Circumference of a Circle
- Introduction
- What is Circumference of a Circle?
- Parts of a Circle
- Formula for Circumference of a Circle
- Circumference to Diameter Ratio
- Solved Examples on the Circumference of a Circle
- Real-Life Applications of the Circumference of a Circle
- Practice Problems
- Frequently Asked Questions
Introduction
Every day, we come across numerous circular items, like coins, buttons, wheels, and wall clocks. Any circular object's boundary is extremely important in mathematics. The boundary that encircles any shape in mathematics is defined by the circumference. In other terms, the circumference is also known as the perimeter, which is used to identify the length of a shape. Like any other shape, we can measure the circumference of a circle too. We will examine the "Circumference of a circle" or "Perimeter of a circle" in this article, along with its definition, formula, and techniques for calculating the circumference of a circle.
What is Circumference of a Circle?
The circumference of a circle, also known as the perimeter of a circle, is used to measure the boundary of a circle. It is the total distance around the outer edge of the circle. We have a formula that helps us find the circumference of a circle. It is measured in units such as feet, inches, centimeters, meters, miles, or kilometers since it indicates length.
When applying the formula to measure the circumference of the circle, we need to consider the radius of the circle. Therefore, in order to calculate the perimeter of a circle, we need to understand the main parts of a circle like radius, diameter and center of a 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 centre. It's essentially twice the length of the radius. It is denoted by `d`, so `d` `= 2``r`.
Formula for the Circumference of a Circle
The radius `(r)` of a circle and the value of `\text{pi}` `(π)` are used to calculate the circumference of a circle. The symbol for the constant is `\text{pi}` (written as `π`). The value of `π`, a unique mathematical constant, is approximately `3.14159`, or `22/7`. Like circumference, several formulas employ the value of `π`.
The circumference of a circle formula is `C``= 2π``r`
If we do not know the radius value while using this circumference formula, we can calculate it using the diameter. That is because the diameter of a circle equals twice the radius. Likewise, if the diameter is known, it may be divided by `2` to determine the value of the radius. Hence, we can write the formula for circumference in terms of the diameter of a circle as follows:
Circumference of a circle formula is `C``= 2π``(\color{#fb8500}{d}/2)` or `C``= π``d`
Circumference to Diameter Ratio
Although a circle's circumference is equal to the length of its boundary, unlike other polygons, it cannot be computed using a ruler or scale. Because a circle is a curved figure, this is the case. Therefore, we utilize a formula that takes the radius or diameter of the circle and the significance of `\text{pi}` to get the circumference of a circle.
When `C = π``d` , `π` is the ratio of circumference to diameter.
Every circle has a fixed ratio between its circumference and diameter. Let `d` be a circle's diameter and `C` be its circumference. Then,
`C/D = π`
This ratio is the same for all circles, no matter how big or tiny. It is about equivalent to `3.14` or `22/7`.
Solved Examples on Circumference of a Circle
Q`1`. Calculate the circumference of a circle whose radius is `56` cm. (Consider `π = 22/7`)
Solution:
Radius `= 56` cm
Circumference of a Circle `= 2\pi r`
`= 2 \times \pi \times r`
`= 2 \times \frac{22}{7} \times 56`
`= 352`
The circumference of the circle is `352` cm.
Q`2`. Determine the circumference of the circle with a diameter of `35` cm. (Consider `π = 22/7`)
Solution:
Diameter `= 35` cm
Circumference of a Circle `= πd`
`= 22/7 × 35`
`= 110`
The circumference of the circle is `110` cm.
Q`3`. Find the radius and diameter of the wheel whose circumference is `132` cm. (Consider `π = 22/7`)
Solution:
Circumference `= 2πr`
Let’s plug in the value of circumference in the formula.
`132 = 2 × 22/7 × r`
`132 = 44/7 × r`
`r = (132 × 7)/44`
`r = 21`
The radius of the wheel is `21` cm.
The diameter is twice the radius. So, the diameter of the wheel is `42` cm.
