5 Ways to Find the Area of a Circle: Formula & Solved Examples

The area of a circle is one of the most fundamental concepts in geometry. Whether you’re working with circular objects in real life or dealing with mathematical problems in school or work, understanding how to calculate the area of a circle is essential. In this guide, we’ll explore 5 ways to calculate the area of a circle using different formulas and approaches. By the end of this article, you can confidently find the area of any given circle, whether you have the radius, diameter, or even the circumference. So, let’s dive into the fascinating world of circles!

What is the Area of a Circle?

The area of a circle refers to the total space occupied by the circle’s surface. It is measured in square units such as square inches, square feet, or square meters, depending on the unit of measurement used for the radius. The circle is a closed curve, where every point on the curve is at a fixed distance from a single point called the center.

In the above figure, we can see a circle, where radius r from the center ‘o’ to the boundary of the circle. The basic formula to find the area of a circle is: Area = π × r²

Where:

• r is the radius of the circle or the distance from the center to any point on the edge.
• π is a mathematical constant, approximately equal to 3.14159.

5 Ways to Calculate the Area of a Circle

Understanding how to calculate the area of a circle is a fundamental skill in geometry, and it is used in many practical situations. Whether you’re measuring a circular field, calculating the space occupied by a circular object, or solving math problems, knowing how to find the area is crucial. Depending on what information you have available, such as the radius, diameter, or circumference, there are several ways to approach the problem.

In this guide, we’ll explore 5 different methods to calculate the area of a circle, each with clear formulas and easy-to-follow steps. Whether you’re a student or someone working on a real-life project, these methods will help you quickly and accurately determine the area of a circle.

Method	Formula	Explanation
1. Using Radius	A = π × r²	Use the radius (r) of the circle to calculate the area.
2. Using Diameter	A = π × (d/2)²	Divide the diameter (d) by 2 to get the radius, then apply the circle formula.
3. Using Circumference	A = C² / (4π)	Use the circumference (C) to calculate the circle’s area.
4. Using Segments	A = (θ / 360) × π × r² – Area of Triangle	Apply when you know the central angle (θ) and radius.
5. Using Sectors	A = (θ / 360) × π × r²	Apply when you have a sector, using the central angle (θ) and radius.

1. Calculate the Area of a Circle with Radius

This is the most direct method to calculate the area of a circle. If you are given the radius (r) of the circle, you can use the area of the circle formula mentioned above.

Formula:

The area of a circle is : π ( Pi ) times the Radius squared: Area = π × r²

Where:

• r is the radius of the circle.

Example: If the radius of the circle is 5 inches, the circle’s area would be:

Area = π × 5² = π × 25 ≈ 78.54 square inches

2. Calculate the Area from Diameter

If you are provided with the circle’s diameter, you can easily find the radius first by dividing the diameter by 2. Once you have the radius, you can then apply the area of circle formula.

The diameter of the circle is double the radius of the circle. Hence the area of the circle formula using the diameter is equal to π/4 times the square of the diameter of the circle.

Formula:

Area = π × (d/2)²

Where: d is the diameter of the circle.

Example: If the diameter of the circle is 10 inches, the radius will be 5 inches. Now, calculate the area:

Area = π × 5² = π × 25 ≈ 78.54 square inches

This method helps when the diameter is directly given but the radius is not.

3. Using Circumference to Calculate Area

The circumference of a circle is the perimeter or boundary length of the circle. You can find the area of a circle if you have the circumference, as they are mathematically related.

The formula to calculate the circle’s area from the circumference is:

Area = C² / (4π)

Where:

• C is the circumference of the circle.

To derive this formula, you need to know that the circumference of a circle is given by:

C = 2π ( Pi ) r

Therefore, by rearranging the equation, you can find the circle’s area.

Example: If the circumference of the circle is 31.42 inches, the radius can be found by dividing the circumference by 2π:

r = C / 2π = 31.42 / 2π ≈ 5 inches

Now, use the formula to find the circle’s area:

Area = π × 5² = π × 25 ≈ 78.54 square inches

4. Find an Area with Segments

A segment of a circle is a region bounded by a chord and the arc it cuts off. To find the area of a segment, you need the radius and the central angle of the segment.

The circle’s area of the segment can be calculated by:

Area of Segment = (θ / 360) × π r² – Area of Triangle

Where:

• θ is the central angle (in degrees).
• r is the radius of the circle.
• The Area of the Triangle can be calculated using basic trigonometry.

