Measurement regarding circle

Today our topic of discussion is Measurement regarding circle.

Measurement regarding circle

 

Measurement regarding circle

 

Measurement regarding circle

  1. Circumference of a circle

 

Measurement regarding circle

 

The length of a circle is called its circumference. Let r be the radius of a circle, its circumference c = 2πr, where = 3.14159265… which is an irrational number. Value of = 3.1416 is used as the approximate value. Therefore, if the radius of a circle is known, we can find the approximate value of the circumference of the circle by using the value of π.

Example 18. The diameter of a circle is 26 cm. Find its circumference

Solution: Let, the radius of the circle is r

∴ diameter of the circle = 2r and circumference = 2πr

As per question, 2r = 26 or, r = 26/2 or, r = 13c

∴ circumference of the circle = 2πr cm (approx.) = 2 x 3.1416 x 13 cm = 81.68

  1. Length of arc of a circle

 

Measurement regarding circle

 

Let O be the centre of a circle whose radius is r and arc AB = s which produces theta deg angle at the centre. .. circumference of the circle = 2πr

Total angle produced at the centre of the circle = 360 deg and arc s produces angle theta deg at the centre. We know, any interior angle at the centre of a circle produced by any arc is proportional to the arc.

θ/(360 deg) = s/( 2πr) * or, s = ( πrθ)/(180 °)

  1. Area of circular region and circular segment

 

Measurement regarding circle

 

The area, surrounded by any circle, is called a circular region and the circle is called the boundary of the circular region.

Circular segment: The area formed by an arc and the radius that joins the end points of that are to the centre of the circle is called circular segment.

If A and B are two points on a circle with centre O, interior to ∠AOB, radius OA and OB, and the arc AB, form a circular segment. In previous class, we have learnt that if the radius of a circle is r, the area = πr².

We know, any angle produced by an arc at the centre of a circle is proportional to the arc.

So, at this stage we can accept that the area of two circular segments of the same circle are proportional to the two arcs on which they stand.

 

Measurement regarding circle

 

Let us draw a radius r with centre O. The circular segment AOB stands on the arc APB whose measurement is . Draw a perpendicular OC on OA

Area of circular segment AOB/Area of circular segment AOC

= Measuremento f ∠AOB/Measurement of ∠AOC

or, Area of circular segment AOB/ Area of circular segment AOC [∠AOC = 90 °]

=θ/(90 °)

or, Area of circular segment AOB x θ/(90 °) area of circular segment AOC 90°

= θ/(90 °) * 1/4 *area of the circle

= θ/(90 °) * 1/4 * πr²

= θ/(360 °) * πr²

So area of circular segment = θ/(360 °) πr²

Example 19. The radius of a circle is 8 cm and a circular segment subtends an angle 56 deg at the centre. Find the length of the arc and area of the circular segment.

Solution: Let, radius of the circle r = 8cm length of arc is s and the angle subtended by the circular segment is θ= 56 °

We know,

s = (πrθ)/(180 °) = (3.1416 * 8 * 56 °)/(180 °) * cm =7.82 cm(approx.)a

Area of circular segment square cm (approx.) = θ/(360 °) * πr² = 56/360 * 3.1416 * 8 ² square cm = 31.28

Example 20. If the difference between the radius and circumference of a circle is 90 cm, find the radius of the circle.

Solution: Let the radius of the circle be r.

∴ Diameter of the circler is 2r and circumference

As per question, 2πr – 2r = 90

or, 2r(π – 1) = 90

or, r = 90/(2(π- 1)) = 45/(3.1416 – 1) = 21.01cm (approx.)

The required radius of the circle is 21.01 cm (approx.)

Example 21. The diameter of a circular field is 124 metre There is a path with 6 metre. width around the field. Find the area of the path.

Solution:

 

Measurement regarding circle

 

Let the radius of the circular field ber and radius of the field with the path be R

r = 124/2 Area of the circular field = r² and area of the circular field with the path = R²

∴ Area of the path = Area of field with path – Area of

metre = 62 metre and R = (62 + 6) metre = 68 metre

the field = (πR² – πr²) = π(R² – r²)

= 3.1416(68² – 62²) = 3.1416(4624 – 3844) = 3.1416 * 780 = 2450.44 square metre (approx.)

