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Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10 PDF Download

Perimeter and Area of a Circle

The perimeter of all the plain figures is the outer boundary of the figure. Likewise, the outer boundary of the circle is the perimeter of the circle. The perimeter of a circle is also called the Circumference of the Circle.

Radius = r and Diametre = d = 2r

Circumference = 2πr = πd (π = 22/7)

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10The area is the region enclosed by the circumference.

Area of the circle = πr2

Example: If a pizza is cut in such a way that it divides into 8 equal parts as shown in the figure, then what is the area of each piece of the pizza? The radius of the circle shaped pizza is 7 cm.
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Solution: The pizza is divided into 8 equal parts, so the area of each piece is equal.
Area of 1 piece = 1/8 of area of circle

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Perimeter and Area of the Semi-circle

The perimeter of the semi-circle is half of the circumference of the given circle plus the length of diameter as the perimeter is the outer boundary of the figure.
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10Area of the semi-circle is just half of the area of the circle.
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Question for Important Definitions & Formulas: Areas Related to Circles
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Area of a Ring

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Area of the ring i.e. the coloured part in the above figure is calculated by subtracting the area of the inner circle from the area of the bigger circle.

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Where, R = radius of outer circle
r = radius of inner circle

Areas of Sectors of a Circle

The area formed by an arc and the two radii joining the endpoints of the arc is called Sector.

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

  • Minor Sector: The area including ∠AOB with point C is called Minor Sector. So OACB is the minor sector. ∠AOB is the angle of the minor sector.
    Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10
  • Major Sector: The area including ∠AOB with point D is called the Major Sector. So OADB is the major sector. The angle of the major sector is 360° – ∠AOB.
    Area of Major Sector = πr2 -  Area of the Minor Sector

Remark: Area of Minor Sector + Area of Major Sector = Area of the Circle

Length of an Arc of a Sector of Angle θ

An arc is the piece of the circumference of the circle so an arc can be calculated as the θ part of the circumference.

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10


Areas of Segments of the Circle

The area made by an arc and a chord is called the Segment of the Circle.

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Minor Segment

The area made by chord AB and arc X is the minor segment. The area of the minor segment can be calculated by

Area of Minor Segment = Area of Minor Sector – Area of ∆ABO

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Major Segment

The other part of the circle except for the area of the minor segment is called a Major Segment.

Area of Major Segment = πr2 -  Area of Minor Segment
Remark: Area of major segment + Area of minor segment = Area of circle

Question for Important Definitions & Formulas: Areas Related to Circles
Try yourself:What is the formula to calculate the area of a ring?
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Areas of Combinations of Plane Figures

As we know how to calculate the area of different shapes, so we can find the area of the figures which are made with the combination of different figures.

Example: Find the area of the colored part if the given triangle is equilateral and its area is 17320.5 cm2. Three circles are made by taking the vertex of the triangles as the centre of the circle and the radius of the circle is the half of the length of the side of the triangle. (π = 3.14 and √3 = 1.73205)

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Solution: Given
ABC is an equilateral triangle, so ∠A, ∠B, ∠C = 60°
Hence the three sectors are equal, of angle 60°.
Required
To find the area of the shaded region.
Area of shaded region =Area of ∆ABC – Area of 3 sectors
Area of ∆ABC = 17320.5 cm2
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10
Side = 200 cm
As the radius of the circle is half of the length of the triangle, so
Radius = 100 cm
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10
Area of 3 Sectors = 3 × 15700/3 cm2 cm2
Area of shaded region = Area of ∆ABC – Area of 3 sectors
= 17320.5 - 15700 cm2
= 1620.5 cm2

Example: Find the area of the shaded part, if the side of the square is 8 cm and the 44 cm.

Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10

Solution: Required region = Area of circle – Area of square

= πr2 – (side)2
Circumference of circle = 2πr = 44
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10
Radius of the circle = 7 cm
Area of circle = πr2
Important Definitions & Formulas: Areas Related to Circles | Mathematics (Maths) Class 10
Area of square = (side) 2 = (8)2 = 64 cm2
Area of shaded region = Area of circle – Area of square
= 154 cm2 - 64 cm2
= 90 cm2

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FAQs on Important Definitions & Formulas: Areas Related to Circles - Mathematics (Maths) Class 10

1. What is the formula for calculating the perimeter of a circle?
Ans. The formula for calculating the perimeter (circumference) of a circle is given by \( C = 2\pi r \), where \( r \) is the radius of the circle and \( \pi \) is approximately 3.14.
2. How do you find the area of a semi-circle?
Ans. The area of a semi-circle can be calculated using the formula \( A = \frac{1}{2} \pi r^2 \), where \( r \) is the radius of the semi-circle. This is half the area of a full circle.
3. What is the formula for the area of a ring?
Ans. The area of a ring can be calculated using the formula \( A = \pi(R^2 - r^2) \), where \( R \) is the outer radius and \( r \) is the inner radius of the ring.
4. How can I calculate the area of a sector of a circle?
Ans. The area of a sector of a circle can be calculated using the formula \( A = \frac{\theta}{360} \times \pi r^2 \), where \( \theta \) is the angle of the sector in degrees and \( r \) is the radius of the circle.
5. What is the formula for finding the length of an arc of a sector with angle θ?
Ans. The length of an arc of a sector can be calculated using the formula \( L = \frac{\theta}{360} \times 2\pi r \), where \( \theta \) is the angle of the sector in degrees and \( r \) is the radius of the circle.
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