Area of a Circle
- Definition of the Area of a Circle
- Parts of a Circle
- Finding the Area of a Circle
- Deriving Area of a Circle
- Finding the Area of a Circle Using Radius
- Finding the Area of a Circle Using Diameter
- Finding the Area of a Circle Using Circumference
- Practice Problems
- Frequently Asked Questions
Definition of the Area of a Circle
The concept of the area of a circle involves the space enclosed within its boundary on a flat surface. Imagine the circle's boundary, known as the circumference, as the edge that outlines the region within. This enclosed space is what we refer to as the area.
- The Area of a Circle formula is `A = \pir^2`.
Here `r` is the radius of the circle and `\pi (\text{pi})` is a special number approximately equal to `3.14` or `\frac{22}{7}`.
The area is written in square units, providing a measure of the space occupied by the circle
- We can also use the diameter `(d)` to determine the area. The diameter is twice the radius.
When using diameter,
The Area of a Circle formula is written as `A = \frac{\pid^2}{4}`.
Understanding the concept of area involves recognizing the relationship between the circumference, diameter, and the special mathematical constant, π, which is the ratio of the circumference to the diameter for any circle.
Parts of a Circle
A circle is a geometric shape with various parts, each contributing to its overall definition. Here are the key components or parts of a circle:
Center: It is the point in the middle of the circle from which all points on the circumference are equidistant. It's like the heart of the circle. It is denoted as `(h, k)`, where `h` is the `x`-coordinate and `k` is the `y`-coordinate of the center.
Radius: It is the distance from the center to any point on the circumference. We use `r` to represent radius.
Diameter: It is the longest distance across a circle, passing through the center. It's essentially twice the length of the radius. It is denoted by `d`, so `d = 2r`.
Circumference: It is the total distance around the outer edge of the circle. It's like measuring the perimeter of the circle. The formula is `C = 2\pir` or `C = \pid`.
Chord: It’s a straight line segment connecting two points on the circumference. The diameter is the longest chord of the circle.
Arc: It can be any portion of the circumference, akin to a slice of the circle. The length of an arc can be calculated using the formula:
- `s = rθ`, where `θ` is the central angle in radians
- `s = 2πr * (θ/360^\circ)`, where `θ` is the central angle in degrees.
Sector: It is the region enclosed by two radii and the arc between them. It's like a slice of pie within the circle. The area `(A)` of a sector can be found using the formula:
- `A = (1/2)r^2θ`, where `θ` is the central angle in radians.
- `A = πr^2 * (θ/360^\circ)`, where `θ` is the central angle in degrees.
Segment: In the context of a circle, a segment refers to the region enclosed by a chord and the arc it subtends. Specifically, a chord divides the circle into two segments – the major segment, lying on the larger part of the circle, and the minor segment, on the smaller part.
Tangent: It is a line that touches the circle at exactly one point, without crossing the circumference. It's like a line that gives the circle a little “tap.”
Secant: It is a line that intersects the circle at two points.
Finding the Area of a Circle
To find the surface area of a circle, you can use the area of a circle formula:
\( A = \pi r^2 \)
Here,
- \( A \) is the area of the circle,
- \( \pi \) (pi) is a mathematical constant approximately equal to `3.14`,
- \( r \) is the radius of the circle.
Here are the steps to find the area of a circle:
- Measure the Radius (\( r \)): The radius is the distance from the center of the circle to any point on its circumference.
- Apply the Formula: Plug the measured radius into the formula \( A = \pi r^2 \).
- Calculate: Square the radius (\( r^2 \)), multiply it by \(\pi\), and you'll get the area of the circle.
Remember to use the value of \( \pi \) appropriate for your calculations, either as `3.14` or as `22/7`, depending on the desired level of precision. The result will be in square units (e.g., square inches, square centimeters) since area is measured in two dimensions.
Deriving Area of a Circle
To derive the formula for the area of a circle, we can consider breaking the circle into infinitesimally small sectors and forming a shape that closely resembles a parallelogram. Here's a step-by-step derivation:
Consider a circle with radius \( r \).
- Divide the Circle into Sectors:
Divide the circle into small sectors by drawing two radii, forming a central angle \( \theta \).
- Arrange the Sectors:
Rearrange these sectors to form a shape resembling a parallelogram. As we take more and more sectors, this shape approaches a rectangle.
- Find the Base of the Parallelogram:
The base of this parallelogram is essentially half the circumference (\( C \)) of the circle. The formula for the circumference is \( C = 2 \pi r \).Hence, length of the base = `\pi r`.
- Determine the Height of the Parallelogram:
The height of the parallelogram is approximately \( r \), the radius of the circle.
- Calculate the Area of the Parallelogram:
The area (\( A_{\text{parallelogram}} \)) of a parallelogram is given by the product of its base and height:
`A_{\text{parallelogram}} = \pi \times r \times r`
- Relate to the Area of the Circle:
As the number of sectors increases, the parallelogram approaches the shape of a rectangle. The area of the rectangle is \( A_{\text{rectangle}} = \pi \times r \times r \). The area of the circle (\( A \)) is approximately equal to the area of this rectangle.
\( A \approx A_{\text{rectangle}} \)
\( A \approx \text{half of }C \times r \)
Substitute the formula for the circumference (\( C = 2 \pi r \)):
\( A \approx \pi \times r \times r \)
Simplify to obtain the formula for the area of a circle:
\( A = \pi r^2 \)
Thus, we have derived the formula for the area of a circle by considering the geometry of a circle and approximating its area with a parallelogram as the number of sectors increases.
