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The circumference of a circle is the distance around its edge. A circle with a diameter of one has an infinite circumference. It's impossible to measure this in real life, thus, different units are used like meters or kilometres, to measure the circumference of a circle.
- The circumference of any shape is the path or boundary which surrounds the object or shape.
- The circumference is also known as the perimeter, which further helps up determine the length of the boundary of any shape or figure.
- The circumference of a circle can be determined by the equation: Circumference = \(\pi\) x diameter.
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Key Terms: Circle, Semi-Circle, Diameter, Circumference, Radius, Perimeter, Area, Perfect Circle, Pi
Circumference of a Circle
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Circles have circumferences, which are the distances they cover. The circumference of a shape is the path or boundary which surrounds it. It means that if you measure the edge of the circle and then walk the same distance around its perimeter, you will have walked 3.14 times as far as you would have.
- The circumference of the circle or the perimeter of the circle is the measure of its boundary, while the area of the circle is the region it occupies.
- The circumference of a perfect circle is constant and equal to π times d, where π ≈ 3.14.
- Because circumferences can be so large or so small, they are usually measured in units like meters or miles rather than cm or inches.
- When a circle is taken and its circumference is measured, it can be found that it has a perimeter always greater than or equal to twice the radius multiplied by ‘pi’.
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Circumference of Circle Formula
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The distance around a circle's edge is C, equal to πr.
To calculate the circumference of a circle you must know how wide the radius of your circle is. Suppose you have a diameter or radius of your circle. In that case, you can find the circumference using the formula: C = 2πr. Thus, the circumference of a circle is found with this equation:
| 2πr |
Here,
- C = circumference of a circle
- π = the constant pi (value of pi = 3.14 or 22/7)
- r = radius of the circle
Circumference of Circle
Note: Pi (π) is often known as a special mathematical constant because it is the ratio of circumference to diameter of a circle.
Thus, it can be represented by: C = πD
- C = Circumference of the circle
- D = Diameter of the circle
Solved ExampleExample: Assuming that the radius of a circle is 4 cm, determine its circumference. Ans: As per the given equation, the Radius is = 4cm Thus, Circumference of a circle = 2πr = 2 x 3.14 x 4 = 25.12 cm |
Area of a Circle
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The area of a circle is the average radius squared, multiplied by pi. The area of a circle formula can be expressed as:
| A = r² |
Where,
- A = Area
- r = Radius
The pattern to find the area is to multiply by pi. There are two possible formulas, one using circumferences and the other using radii. In either case, you need a radius as input and an output value for the radius squared equal to the circle's area.
Radius of a Circle
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The radius of a circle is the distance from the centre point on its circumference to its edge. The radius is calculated by taking one-half of the diameter, and then calculating how far it is out from that point to each side of the radius.
For example, if your diameter were 10 units, you would calculate your radius as 5 units on each side.
If your length for both sides ranged at 4 units out from the midpoint, your final number would be 16 units in total or 2-5/16ths (in case of fractions).
Thus,
- The radius of a circle formula from diameter is: Radius = Diameter/2 or D/2 units
- The radius of a circle formula from circumference is: Radius = Circumference/2π or C/2π units
Circumference of a Circle Infograph
Perimeter of Semi-circle
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Semicircular shapes are formed when two equal parts are divided by a circle. As a result, the perimeter of the semicircle is also half.
⇒ Perimeter = πr +2r
Area of Semi-Circle
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Area of the semi-circle is the region occupied by a semi-circle. In a circle with equal radii, half the area of a semicircle is equal to the area of a circle.
⇒ Area = πr2/2
Circumference to Diameter
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Circular shapes have diameters that are twice their radiuses. Circle circumference to diameter is equal to Pi(π). As a result, this proportion defines the constant π.
⇒ C= 2πr
C = πd (Since, d = 2r)
On dividing both sides by the diameter of the circle, we will get a value close to the value of π.
Thus, C/d = π.
How to find Circumference?
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There are two methods to find the Circumference of a circle. They are as follows:
Method 1
The length of a circle cannot be measured physically by using a scale or ruler. Circumferences of circles can instead be measured with threads. Mark the points on the thread as you trace the path of the circle. Using a ruler, you can measure this length.
Method 2
Find the circumference by multiplying the diameter by the formula C = πd. In this equation, the circumference is "C" and the diameter is "d". By multiplying a circle's diameter by pi, you can determine its circumference.
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Things to Remember
- The circumference of a circle is the number of times a circle is round or the length of a circle.
- (Pi) is a mathematical constant that represents the ratio between the circumference and diameter of a circle. It is approximated to π = 22/7 or 3.14.
- Circular radii become diameters when they reach the boundary of the circle and are extended further. Therefore, Diameter = 2 × Radius.
- The radius or diameter of a circle can be used to find its circumference.
- Circumference formula = π × Diameter; Circumference = 2πr.
