The Circle
A circle is a closed figure bounded by a curved line.
A circle is a path traced out by a point which moves in a plane so that it is always the same distance from a fixed point in that plane.
The fixed point is called the centre of the circle and the distance of the point from the centre of the circle is called the radius.
If the point could move anywhere in space and were not obliged to lie in a plane, it would trace out a sphere.
Circumference:
A curved line that bound a circle is called the circumference. An arc of a circle is part of the circumference. A major arc is an arc greater than half the circumference. A minor arc is an arc less than half the circumference.
Radius:
Radius is the distance from the centre to the circumference.
The fixed point O inside the circle is its centre.
Every point on the curved line is at an equal distance from the centre.
The fixed distance between the centre and any point on a circle is called its radius.
A line segment passing through the centre of the circle, whose endpoints lie on the circle is called the diameter of the circle.
The diameter of a circle is twice the radius.
Diameter = 2 x radius
Chord:
The chord is a line segment that connects two endpoints of an arc.
A line segment whose endpoints lie on the circle is called a chord.
Any part of a circle is called an arc of the circle.
Half a circle is a semicircle.
Diameter:
A diameter is a chord that passes through the centre. The diameter is the largest chord of the circle.
Problem involving Circumference and Area of a Circle:
1. The figure shows a circle of radius 7 cm, touching two sides of a rectangle. The length of the rectangle is 9 cm longer than its width. Calculate
i) the circumference of the circle
ii) the area of the circle
iii) the area of the shaded region.
I) Solution:
We know that,
Circumference of the circle = 2 π r
= 2 x 3.14 x 7
= 44 cm
ii) Solution:
We know that,
Area of the circle = Ï€ r²
= 154 cm2 Ans.
iii) Width of the rectangle = diameter of circle
= 7 x 2
= 14 cm
Length of the rectangle = 14 + 9
= 23 cm
Area of the shaded region = area of the rectangle - area of the circle
= 23 x 14 - 49 π
= 168 cm² Ans.
Ex.2. Find the circumference of a circle of radius 5 cm.
Solution:
We know that,
C = 2 π r
= 2 π x 5 cm
= 10 π cm
= 31.4 cm
Ex.3. Taking π to be 22/7, find the radius of a circle whose circumference is 44 cm.
Solution:
We know that,
C = 2 π r
r = C/ 2Ï€
= 44/ 2 x 22/7
= 7 cm
the radius of the circle is 7 cm.
Area of a circle:
We have seen that the area of similar figures are proportional to the squares of corresponding linear dimensions. Since all circles are similar to each other, the area of a circle must be proportional to the square of its radius. The constant of proportionality is found to be π, and the formula for the area of a circle is
A = Ï€ r²
A/Ï€ = r²
r = √A/Ï€
Q. Find the area of a circle of radius 7 cm. Take π to be 22/7.
Solution:
We know that,
A = Ï€ r²
A = 22/7 x 9 ( 7 )² cm²
= 22 x 7 cm²
= 154 cm² Ans.
MathBook-2
L.Harwood Clarke
Ex- 23 A
6. Find the radius of a circle whose area is 154 cm².
Solution:
We know that,
A = Ï€ r²
Where,
A = area= 154 cm²
r = radius = ?
Ï€ = 22/7
r² = 154/22 x 7
= 49 cm²
= √49 cm²
= 7 cm Ans.
8. Find the area between two concentric circles of radii 9 cm and 2 cm.
Solution:
Area of the circle having radius, r = 9 cm
We know that,
A = Ï€ r²
= 22/7 (9 ) ²
= 1782/7 cm²
Area of the circle having radius, r = 2 cm
A = Ï€ r²
= 22/7 x 2² cm²
= 88/7 cm²
The area between the two concentric circles is
= ( 1782/7 - 88/7 )
= (1782 - 88 ) / 7
= 1694 / 7
= 242 cm² Ans.
9. Find the area of a semicircle of radius 7 cm.
Solution:
The area of a circle having radius r = 7cm is
We know that,
A = Ï€ r²
= 22/7 x 7²
= 154 cm²
The area of a semicircle having radius r = 7 cm is
154 cm² ➗ 2
= 77 cm² Ans.