Finding the Area of a Circle Using Radius
Example `1`: What is the area of a circle with a radius of `6` inches? (Consider `\pi = 3.14`)
Solution:
Area of circle: \( A = \pi r^2 \):
\( A = \pi \times (6)^2 \)
\( A = 36\pi\)
\( A \approx 36\times (3.14)\)
\( A \approx 113.04 \)
Circle area: `36\pi` or `113.04` square inches approximately.
Example `2`: Sophia is planning a garden in the shape of a circular bed in her backyard. The radius of the circular garden is `12` feet. What is the area of the garden meaning how much space will it occupy? (Consider `\pi = 3.14`)
Solution:
Area of circle: \( A = \pi r^2 \):
\( A = \pi \times (12)^2 \)
\(A \approx 3.14 \times 144 \)
\( A \approx 452.16 \, \text{square feet} \)
So, the area of Sophia's circular garden will be approximately `452.16` square feet.
Finding the Area of a Circle Using Diameter
Example `1`: Find the area of a circle with a diameter of `20` centimeters. (Consider `\pi = 3.14`)
Solution:
`r=d/2`
`r=20/2`
`r= 10`
The radius of the circle is `10` centimeters.
Area of circle: \( A = \pi r^2 \):
\( A = \pi \times (10)^2 \)
\( A = 100\pi\)
\( A \approx 100 \times 3.14 \)
\( A \approx 314 \, \text{square centimeters} \)
Circle area: `100\pi` or `314` square centimeters approximately
Example `2`: Alex has a circular table with a diameter of `3.5` feet. He wants to purchase a tablecloth to cover the entire tabletop. How much fabric does Alex need for the tablecloth? (Consider `\pi = 3.14`)
Solution:
The diameter is given as `3.5` feet.
Since the radius (\(r\)) is half of the diameter, \(r = \frac{3.5}{2} = 1.75\) feet.
Apply the Area Formula:
Area of circle: \(A = \pi r^2\):
\( A = \pi \times (1.75)^2 \)
\( A \approx 3.14 \times 3.0625 \)
\( A \approx 9.61 \, \text{square feet} \)
Alex should look for a tablecloth that covers an area of approximately `9.61` square feet to ensure it completely covers the circular tabletop of his table that has a diameter of 3.5 feet.
Finding the Area of a Circle Using Circumference
Example `1`: If the circumference of a circle is \(20 \pi\) units, find its area. Give your answer in terms of `π`.
Solution:
The formula relating the circumference (\(C\)) and radius (\(r\)) is \(C = 2 \pi r\).
Solving for \(r\), we get \(r = \frac{C}{2 \pi}\).
Now, use the radius in the area formula \(A = \pi r^2\).
\( r = \frac{20 \pi}{2 \pi} = 10 \)
\( A = \pi \times (10)^2 = 100 \pi \text{ square units} \)
Example `2`: Maria is planning a circular garden with a circumference of \(36 \pi\) feet. Determine the area of her garden. Give your answer in terms of `π`.
Solution:
Apply the formula \(C = 2 \pi r\) to find the radius:
\( r = \frac{36 \pi}{2 \pi} = 18 \)
Now, use the radius in the area formula \(A = \pi r^2\):
\( A = \pi \times (18)^2 = 324 \pi \text{ square feet} \)
Therefore, the area of Maria's circular garden is \(324 \pi\) square feet.
Practice Problems
Q`1`. Find the area of a circle with a radius of `9` meters. (Consider `\pi = 3.14`)
- `254.34` square meters
- `254.52` square meters
- `254.88` square meters
- `255.06` square meters
Answer: b
Q`2`. A circular swimming pool has a diameter of `14` feet. Calculate its area. (Consider `\pi = 22/7`)
- `150` square feet
- `153` square feet
- `154` square feet
- `155` square feet
Answer: c
Q`3`. If the radius of a circle is `5` inches, determine its area. (Consider `\pi = 3.14`)
- `78.5` square inches
- `79.0` square inches
- `79.5` square inches
- `80.0` square inches
Answer: a
Q`4`. A circular picnic table has a radius of `8` feet. Calculate the area of the tabletop. (Consider ` \pi= 3.14`)
- `201.06` square feet
- `200.96` square feet
- `202.50` square feet
- `203.22` square feet
Answer: b
Q`5`. Sarah is planning a circular flower bed with a diameter of `10` meters. Determine the area of the flower bed. (Consider `\pi = 3.14`)
- `78.54` square meters
- `79.56` square meters
- `80.58` square meters
- `81.60` square meters
Answer: a
Frequently Asked Questions
Q`1`. What is the formula for finding the area of a circle?
Answer: The formula for the area of a circle is \(A = \pi r^2\), where \(A\) is the area and \(r\) is the radius.
Q`2`. How is the radius related to the diameter in the context of the circle's area?
Answer: The radius (\(r\)) is half of the diameter (\(d\)), and the area formula involves squaring the radius. So, the area can also be expressed as \(A = \pi (\frac{d}{2})^2\).
Q`3`. If the circumference of a circle is given, can we find its area directly?
Answer: No, the circumference alone is not sufficient to find the area. To find the area, you need the radius, and the relationship between circumference and radius is \(C = 2\pi r\).
Q`4`. What is the significance of the mathematical constant \(\pi\) in the area formula?
Answer: The constant \(\pi\) represents the ratio of the circumference to the diameter of a circle. In the area formula, \(\pi\) ensures an accurate measurement of the space enclosed within the circle.
Q`5`. Can the area of a circle be negative?
Answer: No, the area of a circle cannot be negative. Area is a measure of space, and it is always a non-negative value. If you encounter a negative result, it indicates an error in the calculation or interpretation.