The Circle Class 10 Notes 

A circle is a shape where every point in its boundary is equidistant from its centre. A circle is a basic concept of geometry that has some key elements used in most mathematical applications, such as radius, diameter, circumference, and area.

1.0Basics of Circle

A circle is actually a collection of all points within a plane found at a fixed distance, which is referred to as a radius, from a fixed point or centre.

Part of a Circle

• Centre: The centre is the fixed point at which every point on a circle is equidistant.
• Radius (r): Distance from the centre to any point on the circle.
• Diameter (d): The longest chord of the circle, passing through the centre. It is twice the radius, i.e., d=2r.
• Circumference: It is the perimeter or boundary of the circle. Given by the formula C=2πr, where π≈3.14 0r 22/7.
• Chord: Any line segment joining two points on the circle. A diameter is a special chord.
• Secant: Any line that intersects the circle at two points.
• Tangent: Any line that touches the circle at only one point. This point is called the point of tangency.
• Arc: A part of the circle's circumference. The length of an arc can be calculated with the following formula:

$\text{Length of an arc}=\frac{\theta }{360}\times 2\pi r$

Arcs are of two types:

• Minor Arc: The shorter arc connecting two points.
• Major Arc: The longer arc connecting two points.
• Sector: The region bounded by two radii and the corresponding arc. It is like a "pizza slice" of the circle. The sector is also divided into two parts.
• • Major Sector: sector corresponding to major arc.
  • Minor Sector: sector corresponding to minor arc.
• Segment: The region bounded by a chord and the corresponding arc. Segments are also of two types.
• • Major segment: region corresponding to chord and major arc.
  • Minor segment: region corresponding to chord and minor arc.

2.0Properties of Circle

1. Perpendicular from the Centre to a Chord: The perpendicular distance from the center of the circle to any chord bisects the chord. 

2. Circle's Tangent and Radius: Tangent to a circle at any point is perpendicular to the radius at that point. This implies that the angle between the radius and the tangent line is always 90°.

3. Angle Subtended by a Chord at the Center: The angle subtended by a chord at the center of the circle is twice the angle subtended at any point on the circumference on the same side of the chord.

4. Equal Chords and Distances from Center: Equal lengths of chords that lie at an equal distance from the center of the circle. If any two chords equal in length lie in the same circle, they will have an equal distance from the center.

5. Length of Two Tangents from an External Point: The two tangents drawn from an external point to a circle are always equal in length. 

3.0Areas Related to Circles Class 10 Notes

In Class 10 Math, Areas related to circles are the group of all the formulae used to calculate the areas of certain parts of the circle. Here is the formula related to the circle: 

1. Area of Circle: $r2\pi r^2$

2. The sector of a circle:  $\frac{\theta }{360}\times \pi r^2$

3. The segment of a circle: It does not have a proper formula, but the segment of a circle can be found by calculating the difference between the corresponding sector and the triangle. 

Area of a segment = Area of the sector – Area of the corresponding triangle 

4.0Examples Related to Circles

Problem 1: A circular garden has a radius of 14 meters. A sector of the garden with a central angle of 60° is to be used for a flower bed. Find: 

• The area of the sector.
• The length of the arc of the sector.

Solution:

1) The area of the sector: $\frac{\theta }{360}\times \pi r^2$

Here: $\theta =60,r=14m$

= $\frac{60}{360}\times \frac{22}{7}\times 14^2$

= $\frac{1}{6}\times 22\times 14\times 2\approx 102.1m^2$

2) Length of arc = $=\frac{\theta }{360}\times 2\pi r$

= $\frac{60}{360}\times 2\times \frac{22}{7}\times 14$

= $\frac{1}{6}\times 44\times 2=14.66m$

Problem 2: A chord of a circle is equal to its radius (r). Now, find the angle subtended by this chord at a point on its minor arc & also at a point on the major arc. (Old NCERT Class 9 Maths Circles 10.5)

To find: $\angle ACBand\angle ADB$

Solution:

According to the question: 

AO = BO = AB

Therefore, ABO is an equilateral triangle. 

Hence, $\angle AOB=60$

$\angle AOB=2\angle ACB$ (The angle subtended by a chord at the centre is always twice the angle subtended at any point on the circle's circumference)

$\angle ACB=\frac{1}{2}\times 60=30$

As we can see, ABCD are points on a circle; hence, ABCD is a cyclic quadrilateral, which means:

$\angle ACB+\angle ADB=180$ (Sum of opposite angles of a cyclic quadrilateral)

$30+\angle ADB=180$

$\angle ADB=180–30=150$

Problem 3: A circle has a radius of 14 cm, and a chord AB subtends a central angle of 90° at the centre of the circle. Find the area of the segment formed by the chord AB.

Solution:

$\text{Area of sector}=\frac{\theta }{360}\times \pi r^2$

$=\frac{90}{360}\times \frac{22}{7}\times 14^2$

$=\frac{1}{4}\times 22\times 14\times 2=154{\text{cm}}^2$

$\text{Area of triangle}=\frac{1}{2}\times b\times h$

$=\frac{1}{2}\times 14\times 14$

$=7\times 14=98{\text{cm}}^2$

$\text{Area of the segment}=154-98=56{\text{cm}}^2$

5.0Areas Related to Circles Class 10 Worksheet with Answers

Segment: The radius of a circle is 12 cm. A chord of length 14 cm is drawn in the circle. Find the area of the segment formed by the chord. Answer: 35.61cm²

Circles and Circumference Worksheet: A sector of a circle has a central angle of 135°. If the length of the arc of the sector is 14 cm, find the circumference of the entire circle. Answer: 168cm 

Word Problem: A person is creating a sector on the edge of a square using a protractor. The square has a side length of 10 cm. The person uses a protractor to measure a central angle of 60° to form a sector at one of the corners of the square.

1. What is the area of the sector formed at the corner of the square? Answer: 5.24cm²

2. What is the length of the arc of this sector? Answer: 10.47cm

For more NCERT solutions, download the areas related to circles class 10 solutions pdf.

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