Mensuration

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.

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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.

What is a circle

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.

What is 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)

Adding all three,
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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