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A Circle is a curved 2-D figure formed by a set of points. A circle is measured in terms of its radius. In geometry, it is a special kind of ellipse where eccentricity is ‘0’ and two foci are coincident. In our daily life, we often come across such objects which have round shapes such as dials of many clocks, bangles, wheels of a vehicle, finger rings, coins, keyrings and so on. In a clock, we notice that the tip of the second’s hand moves in a round path. The path and the path by the tip of the second’s hand is a circle (or circular in motion).
Any line that passes through the centre of the circle and connects two points of the circle is the diameter of the circle. The radius is half the length of the diameter of the circle. The area of the circle describes the amount of space that is covered by the circle and the circumference is the length of the boundary of the circle. This article covers all the essential formulas related to circles, such as the formulas of diameter, area and circumference with respective examples.
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Key Terms: Radius of a circle, the diameter of a circle, the circumference of a circle, area of a circle, Sector of a circle, circumference and Area of a sector of a circle, a segment of a circle, circumference and area of a segment of a circle.
What is a Circle?
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A circle can be defined as the set or locus of all the points which are equidistant from a fixed point. This fixed point is called the ‘centre’. In the figure given below, the centre of a circle and its radius are clearly labelled for your convenience. It can be noted that the radius will be the same when measured from any point on the circle. There are various properties of circles that will be useful throughout your career.
The video below explains this:
Radius Formula Detailed Video Explanation:
What is a Circle?
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Radius of a Circle
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Given in the definition of a circle, the radius is the fixed distance from the centre of a circle to any given point that lies on the circle. Radius is usually denoted as ‘r’ or ‘R’. Any two circles having the same radii are congruent. And the diameter of a circle contains two radii with a central angle of 180 degrees.
Radius of a Circle
Properties of a Tangent drawn to a circle
- Only one tangent can touch a circle at one point of the circle.
- A tangent can not pass through the circle as a tangent never crosses a circle.
- A tangent can never intersect the circle at two points.
- The line of tangency is perpendicular to the radius of a circle.
- The length of two tangents from a common external point is always equal to a circle.
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Diameter of a Circle
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In simple words, diameter is a line segment passing through the centre of a circle whose both endpoints lie on the circle. The diameter is also the longest chord of the circle. The length of a diameter is twice that of the radius of a circle. AB line segment in the given figure is the diameter of the circle with the centre C. The diameter essentially divides the circle into two equal parts each of which is called a semicircle.
Diameter of a Circle
Diameter of a Circle Formula
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Circumference of a Circle
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The circumference of a circle is the distance around a circle. Let us consider a circular ring with radius r. If we take the ring and give it the shape of a straight wire, then the length of the wire is what we call is the circumference. An interesting fact about the circumference of a circle is that irrespective of its size, the ratio of circumference and diameter of any circle is always a constant which is equal to π.
Circumference of a Circle
Formula of Circumference of a CircleCircumference = 2πr |
Area of a Circle
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The space occupied by a circle inside its circumference in two dimensions is called the area of the circle. The shaded portion of the circle given below is its area. Archimedes had invented the area of the circle. The area of a circle and that of the sectors and segments of a circle are described throughout this article.
Formula of Area of a Circle: Area = πr2
Also read: Perimeter and area of a circle
Sector, Perimeter and Area
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The sector of a circle is a part of the circle which is surrounded by two of its radii and an arc. The sector can be of two types: Minor Sector and Major Sector. A sector is called a minor sector when the internal angle between the two radii is less than 180 degrees. In a major sector, the angle between the two radii is greater than 180 degrees. The angle between the two radii in a segment is called the central angle. The central angle here is denoted by a θ.
Formula
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Segment, Perimeter and Area
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The segment is that part of a circle that is bounded by an arc and a chord of the circle. A segment is called a major segment if the centre of a circle lies inside the segment. Otherwise, it is called a minor segment. Refer to the labelled diagram given above. The formula for the area of a segment of a circle can be derived by subtracting the area of the triangle included inside the chord of the segment and two radii from the total area of the sector which contains the segment.
Formula
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Theorems
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The theorem of tangent states that a line will be a tangent to a circle if and only if the line is perpendicular to the radius that is drawn to the point of tangency.
Tangent is perpendicular to the radius at the point of contact
Theorem: This theorem proves that “the tangent drawn to the circle at any point is the perpendicular to the radius of the circle that passes through the point of contact”.
The number of tangents drawn from a given point
- If the point is inside the circle, any line through that point is a secant. So, no tangent can be drawn to a circle that passes through a point that lies inside it.