This method is useful when dealing with concentric circles or sectors and is often applied in real-life scenarios like circular fields or circular objects.

5. Find Area from Sectors of the Circle

A sector is a part of a circle that is enclosed by two radii and the arc between them. To calculate the area of a sector, you can use the following formula:

Area of Sector = (θ / 360) × π r²

Where:

• θ is the central angle (in degrees).
• r is the radius of the circle.

This formula helps when dealing with sectors or pie-shaped slices of a circle, like a pizza or pie. Sectors are commonly found in many real-life circular objects.

Common Mistakes

When calculating the area of a circle, many people make simple but significant mistakes. These errors can lead to incorrect results. Here are some of the most common mistakes to watch out for:

1. Confusing Diameter and Radius:

One of the most common mistakes is confusing the diameter with the radius. The diameter is the distance across the circle through its center, while the radius is the distance from the center to any point on the circle. The radius is always half the length of the diameter. If you’re given the diameter, be sure to divide it by 2 to get the correct radius before using the area of the circle formula.

2. Forgetting to Square the Radius:

Another common error is neglecting to square the radius when calculating the area. The formula for the area of a circle is A = π × r², which means you need to square the radius (multiply it by itself) before multiplying by π\piπ. Forgetting to do this step will lead to incorrect results.

3. Misapplying the Formula:

Ensure you’re using the correct formula for the situation. For instance, the formula for area is different from the formula for circumference. The circumference formula is C = 2πr, while the area of the circle formula is A = π × r². If you accidentally use the wrong formula, you may end up with the wrong result. When the length of the radius or diameter or even the circumference of the circle is already given, then we can use the surface formula to find out the surface area.

4. Incorrect Units:

Be mindful of the units you are using. The area is always given in square units (e.g., square meters, square feet, square inches), so if the radius is in meters, the area will be in square meters. Similarly, if the radius is in inches, the area will be in square inches. It’s important to check that your units are consistent throughout the calculation. For example, if you’re working with the diameter in centimeters but need the area in square meters, you must first convert the units appropriately.

5. Overlooking the Value of π:

Although the value of π is an irrational number, for most practical calculations, it is approximately equal to 3.14159. Always make sure to use this value or its approximation when calculating the area. Sometimes, people round π too early in the calculation, which can lead to a slightly inaccurate result. Use the value of π as accurately as possible to avoid errors, especially for more precise calculations.

By being mindful of these common mistakes, you can ensure that your calculations of the area of a circle are correct and accurate. Always double-check your steps, use the right formulas, and make sure your units and values are correct.

Solved Examples

Example 1: If the radius of the circle is 6 cm, then find the area of the circle.

Solution:
Given: Radius r = 6 cm

Step 1: Write down the formula for the area of a circle:

A = π × r²

Step 2: Substitute the given value of r into the formula:

A = π × 6² = π × 36

Step 3: Multiply by the value of π (approximately 3.14159):

A ≈ 3.14159 × 36 ≈ 113.10

Thus, the area of the circle is approximately 113.10 square centimeters.

Example 2: If the diameter of the circle is 14 units, then find the area of the circle.

Solution:
Given: Diameter d = 14 units

Step 1: Find the radius. The radius is half the diameter.

r = d / 2 = 14 / 2 = 7

Step 2: Write down the formula for the area of a circle:

A = π × r²

Step 3: Substitute the value of r = 7 units into the formula:

A = π × 7² = π × 49

Step 4: Multiply by the value of π (approximately 3.14159):

A ≈ 3.14159 × 49 ≈ 153.94

Thus, the area of the circle is approximately 153.94 square units.

Example 3: If the circumference of the circle is 62.83 units, then find the area of the circle.

Solution:
Given: Circumference 𝐶 = 62.83 units

Step 1: Write down the formula for the circumference of a circle:

C = 2πr

Step 2: Solve for the radius:

r = C / (2π) = 62.83 / (2 × 3.14159) ≈ 10

Step 3: Write down the formula for the area of a circle:

A = π × r²

Step 4: Substitute the value of r = 10 units into the formula:

A = π × 10² = π × 100

Step 5: Multiply by the value of π (approximately 3.14159):

A ≈ 3.14159 × 100 ≈ 314.16

Thus, the area of the circle is approximately 314.16 square units.

