## What Is the Area of a Circle?

**The area of a circle is the region occupied by the circle within its boundary (circumference) in a 2D plane. **

If the radius of a circle is r, the area is defined as A = πr^{2}, where is a mathematical constant with approximate value of 3.14 or $\frac{22}{7}$.

Area of a circle is measured in square units.

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## Parts of a Circle

A circle is a two-dimensional closed geometric shape that is a collection of all points which are at a fixed distance from a fixed point (center). This fixed distance is called the radius of the circle.

Let’s learn about different parts of a circle. It will help us in understanding the different formulas used to calculate the area of a circle.

Part | Definition |
---|---|

Center | The fixed point at the middle that is equidistant from all the points from every point on the boundary of the circle. |

Radius | The fixed distance from the center to any point on the circumference (boundary) of the circle. All radii (plural) of a circle are equal. |

Diameter | A straight line that connects any two points on the circumference while passing through the center of the circle. |

Circumference | The length of the boundary of a circle. |

Chord | A line segment that connects any two points on the circumference of a circle. |

Tangent | A straight line that passes by touching the circumference at one point. |

Arc | A part of the circle’s circumference. |

Segment | A region enclosed by an arc and a chord. |

Sector | A region enclosed by 2 radii and an arc. |

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## Area of Circle Formulas

Description | Formula |
---|---|

Area of a circle using radius | A = πr^{2} |

Area of a circle using diameter | A $= \frac{\pi}{4} d^{2}$ |

Area of a circle using circumference | A $= \frac{C^{2}}{4\pi}$ |

## How to Find the Area of a Circle

Let’s discuss methods to find the area of a circle based on the information available.

**How to Find the Area of a Circle with Radius**

The formula for the area of a circle (A) using its radius r is as follows:

A = πr^{2}

**Example: **A circle has a radius of 5 inches. Find the area of the circle.

A = πr^{2}

A = 3.14 × (5 inches)^{2}

A = 3.14 × 25 inch^{2}

A = 78.5 inch^{2}

**How to Find the Area of a Circle with Diameter**

We know that A = πr^{2} and r $= \frac{d}{2}$

The formula for the area of a circle (A) using its diameter (d) as follows:

A $= \frac{\pi}{4} d^{2}$

**Example:** A circle has a diameter of 10 inches. Find the area of the circle.

A $= \frac{\pi}{4} d^{2}$

A $= \frac{\pi}{4} \times $(10 inches)^{2}

A $= (\frac{3.14159}{4}) \times 100$ inch^{2}

A $= 78.54$ inch^{2}

**How to Find the Area of a Circle with Circumference**

Circumference = C = 2πr

Thus, r $= \frac{C}{2\pi}$

Substitute this value in the formula A = πr^{2}.

A $= \pi (\frac{C}{2\pi})^{2}$

A $= \frac{C^{2}}{4\pi}$

**Example: **A circle has a circumference of 31.42 units. Find the area of the circle.

A $= \frac{C^{2}}{4\pi}$

A $= \frac{(31.42)^{2}}{4 \times 3.14}$

A = 78.6 inch^{2}

## Derivation of Area of a Circle Formula

You can visualize and derive the area of a circle by the following two methods:

**Derivation of the area of a circle formula using the area of a rectangle**

The figure shows a circle divided into 16 equal sectors of equal areas. If the sectors are cut out from the circle and placed as a parallelogram-shaped figure, the area of the circle will be equal to the area of this figure.

The pink-colored sectors will cover half of the circle and thus have half the circumference. The blue-colored sectors will cover the other half.

Radius = r

Circumference = 2πr

Circumference of Pink Sectors $= \frac{1}{2} \times 2\pi r = \pi r$

Circumference of Blue Sectors = πr

We can see that the parallelogram-like shape will appear like a rectangle if the number of sectors is increased.

Length of the rectangle = πr

Breadth = r

Area of rectangle = l × b

Putting the values from the figure into the formula

A = πr × r

A = πr^{2}

Area of a circle = πr^{2}

**Derivation of the area of a circle formula using****the area of a triangle**

The figure below shows concentric circles throughout the radius. Imagine that the circle is cut along a radius and opened up to form a shape that resembles a triangle.

The circumference of a circle will become the base of the triangle, and the radius will become its height.

Circumference = 2πr

Radius = r = Height of the Triangle

So, the area of the triangle (A) will be equal to the area of the circle. We have

Since, Area of a triangle $= \frac{1}{2}$ × base × height

So, putting the values from the figure into the formula

A $= \frac{1}{2} × (2\pi r) × r$

A = πr^{2}

## Real-world Examples of Applications of Area of a Circle

Let us take a couple of real-world applications of the area of a circle.

**Example:** Tracy has a table with a circular tabletop having a diameter of 3 feet. Would a tablecloth with an area 5 square feet be sufficient for covering the tabletop?

The radius of the tabletop will be half of the diameter.

$r = \frac{d}{2} = \frac{3}{2} = 1.5$ feet

Now, the area of the circular tabletop can be calculated using the formula:

A = πr^{2}

A = π(1.5)^{2}

A = 7.065 square feet

Thus, the tablecloth having an area 5 square feet won’t be enough to cover the tabletop.

