### Key Concepts

- Parts of the circle.
- What is an area?
- Area of the circle?
- Area of semicircle, quarter circle.
- Solve problems involving area of the circle.
- Use area to find the radius and diameter.
- Use circumference to find the area of a circle

**8.6 Solve problems involving the area of a circle**

**Circle:**

A circle is a collection of points that are at a fixed distance from the center of the circle. A circle is a closed geometrical shape.

**Example of circles: **Wheel, pizzas, and circular grounds, etc.

**Parts of a circle:**

**Radius: **The distance from center to a point on the boundary of the circle can be defined as radius. It is represented by the letter ‘r’. Radius plays an important role in finding the area and circumference of the circle.

**Diameter: **A line that passes through the center and its endpoints lie on the circle can be defined as diameter. Diameter is twice the length of the radius. It is represented by the letter ‘d’.

Formula:** **d = 2r

d = 2 × radius.

**Circumference: **The circumference of the circle is equal to the length of its boundary. This means that the perimeter of the circle is equal to its circumference.

Formula: Circumference C = 2πr

**Chord: **The length of a line segment joining any two points on the circle is called a chord. Diameter is the biggest chord that is possible in the circle.

**Arc: **The portion of the circumference of the boundary of the circle is called its arc. The smaller portion of the circle’s boundary is called its minor arc, and the larger portion is called the major arc.

**Segment: **The region occupied by an arc and chord of the circle is defined as the segment of the circle.

**Sector:** The region enclosed by two radii and an arc of the circle is called the sector of the circle. Any two radii divide the circle into two sectors.

**Area:**

In geometry, the area can be defined as the space occupied by a flat shape or the surface of an object.

**Area of circle:**

The area of a circle is the amount of space enclosed within the boundary of a circle. The region within the boundary of the circle is the area occupied by the circle. It may also be referred to as the total number of square units inside the circle.

**Formula to find the area of the circle:**

The area of a circle can be determined using the radius or diameter of the circle. Suppose a circle has a radius ‘r’, then the area of the circle is given by πr2 . If the diameter is given, then the area of the circle is given by πd2/4 in square units. Where π = 22 / 7or 3.14.

**Area of semicircle:**

The area of a semi-circle is half the area of the circle.

**Formula to find the area of the semicircle:**

As the area of a circle is πr2 . So, the area of a semicircle is given by

πr2 / 2 or πd2 /8, where *r* is the radius and *d* is the diameter of the circle. Where

π =22 / 7or 3.14.

**Area of quarter circle:**

The area (or) portion that is formed by two radii that are perpendicular to each other and one-fourth portion of the circumference of a circle is known as a quarter circle. This is also known as a quadrant of a circle, i.e., if we divide a circle into 4 equal parts, each part is a quarter circle (or) a quadrant.

**Formula to find the area of the semicircle:**

As the area of a circle is πr2. So, the area of a quarter circle is given by πr2 / 4 or πd2 / 16, where *r* is the radius and *d* is the diameter of the circle. Where π =22 / 7or 3.14.

**8.6.1 Solve problems involving the area of a circle**

**Example 1: **At a school play, there is a spotlight above the center of the floor that covers a lighted area with a diameter of 14 feet. What is the area covered by the spotlight? If the lighted area is to be covered by a carpet that costs $4.95 per square foot. Find the cost of the carpet.

**Solution: **We know that diameter of the lighted area is 14 feet.

**Step 1: **Find the length of radius to determine the area of the lighted area.

Radius =Diameter / 2

=14 / 2

=77feet.

**Step 2: **We know that the area of the circle, A = πr2

A = π× r × r

A =22 / 7× 7 × 7 (Use π as 22/7)

A = 22 × 7

A = 154 square feet.

**Step 3: **Calculate the cost of carpet.

Cost = Area of the lighted area × cost per square foot

Cost = 154 × 4.95

Cost = $762.3

Therefore, the area of the lighted area and the cost of the carpet is 154 square feet and $762.3, respectively.

**Example 2: **A circular display has a diameter of 1.7 m. How much plastic is required to cover the display?

**Solution: **We know that diameter of the display is 1.7 meter.

**Step 1: **Find the length of the radius to determine the area of the display.

Radius = Diameter/2

=1.7 / 2

=17 / 20 meter.

**Step 2: **We know that the area of the circle, A =πr2

A =π× r × r

A =3.14×17 /20 ×17 / 20 (Use π as 3.14)

A = 3.14 × 0.85 × 0.85

A = 2.26865 square meter.

