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# Partial Area of a Circle

Circles are those perfectly round shapes that appear on everything from wheels to pizzas. What happens when we’re not dealing with the entire circle? What if we are interested in finding the area of just a portion of it, like a slice of pizza or a part of a circular garden? That’s where sectors come in.

• Understanding Sectors
• Formula
• Examples
• FAQs

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Understanding Sectors

A sector is like a slice of a pizza, created by two radii (plural of radius) and the arc between them. Just like you might want to know how much pizza is on your plate, you should calculate the area of a sector. A sector divides the circle into two regions, namely Major and Minor Sector.

The Formula

To find the area of a sector, we use the following formula:

Area of Sector = (θ/360) × π × r²

Here is what each part means:

• θ (Theta): It is the central angle of the sector, measured in degrees.
• π (Pi): A unique number (approximately 3.14159 or 22/7) used in circles and other mathematical calculations.
• r (Radius): The distance from the circle’s center to its edge.

Putting It Into Practice

Let’s work through an example to make things clearer. Imagine we have a circle with a radius of 10 units and a central angle of 60 degrees.

1. Given: θ = 60 degrees, r = 10 units
2. Formula: Area of Sector = (60/360) × π × (10²)
3. Calculation: (60/360) × π × 100 ≈ 52.36 square units

So, the area of the sector is approximately 52.36 square units. That’s like finding out how much pizza you’ve got on your slice!

Area of Sector with respect to Length of the Arc

You can also find the area of a sector from its radius and arc length. The formula for the area, A, of a circle with radius, r, and arc length, L, is:

A= r×L / 2

Example

If the length of the arc of a circle with a radius of 14 units is 4 units, then find the area of the sector corresponding to that arc.

Given: L = 4 units, r = 14 units

As we know, A = (rL)/2

Now, substitute the values in the formula, we get

A = [14 × 4]/2

A = 56/2

A = 28

Therefore, the area of the sector corresponding to the arc is 28 square units.

You use an even easier formula if you are given the radians instead of a central angle in degrees.

( A radian is a unit of measurement for angles, just like degrees. However, radians are based on the radius of a circle. In terms of radians, one complete revolution (360 degrees) is equal to 2π radians. It is because the circumference of a circle is 2π times its radius, and if you move along the entire circumference (1 complete revolution), you’ve traveled 2π radians.

Area of Sector = (θ/2) × r²

Here’s what each term means:

• θ (Theta): The central angle of the sector in radians.

Radians are based on π (a circle is 2π radians), so what we did was replace θ/360 ​ with θ/2 π.

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Example:

Problem: Find the area of a sector with a radius of 5 units and a central angle of π/3 radians.

Solution:

1. Given: r = 5 units, θ = π/3 radians
2. Formula: Area of Sector = (π/3 / 2) × 5²
3. Calculation: (π/6) × 25 ≈ 13.09 square units

## FAQs

##### What is a partial area of a circle?

The partial area of a circle refers to the area of a portion or section of a circle, known as a sector. A sector is defined by a central angle and two radii, creating a wedge-like shape within the circle.

##### How is the area of a sector calculated?

The area of a sector can be calculated using the formula:
Area of Sector = (θ/360) × π × r²
Where θ is the central angle of the sector in degrees, π is a constant (approximately 3.14159), and r is the circle’s radius.

##### Can the area of a sector be calculated using radians?

Yes, you can calculate the area of a sector using radians. The formula for radians is:
Area of Sector = (θ/2) × r²
Here, θ is the central angle of the sector in radians, and r is the radius.

##### What's the difference between a sector and a circle?

A circle is a two-dimensional shape with all points equidistant from its center. A sector is a circle portion defined by a central angle and two radii. The sector resembles a slice of the circle.

##### What real-world applications involve sectors of circles?

Sectors are commonly seen in pie charts, clock faces, and circular paths. Architects, engineers, and designers also use sector calculations when planning layouts involving circular structures or motions.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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