Q`4`. The diameter and circumference of a circle differ by `10` feet. Find the radius. (Take `π = 3.14`)
Solution:
In the given question, circumference `-` diameter `= 10`
Circumference `= 2πr` and Diameter `= 2r`
Placing the values of the circumference and diameter in the above equation, we get -
`2πr - 2r` `= 10`
`2r(π - 1 ) = 10`
`r(3.14 - 1 ) = 5`
`2.14 r = 5`
`r = 5/2.14`
`r = 2.34`
The radius of the circle is `2.34` feet.
Q`5`. The ratio of the circumference of two circles is `8:9`. Calculate the ratio of their radii.
Solution: The Circumference of a circle = `2πr`
Assume that the first circle's radius is `r_1` , and the second circle's radius is `r_2` .
Therefore,
Circumference of first circle `= 2πr_1`
Circumference of second circle `= 2πr_2`
According to the question
`\frac{\text{(Circumference of first circle)}}{\text{(Circumference of second circle)}} = \frac{8}{9}`
`(2πr_1)/(2πr_2)` `= 8/9`
`r_1/r_2` `= 8/9`
The ratio of the radii of the two circles is `8:9`.
Real-Life Applications of the Circumference of a Circle
We come across many real-life situations where we need to find the circumference of a circular shape.
- Circumference calculations are commonly used in manufacturing industries, particularly in the production of cylindrical objects such as pipes, tubes, and tires. For example, in the automotive industry, tire manufacturers use circumference calculations to design tires with specific dimensions to fit various vehicle models.
- Athletes and coaches use circumference measurements to select the appropriate size of sports equipment, such as basketballs and soccer balls, based on age, skill level, and league regulations.
- In civil engineering, the circumference of circular roads and roundabouts is calculated to optimize traffic flow and design efficient transportation networks.
- Geographers and cartographers use the circumference formula to calculate the circumference of the Earth and to determine distances between locations on maps and globes. Understanding the circumference of the Earth is crucial for navigation, surveying, and geographic analysis, as well as for developing accurate representations of the planet's surface.
- Healthcare professionals use circumference measurements, such as waist circumference and hip circumference, as indicators of body fat distribution and overall health risk.
Practice Problems
Q`1`. Determine the perimeter of the circle in terms of `π`, whose radius is `12` cm.
- `44π`
- `24π`
- `22π`
- `42π`
Answer: b
Q`2`. Determine the perimeter of the circle in terms of `π`, whose diameter is `63` cm.
- `63π`
- `35π`
- `45π`
- `56π`
Answer: a
Q`3`. Calculate the radius of a circle in terms of `π`, whose circumference is `68` cm.
- `34/π`
- `54/π`
- `56/π`
- `64/π`
Answer: a
Q`4`. Determine the diameter of a circle whose circumference is 628 feet. (Take `π = 3.14`)
- `252` feet
- `200` feet
- `325` feet
- `300` feet
Answer: b
Frequently Asked Questions
Q`1`. What distinguishes the circumference of a circle from its diameter?
Answer: The diameter of a circle is the longest distance across a circle, passing through the center. The circle's circumference is equal to the measurement of its outside perimeter. The diameter and circumference are both length measurements that are stated in terms of linear distance. The circumference of a circle is determined by multiplying the diameter by the mathematical constant `\text{pi}`.
Q`2`. What is the value of `\text{pi}` `(π)`?
Answer: `\text{pi}` is a constant that is used to calculate the circumference and area of circles and other circular shapes. `\text{pi}` has the symbol `π` and a numerical value of `3.14159`. As fractional value `π = 22/7`.
Q`3`. How are spheres and circles different from one another?
Answer: A sphere is a solid, three-dimensional object, whereas a circle is a two-dimensional figure. Consequently, a sphere has a volume but a circle does not.
Q`4`. What is the Earth's circumference?
Answer: The earth's circumference is approximately `24,901` miles.
Q`5`. Can the circumference of a circle be negative?
Answer: Circumference of a circle represents the distance around the circle. So, the circumference of a circle cannot be negative.
Q`6`. What is the ratio of the circumference to a circle to its diameter?
Answer: Every circle has a fixed ratio between its circumference and diameter. This ration is equal to `\text{pi}` `(π)`. `π` is equivalent to `3.14` or `22/7`.