A ≈ 2.27

Therefore, approximately 2.27 square meter of plastic is required for the display.

**8.6.2 Use area to find the radius and diameter**

**Example 1: **Lara takes her little brother to the fishpond at a local park. A bord sign nearby states that the area of the pond is 113.04 square inches. Help Lara in determining the diameter of the pond.

**Solution: **

We know that the largest possible distance between any two points on the boundary of the pond is called the diameter.

**Step 1: **We know that the area of the circle, A = πr2

A = π×r2

113.04 =3.14 ×r2

113.04 / 3.14 = 3.14/ 3.14 xr2(Divide both sides of the equation by 3.14)

36 = r2

6^{2}= r2

6 = r

r = 6 inches.

We got the radius of the pond as 6 inches.

**Step 2: **Calculate the diameter of the pond.

We know that the diameter is twice the length of the radius.

So, d = 2 × r

d = 2 × 6

d = 12

Therefore, the diameter of the pond is 12 inches.

**Example 2: **If the area of the circle is 110 yd2 . Determine the diameter.

**Solution: **

We know that the largest possible distance between any two points on the boundary of the pond is called the diameter.

**Step 1: **We know that the area of the circle, A =πr2

A =π×r2

154 =22 / 7× r2

154 ×7 / 22 = 22 / 7 × r2 × 7 /22 (Multiply both sides of the equation by 7 / 22)

49 =r2

7^{2} = r2

7 = r

r = 7 yards.

We got the radius of the circle as 7 yards.

**Step 2: **Calculate the diameter of the pond.

We know that the diameter is twice the length of the radius.

So, d = 2 × r

d = 2 × 7

d = 14

Therefore, the diameter of the circle is 14 yards.

**8.6.3 Use circumference to find the area of a circle**

**Example 1: **What circular area is covered by the signal of a radio station if the circumference is 748 miles?

**Solution: **We know that circumference of the circle is 748 miles.

**Step 1: **Find the length of the radius from the circumference.

Circumference, C =754

748 = 2 ×π× r

748 = 2 ×22 / 7 × r

748 = 44 / 7× r

Multiply by 7 / 44 on both sides of the equation

748 × 7 / 44 = 44 / 7 × r ×7 / 44

17 × 7 = r

r = 119 miles.

**Step 2: **Find the area of the station.

We know that the area of the circle, A = πr2

A =π× r × r

A = 22 / 7 ×119 ×119 (Use π as 22 / 7 )

A =22 / 7×119 ×119

A = 22 × 17 × 119

A = 18326 square miles

Therefore, we conclude that signal covers an area of 18326 square miles.

**Example 2: **Find the area of a circle whose circumference is the same as the perimeter of square of side 22 feet.

**Solution: **We know that the circumference of the circle is equal to the perimeter of square.

**Step 1: **Find the perimeter of the square

Perimeter of the square = 4 × a

= 4 × 22

= 88 feet.

Circumference, C =88

88 =2 ×π× r

88 =2 ×22 / 7 × r

88 =44 / 7 × r

Multiply by 7 / 44 on both sides of the equation

88 × 7 / 44 = 44 / 7 × r × 7 / 44

2 × 7 = r

r = 14 feet.

**Step 2: **Find the area of the circle.

We know that the area of the circle, A =πr2

A = π × r × r

A = 22 / 7 × 14×14 (Use π as 22/ 7 )

A =22×2 ×14

A = 616 square feet

Therefore, the area of the circle is 616 square feet.

## Exercise:

- Determine the area if the radius of the circle is 35 feet.
- Determine the area of the circle if the diameter of the circle is 119 inches.
- If the diameter of the circle is doubled. What will happen to the area of the circle?
- What is the area of the circle with an area of 28.26 square meter?
- Find the diameter of the circle if its area is 616 .
- The area of the circle is found to be 12.56 . Determine the diameter of the circle.
- The area of a circle is 81π square units. What is the radius of this circle?
- An asteroid hits the Earth and creates a huge round crater. Scientists measures the distance around the crater as 78.5 miles. What is the area of the crater?
- The distance around a carousel is 21.98 yards. Determine the area of the carousel.
- A circular plate has a circumference of 16.3 inches. What is the area of this plate? Use 3.14 for π. Round to the nearest whole number.

### What have we learned:

- What are parts of the circle?
- What is an area?
- What is an area of the circle?
- What is an area of semicircle and quarter circle?
- Solving problems involving area of the circle.
- Using area to find the radius and diameter.
- Using circumference to find the area of the circle.

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: