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State Board · Class 5 · Mathematics

Lesson 7: Circles

5 concepts · 40 practice questions with answers · Free to read

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Circle, centre and radius

From daily life

A bangle, a coin and a clock face are all round. If you pin a thread at the middle of a clock face and swing it, its tip reaches every point on the edge at the same distance. That is a circle.

Simple definition

A circle (वर्तुळ) is a round shape in which every point on the boundary is at the same distance from one fixed point inside it. The fixed point is the centre (केंद्र). A line segment from the centre to any point on the circle is a radius (त्रिज्या). The boundary is the circumference (परिघ).

A circle with its centre, a radius, a diameter, a chord and the circumference marked.centreradiusdiameterchordcircumference

In the figure : A circle with its centre, a radius, a diameter, a chord and the circumference.

Rule 1: Radius and centre — All radii of one circle are equal.

  • Easy : A bangle with centre O: OA = OB = OC, since they are all radii.
  • Medium : If one radius is 6 cm, every other radius of that circle is also 6 cm.
  • Exam-type : Two circles with the same centre but different radii are different circles. The radius decides the size.
A circle of radius 5 cm with three points: 3 cm from the centre is inside, 5 cm is on the circle, 8 cm is outside.r=53 cm → inside5 cm → on the circle8 cm → outside(distance from the centre)

In the figure : For a circle of radius 5 cm: a point 3 cm from the centre is inside, 5 cm is on the circle, 8 cm is outside.

Rule 2: Inside, on or outside — Compare the distance from the centre with the radius.

  • Easy : Distance less than the radius → inside.
  • Medium : Distance equal to the radius → on the circle.
  • Exam-type : Distance more than the radius → outside. With radius 7 cm, a point 9 cm away is outside.
Remember

Every point on the circle is at the same distance from the centre: the radius.

Common mistake

Thinking that the centre is a point on the circle. The centre lies inside and is not part of the boundary.

Shortcut

Memory clue: Radius = Ray from the centre.

Spot the difference

PartMeaning
centrefixed point inside
radiuscentre to circle
circumferencethe boundary

Practice questions

Easy
  1. 1What is the fixed point in a circle called?
  2. 2What is the boundary of a circle called?
  3. 3A radius of a circle is 6 cm. What is another radius?
Medium
  1. 4A circle has radius 5 cm. A point is 3 cm from the centre. Where is it?
  2. 5Radius 7 cm. A point is 9 cm from the centre. Where is it?
  3. 6Radius 4 cm. A point is 4 cm from the centre. Where is it?
Challenging
  1. 7Points A, B and C are at distances 2 cm, 5 cm and 8 cm from the centre of a circle of radius 5 cm. Place each.
  2. 8Meera says the centre of a circle is a point on the circle. Is she right? Explain.
Show answers (try them yourself first)
  1. 1Centre.
  2. 2Circumference.
  3. 36 cm
  4. 4Inside the circle.
  5. 5Outside the circle.
  6. 6On the circle.
  7. 7A inside, B on the circle, C outside.
  8. 8No. The centre is inside, at the distance of one radius from every point of the circle.

Diameter

From daily life

A cartwheel is measured by laying a stick across its widest part, through the middle. The length of the stick is the diameter.

Simple definition

A diameter (व्यास) is a line segment through the centre with both ends on the circle. It is made of two radii, so diameter = 2 × radius and radius = diameter ÷ 2. All diameters of a circle are equal, and a diameter cuts the circle into two equal semicircles (अर्धवर्तुळे).

A circle whose diameter is made of two radii: d = 2 times r.rrd = r + rd = 2 × rr = 6 → d = 12d = 18 → r = 9

In the figure : The diameter is two radii end to end: d = 2 × r.

Rule 1: Diameter from radius — Double the radius.