The required area of the path is 2450.44 square metre (approx.)

Work: Circumference of a circle is 440 metre. Determine the length of the sides of the inscribed square in it.

Example 22. The radius of a circle is 12 cm and the length of an arc is 14 cm. Determine the angle subtended by the circular segment at its centre.

Solution: Let, radius of the circle is r = 12 cm, the length of the arc is s = 14 cm and the angle subtended at the centre is θ.

We know, s = (πrθ)/180

or, πrθ = 180s

or, θ= (180s)/(πr) = (180 * 14)/(3.1416 * 12) = 66.84 °(approx.)

The required angle is 66.84° (approx.)

Example 23. Diameter of a wheel is 4.5 metre. To traverse a distance of 360 metre, how many times the wheel will revolve?

Solution: Given that, the diameter of the wheel is 4.5 metre.

∴The radius of the wheel, r = 4.5/2 = 2.25 metre and circumference = 2πr

Let, for traversing 360 metre, the wheel will revolve ʼn times,

As per question, n * 2πr= 360 n = 360/(2πr) = 360/(2 * 3.1416 * 2.25) = 25.46 (approx.)

therefore The wheel will revolve 25 times. (approx.)

Example 24. Two wheels revolve 32 and 48 times respectively to cover a distance of 211 metre 20cm. Determine the difference of their radii.

Solution: 211 metre 20 cm = 21120 cm

Let, the radii of two wheels are R and r respectively; where R > r .. Circumferences of two wheels are 2R and 2r respectively and the difference of radii is (R – r)

As per the question, 32 * 2πr = 21120

or, R = 21120/(32 * 2π) = 21120/(32 * 2 * 3.1416) = 105.04 and 48 * 2πr = 21120 cm (approx.)

,r= 21120 48*2π = 21120 48*2*3.1416 =70.03 cm (approx.)

or, R – r = (105.04 – 70.03) = 35.01cm = 0.35m (approx.)

The difference of radii of the two wheels is 0.35 metre (approx.)

Example 25. The radius of a circle is 14 cm. The area of a square is equal to the area of the circle. Determine the length of the square.

Solution: Let the radius of the circle, r = 14 cm and the length of the square is a.

∴ Area of the square region = a² and the area of the circle = πr²

According to the question, a² = πr²

or, a = √(π) * r = √(3.1416) * 14 = 24.81 (approx.)

The required length is 24.81 cm (approx.)

Example 26. In the figure, ABCD is a square whose length of each side is 22 metre and region AED is a half circle. Determine the area of the whole region. 

Solution:

 

Measurement regarding circle

 

Let the length of each side of the square ABCD be a.

∴ Area of square region = a² Again, AED is a half circle..

∴ Radius of the half circle r = 22/2 met re = 11 metre. Therefore, area of the half circle = 1/2 * πr²

∴ Area of the whole region = Area of the square ABCD + area of the half circle AED

= (a² + 1/2 * πr²)

= (22²+ 1/2 * 3.1416 * 11²) = 674.07 square metre (approx.)

The required area is 674.07 square metre (approx.)

Example 27. In the figure, ABCD is a rectangle whose length is 12 metre the breadth is 10 metre. and DAE is a circular segment. Determine the length of the arc DE and the area of the whole region.

Solution:

 

Measurement regarding circle

 

The radius of the circular segment, r = AD = 12 metre and the angle subtended at centre theta = 30 °

.. length of the arc DE = (πrθ)/180

= (3.1416 * 12 * 30)/180 = 6.28 metre (approx.)

Area of the circular segment ADE θ/ 360 x πr²

= 30/360 * 3.1416 * 12² = 37.7 . square metre (approx.)

The length of the rectangle ABCD is 12 metre and breadth is 10 metre.

.. Area of the rectangle = length x breadth = 12 * 10 = 120 square metre

.. Area of the whole region (37.7+120) square metre = 157.7 square metre (approx.)

The required area is 157.7 square metre (approx.).

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