- There is exactly one tangent to a circle that passes through it if a point of tangency lies on the circle,
- There are accurately two tangents to a circle through it, when the point lies outside of the circle.
Tangent is perpendicular to the radius at the point of contact
Lengths of tangents drawn from an external point
Theorem: The second theorem states that two tangents will be of equal length when the tangent is drawn from an external point to a circle. The tangent segment refers to the line that joins the external point and the point of tangency.
Lengths of tangents drawn from an external point
Tangent as a special case of Secant
When the two endpoints of its corresponding chord coincide, the tangent to a circle may be considered as a special case of the secant.
Moreover, a circle can possess two parallel tangents at most for a given secant.
There are exactly two tangents that are parallel to that and intersect the circle at two diametrically opposite points, for every given secant of a circle.
Tangent as a special case of Secant
Solved Examples
Example 1: Find the area, circumference, and diameter of a circle that is of radius 5 cm.
Solution: As we know, area of a circle = π × r²
Given, r = 5 cm, π = 3.14
Therefore, area of a circle = 3.14 × 5² = 78.5 cm
Circumference of a circle = 2πr
Given, r = 5 cm, π = 3.14
Hence, circumference of a circle = 2 × 3.14 × 5 = 31.4 cm
Diameter of a circle = 2 × radius
Given, r = 5 cm,
Thus, diameter of a circle = 2 × 5 = 10 cm
Example 2: Find the equation of the circle whose centre is (2,6) and the radius is 4 units.
Solution: Given, the centre of the circle is not an origin. Therefore, the general equation of the circle will be applied.
The general equation of a circle is (x – x?)² + (y – y?)² = r²
Substituting the values:
(x - 2)² + (y - 6)² = 4²
Therefore, the required equation of a circle is (x - 2)² + (y - 6)² = 4²
Things to Remember
- A circle is the locus of all the points that are equidistant from a fixed point. The distance between a point on the circle and the centre is called the radius.
- Diameter is a line segment passing through the centre whose endpoint lies on the circle. The length of a diameter is two times that of the radius. The midpoint of the diameter is the centre of the circle.
- The circumference of a circle is the distance around the circle. The area of a circle is the region bounded in the interior part of the circumference of the circle.
- A segment is bounded by two radii and an arc of the circle. A sector is bounded by a chord and an arc of the circle.
Sample Questions
Ques: In the figure, MN and PQ are the arcs of two concentric circles of radii 7 cm and 3.5 cm respectively and ∠MON = 30°. Find the area of the shaded region. (4 marks)
Ans: Given MN and PQ are the arcs of two concentric circles with centre O. Central angle of the arcs is 30 degrees. Radii of the two circles are given by R1 = 7 cm and R2 = 3.5 cm.
It is clear from the figure that the region OMN is a sector of the bigger circle and area OPQ is that of the smaller circle.
We need to find the area of the shaded region.
Area of shaded region QPMN = Area of sector OMN - Area of sector OPQ = (θ/360)πR12 - (θ/360)πR22= (30/360)*(22/7)(7*7 - 3.5*3.5) = (1/12)(22/7)(36.5) = 9.55 square cm.
Therefore the area of the shaded region is 9.55 square cm.
Ques: If the area of a circle is equal to the sum of the areas of two circles of diameters 10 cm and 24 cm, calculate the diameter of the larger circle (in cm). (2 marks)
Ans: Let the radii of the circle be R.
πR2 = πr21 + πr22
πR2 = π(r21 + πr22)
Let Radii of the two given circles are r1
10/2 = 5cm, r2
24/2 = 12 cm
R2 = 52 + 122 = 25 + 144
R2 = 169 = 13 cm
∴ Diameter of the required circle = 2(13) = 26 cm.
Ques: Find the area of the circle whose circumference is 44cm. (2 marks)
Ans: Given that the circumference of the circle is 44cm.
Which means 2πr = 44 or r = 44*7/44 = 7 cm.
Now the area of the circle is given by πr2
Therefore, area = (22/7)(7*7)= 154 square cm.
Ques: A steel wire when bent in form of a square has an area of 121 square cm. Find the area of the circle if the same wire is bent in form of a circle. (2 marks)
Ans: Given, when the wire is bent in form of a square area =121
Where a is the side of the square, a = 11 cm
Let us find the length of the wire, which is equal to the perimeter of the square with side a = 4 x a = 44.
Hence the perimeter of the circle is also the same as that of the square. This means 2πr = 44
Now the radius of the circle is 7cm.