Sample Questions
Ques: If the difference between the circumference and the radius of a circle is 37 cm, then using π = 22/7, calculate the circumference (in cm) of the circle. (Delhi 2013) (2 Marks)
Ans: Thus,
Ques: Find the area of a quadrant of a circle, where the circumference of circle is 44cm. (All India 2011) (2 Marks)
Ans: It can be said that:
Ques: Calculate the circumference of a circle if its radius is 5 cm. (2 Marks)
Ans: The radius of the circle = 5 cm
As we know, the value of π = 3.14
Circumference of Circle= 2πr
= 2 x 3.14 x 5
=31.4 cm
Ques: Calculate the area of a circle with diameter 4 cm. (2 Marks)
Ans: The diameter of the circle d= 4cm
The radius r= d/2=4/2= 2cm
Area= A=πr2
=3.14x2x2
= 12.56 cm2
Ques: What is the radius of a circle with a circumference 22 cm. (2 Marks)
Ans: Given that circumference of circle = 22 cm
Circumference= 2πr = 22
r=22/2π
r=22x7/22x2
So, the radius of the circle=3.5 cm
Ques: Calculate the perimeter and area of a semicircle with radius 3 cm. (3 Marks)
Ans: We are given the radius r=3cm
The perimeter and area of the semi-circle can be calculated as:
Perimeter = πr +2r
=3.14 x 3 + 2 x 3
=9.42+6
=15.42 cm
Area = πr2/2
=3.14 x 3 x 3/2
=14.13 cm2
Ques: Bella works at a lab with a huge circular particle accelerator. It has a radius of 6 metres. What is the accelerator’s circumference? (1 mark)
Ans: Given, the radius of the accelerator =6m
Then the accelerator’s circumference =2πr,
=2π×6=12π
Hence, the accelerator’s circumference is 12π.
Ques: A 15-inch (diameter) pizza is served to Tim and his friends. Calculate its circumference. (2 marks)
Ans: Diameter of pizza (d) = 15 inches
The formula for the circumference of the circle in terms of diameter C=πd
C = π × 15 = 15π inches
Hence, the circumference of pizza is 15π inches.
Ques: Wheelchair wheels have a diameter of 14 millimetres. How far does the wheelchair move if the wheel rotates once? (2 marks)
Ans: If the wheel rotates once, the wheelchair will move by a distance equal to the circumference of the wheel.
Given, the diameter of the wheel (d)=14m
We know that the circumference of a wheel C=πd
C = 22/7 × 14
= 44m
Thus, the wheelchair moves 44m in one revolution of the wheel.
Ques: The diameter of a semicircular shape is 14 cm. What will be the perimeter of this shape? (2 marks)
Ans: Given,
Diameter of semicircle = d = 14 cm
Radius = r = d/2 = 14/2 = 7 cm
Perimeter of semicircle = (Perimeter of circle/2) + d
= (2πr/2) + d
= πr + d
= (22/7) × 7 + 14
= 22 + 14
= 36 cm
Ques: Find the area of a circular region whose radius is 21 m. (2 marks)
Ans: Given,
Radius of the circular region = r = 21 m
Area of a circle = πr2
= (22/7) × 21 × 21
= 22 × 3 × 21
= 1386 sq. m
Therefore, the area of the circular region is 1386 sq. m.
Ques: Given the radius of a circle is “r” cm and if it is doubled then what will be the circumference of the new circle? (2 marks)
Ans: Given, the radius of the circle = rcm
Then the circumference of the circle = 2πr
If the radius of the circle is doubled then, the new radius R = 2rcm
Therefore, the circumference of the new circle = 2πR
= 2π×2r
= 4πr
Hence, the circumference of the new circle is 4πr
Ques: Find the circumferences of the circles from their diameter of 6.3 m. (2 marks)
Ans: The diameter of the circle is 6.3 m.
Circumference = πd
Circumference = π × 6.3 m
=22/7×6.3 = 22×0.9
(Taking π=22/7)
= 19.8 m
∴ Circumference of the circle is 19.8 m.
Ques: From a circular sheet with a radius of 5cm, a circle of radius 3 cm is removed. Find the circumference of the remaining sheet. (2 marks)
Ans: Given, the radius of the outer sheet r1=5cm
The radius of the inner sheet r2=3cm
Circumference of the remaining part =2π(r1–r2)
=2π (5–3)
=2π×2
=4π
Hence, the circumference of the remaining part is 4π.
Ques: Ratio of radii of two circles 4:5. What is the ratio in their areas? (3 marks)
Ans: Let the radius of the first the circle r1=4x
The radius of the second circle r2=5x
Thus,
Ques: Find the circumferences of the circles from their diameter of 3.5 cm. (3 marks)
Ans: The diameter of the circle is 3.5 cm.
Circumference of a circle = 2πr
Or circumference = πd
Circumference = π × 3.5 cm
= 22/7×3.5 = 22×0.5
(Taking π=22/7)
= 11 cm
∴ Circumference of the circle is 11 cm.
Ques: Determine the perimeter and area of circle that has radius of 5 cm. [Note: π = 3.14] (5 marks)
Ans: As per the question,
Radius, r = 5 cm
And, π = 3.14
As we are aware, the circumference (or) perimeter of a circle = 2πr units
Area of a circle = πr2 square units.
Upon substitution of the values in the perimeter and area of circle formula, we can obtain:
The area of circle = πr2 = 3.14(5)2
Thus, A = 3.14(25)
A = 78.5 cm2
Therefore, circumference of a circle = 2πr = 2(3.14)(5)
Circumference = 3.14(10) = 31.4 cm.
Thus, the perimeter and area of circle are 31.4 cm and 78.5 cm2 respectively.
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