  • Easy : Radius 5 cm → diameter 10 cm.
  • Medium : A wheel of radius 35 cm is 2 × 35 = 70 cm across.
  • Exam-type : A circular table of radius 60 cm has a diameter of 120 cm, which is 1 m 20 cm.
Live figure

Diameter

In the app you can move this figure with your finger.

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Rule 2: Radius from diameter — Halve the diameter.

  • Easy : Diameter 18 cm → radius 9 cm.
  • Medium : A plate with a diameter of 26 cm has a radius of 13 cm.
  • Exam-type : The longest straight line inside a circle is the diameter. A rod of 50 cm cannot lie inside a circle of diameter 40 cm.
Remember

d = 2 × r and r = d ÷ 2. The diameter passes through the centre.

Common mistake

Using the radius as the diameter. A circle of "radius 4" has a diameter of 8.

Shortcut

Memory clue: Diameter is Double the radius (D for Double).

Spot the difference

GivenFindRule
radiusdiameter× 2
diameterradius÷ 2

Practice questions

Easy
  1. 1The radius of a circle is 8 cm. Find its diameter.
  2. 2The diameter is 30 cm. Find the radius.
  3. 3What does the diameter of a circle divide it into?
Medium
  1. 4Find the diameter of a wheel of radius 45 cm.
  2. 5A round table is 1 m 20 cm across. What is the radius in cm?
  3. 6True or false: all diameters of a circle are equal.
Challenging
  1. 7A rod 50 cm long is to be kept in a circular tray of diameter 40 cm. Will it fit lying flat? Why?
  2. 8The sum of the radius and the diameter of a circle is 18 cm. Find the radius.
Show answers (try them yourself first)
  1. 116 cm
  2. 215 cm
  3. 3Two equal semicircles.
  4. 490 cm
  5. 5120 ÷ 2 = 60 cm.
  6. 6True.
  7. 7No. The longest line in the tray is its diameter, 40 cm, which is less than 50 cm.
  8. 8r + 2r = 3r = 18, so r = 6 cm.

Chord, arc and semicircle

From daily life

A rope tied between two points on a round pond is a chord. The longest rope that can stretch across the pond passes through the centre.

Simple definition

A chord (जीवा) joins any two points on a circle. A piece of the circumference is an arc (कंस). The diameter is the longest chord.

Rule 1: Chord and diameter — Every diameter is a chord. A chord is a diameter only if it passes through the centre.

  • Easy : In a circle with diameter 10 cm, no chord is longer than 10 cm.
  • Medium : A chord of length 8 cm in this circle is not a diameter, because 8 cm is less than 10 cm.
  • Exam-type : A chord that passes through the centre is exactly the diameter; there can be many of them, all of equal length.

Rule 2: Arcs and semicircles — A diameter divides the circle into two equal arcs, each a semicircle.

  • Easy : A semicircle is half of a circle.
  • Medium : Two cuts through the centre at right angles make four quarter circles.
  • Exam-type : A round roti cut by two diameters at a right angle gives 4 equal quarters; each quarter has two radii as sides and an arc.
Remember

Diameter = longest chord. A chord not through the centre is shorter.

Common mistake

Saying every chord is a diameter. Only a chord through the centre is a diameter.

Shortcut

Memory clue: Chord through the centre = diameter.

Spot the difference

TermMeaning
chordjoins two points on the circle
diameterchord through the centre
arcpart of the circumference
semicirclehalf of the circle

Practice questions

Easy
  1. 1What is a line joining two points on a circle called?
  2. 2What is the longest chord of a circle?
  3. 3What is a piece of the circumference called?
Medium
  1. 4Is every diameter a chord? Is every chord a diameter?
  2. 5A circle has diameter 12 cm. Can it have a chord of 14 cm?
  3. 6Into how many equal arcs does a diameter divide a circle?
Challenging
  1. 7Two diameters at right angles cut a pizza. How many pieces are formed and what part of the pizza is each?
  2. 8Ravi draws a chord of 9 cm in a circle of radius 5 cm and says it is a diameter. Is he right?
Show answers (try them yourself first)
  1. 1Chord.
  2. 2The diameter.
  3. 3Arc.
  4. 4Yes; no.
  5. 5No, because no chord is longer than the diameter.
  6. 6Two (semicircles).
  7. 74 pieces; each is 14 of the pizza.
  8. 8No. The diameter is 10 cm; a 9 cm chord is shorter, so it does not pass through the centre.