Finally the area of the circle is = πr2 = (22/7)(7*7)= 154 square cm
Ques: The area of a circular playground is 22176 square meters. Find the cost of fencing this ground at the rate of Rs. 50 per metre. (2 marks)
Ans: Area of the playground is = 22176 sq metre
The radius of the playground is given by, πr2 = 22176. Or r = 84 m
Now circumference of the playground = 2πr = 2 x 22/7 x 84 = 528 m
Cost of fencing the entire playground at a rate of Rs. 50 per metre = 528 x Rs. 50 = Rs. 26,400
Ques: A square is inscribed in a circle. Find the ratio of the area of the square and the circle. (3 marks)
Ans: Given that a square is inscribed in a circle. As shown in the figure, the diameter of the circle and diagonal of the square are the same. Hence, let the length of AC = 2r.
Which implies that the radius of the circle is r.
Now in the triagle ABC, AB = BC and AC = 2r.
Using Pythagoras theorem,
AB2 + BC2 = AC2
2AB2 = 4r2
AB2 = 2r2
AB = Sqrt(2)r
Now ratio of area of the square and the circle = Area of square / Area of circle = AB2 / πr2 = 2r2/πr2 = 2/π.
Ques: A horse is tied to a pole with a 28 m long string. Find the area where the horse can graze. (2 marks)
Ans: Given that the length of the rope = 28m
The horse can graze in a circular area with the radius of circle = 28m
Therefore, the area required is = πr2 = 22/7 x 28 x 28 = 2464 square metre.
Ques: How many tangents can be drawn from the external point to a circle? (1 mark)
Ans. Two tangents will be drawn from the external point to a circle.
Ques: Given: A triangle OAB which is an isosceles triangle and AB is tangent to the circle with centre O. Find the measure of ∠OAB. (1 mark)
Ans. The measure of ∠OAB in the given isosceles triangle OAB will be 45 degrees.
Ques: What should be the angle between the two tangents which is drawn at the end of two radii and are inclined at an angle of 45 degrees? (1 mark)
Ans. The angle between them shall be 135 degrees.
Ques: Given a right triangle PQR which is right-angled at Q. QR = 12 cm, PQ = 5 cm. The radius of the circle which is inscribed in triangle PQR will be? (1 mark)
Ans. The radius of the circle will be 2 cm.
Ques: Define Tangent and Secant. (1 mark)
Ans. A tangent is a line that meets the circle only at one point.
A secant is a line that meets the circle at two points while intersecting it. These two points are always distinct.
Ques: What is a circle? (1 mark)
Ans. If we collect all the points given on a plane and are at a constant distance, we will get a circle. The constant distance is the radius and the fixed point will be the centre of the circle.
Ques: Prove that the tangents that are drawn at the ends of the diameter of a circle are parallel. (2021OD)
Ans.
Proof: ∠1 = 90° … (i)
∠2 = 90° … (ii)
From (i) and (ii), ∠1 = ∠2
However, these are alternate interior angles.
Therefore, PQ || RS.
Ques: In the given figure, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, then find ∠ATQ. (2015D)
Ans.
∠ABQ =½ , ∠AOQ = 58°/2 = 29°
∠BAT =90° [Tangent is perpendicular through the point of contact to the radius]
∠ATQ = 180° - (∠ABQ + ∠BAT)
180 - (29 + 30) = 180° - 119° = 61°
Ques: In the given figure, the chord AB of the larger of the two concentric circles with centre O touches the smaller circle at C. Prove that AC =CB. (2021D)
Ans.
Join OC.
![The chord AB of the larger of the two concentric circles with centre O touches the smaller circle at C [Join OC]](https://images.collegedunia.com/public/image/05de0860076c44d04f86342a64ce3966.png)
Proof: AB is a tangent to a smaller circle where OC is the radius.
Therefore, ∠OCB = 90° from the above theorem.
In the larger circle, AB is a chord and OC perpendicular to AB.
AC = CB … [ perpendicular from the centre bisects the chord].
Ques: In the figure, a right triangle ABC circumscribes a circle of radius r. If AB and BC are of lengths 8 cm and 6 cm respectively, what is the value of r? (2012 OD)
Ans. Join AO, OB and CO.
Proof: area of Δ ABC
From (i) and (ii), we get, 12r = 24
Therefore, r = 2 cm.
Ques: In the figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q.If PA = 12 cm, QC =QD = 3 cm, then find PC + PD. (2017D)
Ans. PA = PB =12 cm (i)
QC =AC =3 cm … (ii)
QD =BD = 3 cm … (iii)
PC = PD = ?
= (PA - AC) + (PB - BD)
= (12 - 3) + (12 - 3) [From equation (i), (ii) and (iii)
= 9 + 9 = 18 cm.
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