Area of Rectangles

An Introduction to Rectangles: What They Are and How to Find Their Area

A rectangle is a four-sided shape that belongs to a family of shapes called quadrilaterals. It is a unique shape with four internal angles, all of which measure ninety degrees. You can find rectangles in everyday objects like books, windows, and suitcases.

Calculating the Area of a Rectangle: A Simple Equation

If you need to determine the total space covered by a rectangle, you will need to find its area. As mentioned earlier, a rectangle has four right angles, and its two-dimensional space is referred to as its area.

The Connection between Rectangles and Parallelograms

A rectangle is a type of parallelogram, a quadrilateral with two pairs of parallel opposite sides. Because all angles of a rectangle are right angles, its opposite sides are always parallel. This unique characteristic makes rectangles a special kind of parallelogram.

The Formula for Calculating the Area of a Rectangle

The formula for determining the area of a rectangle is:

Area = base × height

In this formula, the base (b) represents the length of one of the longer sides of the rectangle, while the height (h) is the length of the side perpendicular to the base. In some cases, the base and height may also be referred to as the length and breadth of the rectangle.

Special Case: Finding the Area of a Square

A square is a special case of a rectangle as it also has four right angles. However, in addition to this, all four sides of a square are equal in length. Therefore, the formula for finding the area of a square can be simplified to:

Area = side × side

Units of Measurement for Area

The area of a shape is measured in square units, such as square centimeters (cm²), square feet (ft²), or square inches (in²). This measurement refers to the number of squares needed to cover the entire surface of the shape.

Using the Pythagorean Theorem to Calculate the Unknown Side of a Rectangle

If you know the length of one side and the diagonal of a rectangle, you can use the Pythagorean Theorem to find the length of the other side. This theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Example Problems to Help Understand Area Calculations

Problem 1: Find the height of a rectangle with an area of 60 m² and a base length of 20 m.

Solution:

Area = base × height
60 m² = 20 m × height
height = 60 m² ÷ 20 m
height = 3 m

Problem 2: A rectangle has a base length of 9 ft. and a diagonal length of 15 ft. Find its area.

Solution:

Since the internal angles of a rectangle are right angles, the diagonal is the hypotenuse of one of its right-angled triangles.
Using the Pythagorean Theorem:
AD² + AB² = BD²
AD² + 9² = 15²
AD² = 225 - 81
AD = 12 ft.

Area of the rectangle = base × height
= 12 ft. × 9 ft.
= 108 ft²

Problem 3: Find the area of a square with side lengths of 10 ft.

Solution:

Area = side × side
= 10 ft. × 10 ft.
= 100 ft²

Key Takeaways: Remember These Points

• A rectangle is a four-sided shape with four internal right angles.
• The area of a rectangle is found using the formula: Area = base × height.
• A square is a special type of rectangle with all four sides being equal, making its area formula: Area = side × side.
• Area is measured in square units like cm², ft², or in².
• The Pythagorean Theorem is useful in calculating the unknown side of a rectangle when given the length of one side and the diagonal.

Understanding Rectangles: Finding Area with Base and Height

A rectangle's dimensions can be a bit confusing, but let's break it down. The base represents the length of the rectangle, while the height indicates the width. So, if you have a rectangle with a base of 5 inches and a height of 3 inches, the area would be 15 square inches. This is calculated by multiplying the base and height, giving you 5 inches x 3 inches = 15 square inches.

It's crucial to note that the units for the base and height must be the same for the formula to work accurately. For instance, if you have the base measurement in inches, the height must also be measured in inches.

But why does this formula work? The area of a rectangle is simply the total space inside the shape. Think of the rectangle as a grid, with each square representing one unit of area. By multiplying the base and height, we are essentially counting the squares in the grid, giving us the total area.

This formula has another helpful use – finding the area of squares. Since squares are a type of rectangle with equal sides, the same formula applies. Whether you're dealing with a square or a rectangle, this formula remains constant.

Next time you need to determine the area of a rectangle, remember the simple formula: Area = base x height. Armed with this knowledge, you can confidently calculate the area of any rectangle and impress your peers with your mathematical abilities.