Such a calculation is useful in day-to-day life scenarios to know the precise measurements of circular objects, thus minimizing the wastage of money and resources.

## Facts about Area of a Circle

- The word “circle” itself comes from the Latin word “circulus,” meaning “a small ring.”
- Archimedes is known for his method of exhaustion, which involved inscribing and circumscribing polygons around a circle to approximate its area.

## Conclusion

In this article, we learned how to calculate the area of a circle. We discussed formulas used to find the area of a circle based on the information known to us. We also derived the formula for the area of a circle using the area of a rectangle and area of a triangle. Now, let’s solve a few examples and practice MCQs.

## Solved Examples on Area of a Circle

**Example 1: If the diameter of a circular garden is 50 feet, find the area of the garden.**

**Solution:**

Diameter of the circular garden (d) = 50 feet

The radius of the circular garden (r) $= \frac{d}{2} = 25$ feet

The area of the circular garden (A) = πr^{2}

A = π(25)^{2}

A = 625π square feet

Therefore, the area of the circular garden is 625π square feet.

**Example 2: Calculate the area of a circle with a radius of 10 feet in terms of **π**.**

**Solution: **

Radius (r) = 10 feet

The area of the circle (A) = πr^{2}

A = π(10)^{2}

A = 100π

Therefore, the area of the circular swimming pool is 100π square feet.

**Example 3: What is the area of a circular window with a diameter of 3 feet?**

**Solution:**

Diameter (d) = 3 feet

Radius (r) $= \frac{d}{2}$

r = 1.5 feet

So, A = πr^{2}

A $= \pi \times (1.5)^{2}$

A = 2.25π

Therefore, the area of the circular window is 2.25π square feet.

**Example 4: What is its area if a circular table measures 12π feet around its edge?**

**Solution:**

Circumference of the circular table (C) = 12π

The area of the circular table when the circumference is given $= A = (\frac{C}{2π})^{2}$

A $= (\frac{12\pi}{2\pi})^{2}$

A = 6^{2}

A = 36 feet^{2}

So, the area of the circular table is 36 feet^{2}. We have another way to find out the area of the circular table when its circumference is provided.

A $= \frac{C^{2}}{4\pi}$

A $= \frac{12\pi^{2}}{4\pi}$

A $= \frac{144\pi}{4\pi}$

A = 36 square feet

**Example 5: If a pizza restaurant charges $0.10 per square inch of pizza, what would be the cost of a pizza with a 14-inch diameter? (Round to the nearest whole number.)**

**Solution:**

Diameter of the pizza = 14 inches

Radius of the pizza $= \frac{14}{2} = 7$ inches

We need to calculate the area first. For the area of the pizza, we can use the formula (A) = πr^{2}

A = π × 7^{2} = 49π square inches

Cost per square inch = $0.10

Cost of a pizza with 49π square inches area $= 49\pi \times \$0.10 = \$15.386 = \$15$ (Rounded to the nearest whole number.)

Therefore, the cost of the pizza is $\$15$.

## Practice Problems on Area of a Circle

## Area of a Circle - Definition, Formula, Derivation, Examples, FAQs

### What is the area of a circle whose diameter is 8 units?

d = 2 units

r = 4 units.

A $= \pi r^{2} = \pi (4)^{2} = 16\pi$ square units

### What is the radius of a circle with an area of 36π square inches?

A $= \pi r^{2}$

$36\pi = \pi r^{2}$

$r = \sqrt{36} = 6$ inches.

### What is the area of a circle with a circumference of 18π units?

$C = 2\pi r$

Thus, $r = \frac{C}{2\pi} = \frac{18\pi}{2\pi} = 9$ units

Now, we can put the value of r into the area formula to calculate the area of the circle.

A $= \pi r^{2}$

A $= \pi(9)^{2} = 81\pi\; unit^{2}$

### What is the area of a unit circle?

Radius of unit circle = 1 unit

A $= \pi r^{2} = \pi (1)(1) = \pi$ square units

## Frequently Asked Questions about Area of a Circle

**What is the formula for the area of a circle?**

Area of a circle $= \pi r^{2}$, where r is the radius of the circle.

**Can the area of a circle be negative?**

The area of a circle can never have a negative value. The area of a circle is the amount of space it encloses given by A = πr^{2 }. Both π and r are positive, thus area is always a positive value.

**Is the radius of a circle a negative or a positive value?**

The radius of a circle is always a positive value and not a negative one. The radius of a circle is the distance from the center to any point on its boundary. It is always a positive value as per its definition.

**Is the area and circumference of a circle directly proportional to the radius?**

The area of a circle is directly proportional to the square of radius. The circumference of a circle is directly proportional to the radius.

**What is meant by the surface area of a circle?**

The surface area is a term used in reference with three-dimensional objects. It refers to the total area of all its faces or surfaces. A circle is a plane two-dimensional shape, so it does not have a “surface area”; instead, it just has an area. So, the surface area of a circle is nothing but its area.