Example 4: If the radius of the circle is 15 inches, then find the area of the circle.

Solution:
Given: Radius r=15 inches

Step 1: Write down the formula for the area of a circle:

A = π × r²

Step 2: Substitute the given value of r = 15 inches into the formula:

A = π × 15² = π × 225

Step 3: Multiply by the value of π (approximately 3.14159):

A ≈ 3.14159 × 225 ≈ 706.86

Thus, the area of the circle is approximately 706.86 square inches.

Example 5: If the central angle of the sector is 120° and the radius of the circle is 9 cm, then find the area of the sector.

Solution:
Given:

• Central angle θ=120°
• Radius r = 9 cm

Step 1: Write down the formula for the area of the sector:

A = (θ / 360) × π × r²

Step 2: Substitute the given values θ=120° and r=9 cm into the formula:

A = (120 / 360) × π × 9² = (1 / 3) × π × 81

Step 3: Multiply by the value of π (approximately 3.14159):

A ≈ (1 / 3) × 3.14159 × 81 ≈ 84.78

Thus, the area of the sector is approximately 84.78 square centimeters.

Conclusion

The area of a circle formula is useful for measuring the region occupied by a circular field or a plot. Understanding how to find the area of a circle is a crucial skill in mathematics and in many real-life applications.

Whether you’re calculating the surface area of a circular field, measuring the space occupied by a circular object, or solving a geometry problem, knowing these 5 methods will make your calculations more accurate and efficient. From using the radius, diameter, or circumference, to applying formulas for segments and sectors, you’re now equipped with all the tools to master the area of a circle. Practice with different problems, and you’ll soon become an expert!

FAQs

Q1: How to work out area from diameter?

The diameter of the circle is twice the radius of the circle. d = 2r. To calculate the area of a circle from the diameter, the first step is to find the radius, since the radius is always half of the diameter. The formula to calculate the radius is: r = d / 2.

Once you have the radius, you can use the formula for the circle: A = π × r²

For example, if the diameter of the circle is 10 units, first divide the diameter by 2 to find the radius: r = 10 / 2 = 5 units

Then, apply the formula: A = π × 5² = π × 25 ≈ 78.54 square units

Thus, the area of the circle is approximately 78.54 square units. Always remember to first convert the diameter into the radius before applying the formula.

Q2: What is 2πr?

The expression 2π ( Pi ) r represents the circumference of a circle, which is the total distance around the circle. It is calculated by multiplying 2, the constant π (pi), and the radius r. The formula is: C = 2π ( Pi ) r

For example, if the radius r = 5 units, the circumference is: C = 2π × 5 = 10π ≈ 31.42 units

This formula gives you the perimeter or the boundary length of the circle. It’s essential for finding the circumference of a circular object or circular ground.

Q3: How to work out m² of a circle?

To calculate the area of a circle in square meters (m²), you simply need to apply the formula: A = π × r²

Where r is the radius of the circle in meters. If the radius is already in meters, the resulting area will be in square meters.

For example, if the radius r = 3 meters: A = π × 3² = π × 9 ≈ 28.27 m²

Thus, the area of the circle is approximately 28.27 square meters. If the radius is provided in other units, make sure to convert them to meters before applying the formula to ensure your answer is in square meters.

Q4: How to find the area of a circle?

To find the area of a circle, you use the area formula: A = π × r²

Where r is the radius of the circle. If you are given the diameter, you can first divide it by 2 to find the radius and then use the formula.

For example, if the radius r = 4 units: A = π × 4² = π × 16 ≈ 50.27 square units

Therefore, the area of the circle is approximately 50.27 square units. Remember to always square the radius before multiplying it by π to find the correct area.

Q5: How to find the area with just the circumference?

When the circumference of a circle is given, you can find the circle’s radius using the formula for circumference: C = 2π ( Pi ) r

Rearrange the formula to solve for r: r = C / 2π

Once you have the radius, use the area formula to find the area: A = π × r²

For example, if the circumference is 31.42 units, first find the circle’s radius: r = 31.42 / 2π ≈ 5 units

Then, apply the area formula: A = π × 5² = π × 25 ≈ 78.54 square units

Thus, the area of the circle is approximately 78.54 square units. This method allows you to find the area even if you only have the circumference of the circle.