 # Circles

Out of all the geometric shapes and figures, the circle is a fascinating form present around us. The pizza, coins, the sun are all circular in shape (two-dimensional). Just a few important things in our lives that are circular in shape. And have you ever thought about how many sides a circle has? Let us study what is a circle in detail.

### Suggested Videos        Area of Rectangle and Square Introduction to Mensuration Area of polygons ## What is a Circle?

circle is a shape wherein all points have the same distance from the centre. Few real-world examples include a wheel, dining plate, coin etc. Drawing it properly isn’t easy with a running hand. The availability of a compass (a geometric tool) is preferred by most people, be it at school or at the workplace. 2D drawing of a sphere is a circle

## Important Terms Related to Circle

We know what is a circle. But what are the parts of a circle? Here are a few important terms that make up for the parts of a circle. • Diameter: The diameter is a line which is drawn across a circle passing through the centre.
• Radius: The distance from the middle or centre of a circle towards any point on it is a radius. Interestingly, when you place two radii back-to-back, the resultant would hold the same length as one diameter. Therefore, we can call one diameter twice as long as the concerned radius.
• Area of CircleIn a circle, the area can be stated as π times the square of the radius. It is, A = π r2. Taking into consideration the diameter: A = (π/4) × D2
• Chord: A line segment that joins two points present on a curve is the chord. In geometry, the usefulness of a chord is focused on describing a line segment connecting two endpoints which rest on a circle.
• Tangent & Arc: A line which slightly touches the circle on its travel to a different direction is Tangent. On the other hand, a part of the circumference is an Arc.
• Sector & Segment: A sector is a part of a circle surrounded by two radii of it together with their intercepted arc. The segment is that region which is enclosed by a chord together with the arc subtended by the chord.

## Properties of the Circle

• Circles with equal radii are congruent.
• Also, the circles with different radii appear to be similar.
• The chords that are equidistant from the centre are of the same length.
• All points on the circle are equidistant from the centre point.
• The longest chord in the circle is the diameter.
• A diameter of a circle divides it into two equal arcs. Each of the arcs is s a semi-circle.
• If the radii of two circles are exactly the same value, then the circles are congruent.
• Two or more circles that have different radii but the same centre are concentric circles.

## Solved Questions

Q1. Three circles are mutually tangent externally Their centres form a triangle whose sides are of lengths 3, 4 and 5 The total area of the circles (in square units) is

1. 9 π
2. 16 π
3. 21 π
4. 14 π

Solution: D. Let the respective radii of the circles be a, b and c. then,

a + b = 3             (1)
b + c = 4             (2)
c + a = 5             (3)

a + b + c = 6      (4)
From the above equations, we have
c = 3, a = 2, b = 1
Now the area of three circles = π (1²) + π (2²) + π (3²) = π + 4π + 9π = 14π

Q2. A horse is tethered by a rope 10 m long at a point. Find the area of the region where it can graze (π = 3.14)

Solution: The area of the region the horse can graze is circular with a radius equal to the length of the rope.

Area of the circle is πr²
= 3.14 × 10²
= 3.14 × 100
=314
Hence the area of the region the horse can graze is 314cm²

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