter = a+b+c+dWhere,

a,b are lengths of two parallel sides

c,d are length of two unparallel sides

Note:For isosceles trapezoid c = d (Unparallel side lengths are equal)

Let’s look into a few examples to get more understanding.

### Sample Problems on Perimeter of Isosceles Trapezoid

**Question 1: What is the perimeter of an** **isosceles trapezoid if the length of sides are 7cm, 5cm, 3cm, 3cm.**

**Solution:**

Given,

Length of parallel sides (a) = 7cm, (b) = 5cm

Length of un parallel sides (c) = 3cm, (d) = 3cm

Perimeter = a + b + c + d

= 7+5+3+3

= 18cm

So perimeter of given isosceles trapezium is 18cm.

**Question 2: What is the perimeter of an** **isosceles trapezoid if the length of parallel sides are 8cm, 4cm and length of sides of equal lengths 2cm.**

**Solution:**

Given,

Length of parallel sides (bases) (a) = 8cm, (b) = 4cm

Length of un parallel sides (legs) (c) = 2cm, (d) = 2cm

Perimeter = a + b + c + d

= 8 + 4 + 2 + 2

= 16cm

So perimeter of given isosceles trapezium is 16cm.

**Question 3: What is the area and perimeter of an isosceles trapezium with base lengths are 3cm, 6cm and the length of the** **other 2 sides which are equal in length is 2.5cm and height is 1.5cm.**

**Solution:**

Given

Length of bases (a) = 6cm, (b) = 3cm

Length of legs i.e., sides with equal lengths (c) = 2.5cm, (d) = 2.5cm

Height (h) = 1.5cm

Area =((a+b)/2) × h

= ((6+3)/2) × 1.5

= (9/2) × 1.5

= 4.5 × 1.5

= 6.75cm

^{2}Perimeter = a + b + c + d

= 6 + 3 + 2.5 + 2.5

= 14cm

So for the given data, Area is 6.75cm^{2}& perimeter is 14cm.

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