Circles and Area Related to Circles for Class 10

A circle is made by joining all the points in a plane which are at a fixed distance from a given point. This chapter includes a brief about different terms related to circles, theorems on circles, theorems related to tangents and areas related to circles.

Circle

A circle is a two-dimensional figure made by joining all the points in a plane which are at a fixed distance called the radius and from a given point called the centre.

Radius

The radius of a circle is the distance from the centre of the circle to any point on its edge.

Centre

The fixed point is called the centre.

Circumference

Circumference is the perimeter of the circle.

Chord

A chord is a straight line that connects two points on the circumference (edge) of a circle. 

Diameter

The diameter is the longest chord of the circle which passes through the centre. It is twice the radius.

Arc

An arc is a part of the circle's circumference

Minor Arc

A minor arc is a section of a curve on a circle that's smaller than a semicircle. It's like a piece of the circle's circumference that's less than half of the full circle.

Major Arc

A major arc is a section of a curve on a circle that's larger than a semicircle. It's like a piece of the circle's circumference that's more than half of the full circle.

Sector

An area made by an arc and two radii of the circle, by joining the centre to the endpoints of the arc.

Minor Sector

A sector is called a minor sector when the angle at the centre of the circle is less than 180°.

Major Sector

A sector is called a major sector when the angle at the centre of the circle is more than 180°.

Segment

A segment refers to the region enclosed by a chord and the arc.

Minor Segment

A minor segment is the smaller part of a circle that's formed by a minor arc and the chord connecting its endpoints.

Major Segment

A major segment is a larger portion of a circle formed by a major arc and the chord connecting its endpoints. 

Theorems Related to Circles

Theorem 1

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Theorem 2

Angles in the same segment of a circle are equal.

Theorem 3

An angle in a semicircle is a right angle.

Cyclic Quadrilateral

A cyclic quadrilateral is a four-sided polygon inscribed in a circle. It has all its four vertices lying on a circle.

ABCD is a cyclic quadrilateral.

→ Concyclic points are those that lie on the same circle's circumference.

→ The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. Its converse is also true.

Theorem

In a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

Thus, ∠BAD = ∠BCE

The Relation between a Circle and a Line in a Plane

In (i), the line PQ and the circle have no common point. In this case, PQ is called a non-intersecting line with respect to the circle.

In (ii), there are two common points A and B that the line PQ and the circle have. In this case, PQ is called a secant of the circle.

In (iii), there is only one point A which is common to the line PQ and the circle. In this case, PQ is called a tangent to the circle.

Secant

A secant is a line that intersects the circle at two distinct points.

Tangent to a Circle

A tangent is a line or a straight segment that touches the circle at exactly one point.

Note:

→ The common point of the tangent and the circle is called the point of contact.

→ Tangent could not be drawn from any point inside the circle.

→ At any point in a circle, there can be one and only one tangent.

As shown in the figure, P is a point from where only one tangent can be drawn to this circle.

→ Two tangents can be drawn to a circle from a point outside the circle.

KS and KR are the two tangents to the circle from point K.

Theorem on Tangent

Theorem 1

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

OP is the radius of the circle and Q is any point on the line XY which is the tangent to the circle at point P. As OP is the shortest line of all the distances from point O to the points on XY. So OP is perpendicular to XY.

Example: If the length of the tangent PQ is 8 cm and the distance of point Q from the centre is 17 cm, then what is the length of the radius OP?

a) 14 cm
b) 15 cm
c) 16 cm
d) 10 cm

Answer: b) 15 cm

Explanation: Consider the given figure,

We are given,

OQ = 17 cm

PQ = 8 cm

To find: OP

We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Thus, △OPQ is a right-angled triangle.

Using Pythagoras theorem,

OQ² = OP² + PQ2
(17)² = OP² + (8)²
289 = OP² + 64
OP² = 289 − 64
       = 225
OP = √225
OP = 15 cm

Theorem 2

The lengths of tangents drawn from an external point to a circle are equal.

Here, two tangents are drawn from the external point P. As the tangent is perpendicular to the radius, it forms the right-angle triangle.

So △AOP and △BOP are congruent right-angle triangles.

Hence AP = BP.

Theorem 3

The point of contact between two circles that touch each other is located along the straight line that connects their centres.

Case I: When circles touch externally

Two circles with centres A and B touch each other externally at point P which lies on the AB.

Case II: When circles touch internally

O₁ is the centre of circle S₁ and O₂ is the centre of circle S₂. These two circles touch each other internally at point P which lies on the line O₁O₂ produced.

Note

If r₁ and r₂ are the radii of two circles and d is the distance between their centres, then
(i) When circles touch externally,
d = r₁ + r₂

(ii) When circles touch internally,

d = r₁ − r₂, if r₁ is greater
d = r₂ − r₁, if r₂ is greater

Theorem 4

If two chords of a circle intersect internally or externally, then the product of the lengths of their segments is equal.

Case I: When chords intersect internally

Chords AB and CD of a circle intersect at point P inside the circle.

Thus, PA × PB = PC × PD

Case II: When chords intersect externally

Chords ON and PQ of a circle intersect each other at point M outside the circle.

Thus, MO × MN = MQ × MP

Theorem 5

An angle in the alternate segment is equal to the angle formed by a tangent and a chord through the point of contact.

Tangent AB touches the circle at point P. From P, chords PR and PQ are drawn.

∠RPB = ∠PQR

∠RPA = ∠PSR

Theorem 6

When a tangent and a chord intersect externally, the square of the tangent's length from the point of contact to the intersection is equal to the product of the lengths of the chord's segments.

Tangent PT and chord AB of the circle intersect each other at a point P outside the circle.

Thus, PT² = PA × PB

Area of a Circle

Area = πr²

Circumference = 2πr or πd

Radius(r) = 2 × Diameter(d)

Where

r = radius of a circle

d = diameter of a circle

Example: What is the circumference of a circle of radius 21 cm?

a) 132 cm
b) 133 cm
c) 134 cm
d) 136 cm

Answer: a) 132 cm

Explanation: We know that the circumference of a circle is given by 2πr.

We are given r = 21 cm

Circumference = 2πr
= 2 × ²²⁄₇ × 21
= 2 × 22 × 3
= 132 cm

Area of a Sector

Length of an arc of a sector of angle θ = ^θ⁄₃₆₀ × 2πr

Area of the sector of angle θ = ^θ⁄₃₆₀ × πr²

Example: What is the area of the sector of a circle with a radius of 8 cm and of angle 75°? (use π = 3.14)

a) 41.87 cm²
b) 43.87 cm²
c) 44.87 cm²
d) 42.87 cm²

Answer: a) 41.87 cm²

Explanation: We know that 

Area of the sector of angle θ =  ^θ⁄₃₆₀ × πr²

Here,

r = 8 cm

θ = 75°

Thus, 

Area of the sector of angle 75° = ⁷⁵⁄₃₆₀ × π(8)²
= ⁷⁵⁄₃₆₀ × 3.14 × (8)²
= ⁷⁵⁄₃₆₀ × 3.14 × 64
= 41.87 cm²

Area of a Segment

→ Area of segment of a circle = Area of the corresponding sector − Area of the corresponding triangle

→ Area of the minor segment APB of the circle = Area of sector OAPB − Area of △OAB.

Area of the major sector of a circle of radius r = πr² − Area of the corresponding minor sector.