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# Area of Rectangles

A rectangle is a special case of a quadrilateral, which is a four-sided plane figure. All 4 internal angles of a rectangle are right angles. A book, a football field, a window, a traveling suitcase are all examples of rectangles.

Now suppose you want to calculate the total space covered by a football field. Then, you would need to know how to calculate the area of a rectangle.

A rectangle is a quadrilateral with internal angles that are all right angles. The two-dimensional space occupied by a rectangle is the area of a rectangle.

A quadrilateral with 2 pairs of parallel opposite sides is called a parallelogram. Since all angles of a rectangle are right angles, it turns out that the opposite pairs of sides of a rectangle are always parallel. This makes every rectangle a parallelogram. In fact, a rectangle is considered a special type of parallelogram.

## Area of rectangles: Formula

Consider the following rectangle.

Rectangle illustration, Nilabhro Datta – StudySmarter Originals

The area of a rectangle is given by the formula:

Area = b × h

where b = length of base, h = length of height

Now the value, b, is the length of the side AB, which is considered to be the base here. Conventionally, one of the longer sides of the rectangle is taken to be the base, and one of the sides perpendicular to the base is considered to be the height. In this rectangle, the height is equal to the length of AD.

In some conventions, the base and height are referred to as the length and breadth of the rectangle.

### Special case: Formula for the area of a square

A square is a special case of a rectangle. In addition to all 4 internal angles being right angles, all 4 sides of a square are equal.

Square illustration, Nilabhro Datta, StudySmarter Originals

Look at the above square and recall the formula for the area of a rectangle: Area = base × height.

Since all 4 sides of a square are equal, the base and height are equal. Just knowing the length of the sides of a square is enough to calculate its area. Thus, in the case of a square the formula can be reduced to:

$Area=lengthofside×lengthofside={\left(lengthofside\right)}^{2}$

## Area of rectangles: Square units

When considering the area of a figure, remember that area is measured in square units, such as square centimeters (cm2), square feet (ft2), square inches (in2), etc.

If you are unfamiliar with the square unit, it is helpful to consider the concept as it is represented visually in the figure below. Consider how many square units are needed to exactly and exhaustively cover the entire surface of a closed figure. This amount is the figure's area.

Square units, Jurgensen & BrownGeometry

## Area of rectangles: Example problems

A rectangle with an area of 60 m2 has a base of length 20 m. What is the height of the rectangle?

Solution

Area = b × h

⇒60 m2 = 20 m × h

⇒ h = 60 m2 ÷ 20 m

⇒ h = 3 m

If you are given the length of 1 of the sides (base or height) of a rectangle and the length of the diagonal, you can calculate the unknown side length (height or base) using the Pythagoras' Theorem. The Pythagoras' theorem states that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other 2 sides.

The following figure shows how the diagonal of a rectangle divides it into 2 right angled triangles, thus allowing us to use the Pythagoras' theorem. Then, once both the base and height of the rectangle are known, the area can be calculated.

The diagonal of a rectangle divides it into 2 right angles triangles, Nilabhro Datta - StudySmarter Originals

In the following rectangle ABCD, AB = 9, BD = 15. Find the area of the rectangle.

Solution

Since the internal angles of a rectangle are right angles, BD is the hypotenuse of the right angled triangle, ΔABD.

So,

According to the Pythagorean Theorem,

$A{D}^{2}+A{B}^{2}=B{D}^{2}\phantom{\rule{0ex}{0ex}}⇒A{D}^{2}+{9}^{2}={15}^{2}\phantom{\rule{0ex}{0ex}}⇒A{D}^{2}={15}^{2}-{9}^{2}\phantom{\rule{0ex}{0ex}}⇒A{D}^{2}=144\phantom{\rule{0ex}{0ex}}⇒AD=12$

Area of the rectangle = b × h

= 12 ft. × 9 ft.

= 108 ft2

A square has sides of length 10 ft. What is the area of the square?

Solution

Area = side × side

= 10 ft. × 10 ft.

= 100 ft2

## Area of rectangles - Key takeaways

• A rectangle is a quadrilateral with internal angles that are all right angles.
• The area of a rectangle is given by the formula:

Area = b × h

where b = base, h = height.

• A square is a special case of a rectangle. In addition to all 4 internal angles being right angles, all 4 sides of a square are equal.

• The area of a square is given by the formula: Area = side × side

The area of a rectangle is given by the formula:

Area = b × h

where b=base, h=height.

The area of a rectangle is given by the formula:

Area = b × h

where b=base, h=height.

## Final Area of Rectangles Quiz

Question

What is a rectangle?

A rectangle is a quadrilateral whose internal angles are all right angles.

Show question

Question

What is the formula for finding the area of a rectangle?

Area = b × h

where b=base, h=height.

Show question

Question

A square has sides of length 7. What is the area of the square?

49

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Question

A square has sides of length 12. What is the area of the square?

144

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Question

A square has sides of length 4. What is the area of the square?

16

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Question

A square has diagonals of length 10. What is the area of the square?

50

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Question

A square has diagonals of length 8. What is the area of the square?

32

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Question

A rectangle with an area of 16 has a base of length 8. What is the height of the rectangle?

2

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Question

A rectangle with an area of 36 has a base of length 9. What is the height of the rectangle?

4

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Question

What is the area of a rectangle with base 10 and height 5?

50

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Question

What is the area of a rectangle with base 4 and height 1?

4

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Question

A rectangle has a base of length 4 and diagonals of length 5. What is the area of the rectangle?

12

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Question

A rectangle has a base of length 6 and diagonals of length 10. What is the area of the rectangle?

48

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Question

A rectangle with an area of 108 has a base of length 12. Find the length of the diagonal of the rectangle.

15

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Question

A rectangle with an area of 12 has a height of 3. Find the length of the diagonal of the rectangle.

5

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