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.

Area of Rectangles Area of Rectangles

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Contents
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    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

    The area of a rectangle is the measure of the amount of space enclosed by the rectangle. It is calculated by multiplying the length of the rectangle by its width. The area of a rectangle is calculated by multiplying its length by its width. Consider the following rectangle.

    Rectangle example - StudySmarterFinding the area of the Rectangle through this illustration example

    The area of a rectangle is given by the formula:

    \[Area = b \times h\]

    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 example - StudySmarterAn example of the area of the Square.

    Look at the above square and recall the formula for the area of a rectangle: \[Area = base \times height.\]

    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:

    \[Area = length\,of\,side \times length\,of\,side = (length\,of\,side)^2\]

    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.

    Area of rectangles square unit StudySmarterSquare units, Jurgensen & BrownGeometry

    Area of rectangles: Example problems

    The following examples show how to find the area of a rectangle.

    Example 1: Suppose you have a rectangle with a length of 10 units and a width of 5 units. To find the area:

    \[Area=10\,units\times5\, units=50\, square\, units\]


    Example 2: Imagine a garden plot shaped as a rectangle with a length of 15 meters and a width of 8 meters. To determine the area:

    \[Area=15\,meters\times8\, meters=120\, square\, meters\]


    Example 3: Consider a rectangular swimming pool with a length of 25 meters and a width of 10 meters. To find the area:

    \[Area=25\,meters\times10\, meters=250\, square\, meters\]

    Here is a more in-depth breakdown on how to find the area of a rectangle:

    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.

    Area of Rectangles, Pythagoras Theorem, StudySmarterThe diagonal of a rectangle divides it into 2 right angles triangles.

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

    Area of Rectangles, Example, StudySmarter

    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,

    \[AD^2 + AB^2 \Rightarrow AD^2 + 9^2 = 15^2 \Rightarrow AD^2 = 15^2 - 9^2 \Rightarrow AD^2 = 144 \Rightarrow AD = 12 \]

    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 with fractions

    If the length and width of a rectangle are given as fractions, you can still calculate its area by multiplying these fractions.

    Let's break it down with an example:

    Suppose the length \( b \) of a rectangle is \( \frac{3}{4} \) units and the width \( h \) is \( \frac{2}{5} \) units.

    To find the area \( A \) of the rectangle: \[ A = b \times h \] Substitute in the given fractions: \[ A = \frac{3}{4} \times \frac{2}{5} \] To multiply the fractions: \begin{align*} \text{Multiply the numerators:} & \quad 3 \times 2 = 6 \\ \text{Multiply the denominators:} & \quad 4 \times 5 = 20 \\ \end{align*} This gives: \[ A = \frac{6}{20} \] Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor: \[ A = \frac{3}{10} \] Thus, the area of the rectangle is \( \frac{3}{10} \) square units.

    Perimeter and area of rectangles

    The perimeter and area are two fundamental properties of a rectangle.

    Perimeter of a Rectangle: The perimeter is the total distance around the rectangle, or the sum of all its sides. Since opposite sides of a rectangle are equal in length, the perimeter \(P\) can be found using the formula: \[P=2l+2w\]

    In the same example, with a length of 5 units and a width of 3 units, the perimeter would be

    \[2(5)+2(3) = 10+6 = 16\,units \]

    In summary:

    • The area gives the total space enclosed by the rectangle and is measured in square units (e.g., square centimeters, square meters, square inches).
    • The perimeter gives the total distance around the rectangle and is measured in linear units (e.g., centimeters, meters, inches).

    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

    • When the dimensions of a rectangle are given in fractions, the process remains the same: simply multiply the fractional length by the fractional width.

    Frequently Asked Questions about Area of Rectangles

    How to find the area of a rectangle?

    The area of a rectangle is given by the formula:

    Area = b × h

    where b=base, h=height.

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

    The area of a rectangle is given by the formula:

    Area = b × h

    where b=base, h=height.

    What is a rectangle?

    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

    How to calculate the surface area of a rectangle?

    The concept of "surface area" is generally associated with three-dimensional objects, like prisms, cylinders, spheres, etc. However, a rectangle is a two-dimensional shape, so it doesn't have a "surface area" in the traditional sense. Instead, it has an "area."

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