Drawing circles with a compass

From daily life

A potter uses a compass-like stick: he fixes one end on the wheel's centre and marks the edge with the other end. This is how a compass draws a circle.

Simple definition

Fix the needle of the compass (कंपास) on the centre, open it to the radius you need and turn the pencil end around once. Circles that share one centre but have different radii are concentric circles (एककेंद्री वर्तुळे).

Three concentric circles with radii 2, 4 and 6 cm sharing the centre O.O246 cmSame centre O,different radii:concentric circles

In the figure : Concentric circles of radii 2, 4 and 6 cm share the centre O.

Rule 1: Open the compass — Open the compass to the radius, not the diameter.

  • Easy : For radius 3 cm, open the compass to 3 cm.
  • Medium : To draw a circle of diameter 8 cm, open the compass to 4 cm (half of 8).
  • Exam-type : For a circle of diameter 11 cm, open to 5.5 cm; the radius can be a decimal.

Rule 2: Concentric circles — Keep the needle fixed and change only the opening.

  • Easy : Ripples in a pond and a target board are concentric.
  • Medium : Circles of radii 2, 4 and 6 cm with centre O are concentric; their diameters are 4, 8 and 12 cm.
  • Exam-type : Two circles with different centres, even of equal radius, are not concentric.
Remember

Fix the needle, open to the radius, turn once.

Common mistake

Moving the needle while drawing. Hold it steady so that the radius stays the same.

Shortcut

Memory clue: Needle on the centre, pencil on the edge.

Spot the difference

CircleRadiusDiameter
Inner2 cm4 cm
Middle4 cm8 cm
Outer6 cm12 cm

Practice questions

Easy
  1. 1Where is the needle of the compass fixed?
  2. 2To draw a circle of radius 4 cm, what is the compass opening?
  3. 3To draw a circle of diameter 10 cm, what is the compass opening?
Medium
  1. 4What are circles with a common centre called?
  2. 5Name two real-life examples of concentric circles.
  3. 6The radii of three concentric circles are 3 cm, 6 cm and 9 cm. Find their diameters.
Challenging
  1. 7A compass is opened to 7.5 cm. What is the diameter of the circle drawn?
  2. 8Two circles of radius 5 cm are drawn with different centres. Are they concentric? Give a reason.
Show answers (try them yourself first)
  1. 1On the centre.
  2. 24 cm
  3. 35 cm
  4. 4Concentric circles.
  5. 5Ripples in a pond; a target board.
  6. 66 cm, 12 cm, 18 cm
  7. 715 cm
  8. 8No. Concentric circles must share the same centre.

Circles around us

From daily life

A bicycle wheel, a coin, a plate and a bangle are circles. The axle of the wheel is at the centre and the spokes are like radii.

Simple definition

Use what you know about circles in daily life: the diameter of a round object is twice its radius, and to fit one object into another space, compare diameters.

A wheel with 12 spokes: each spoke is a radius of 30 cm, so the wheel is 60 cm across.30 cmEach spoke is a radiusradius = 30 cmdiameter = 60 cm

In the figure : Each spoke of this wheel is a radius of 30 cm, so its diameter is 60 cm.

Rule 1: Wheels and spokes — The spokes meet at the centre; each spoke is a radius.

  • Easy : A spoke of 30 cm gives a wheel diameter of 60 cm.
  • Medium : A cycle wheel with a 35 cm radius is 70 cm across.
  • Exam-type : A circular park has a diameter of 40 m. A path from one point of the boundary through the centre to the opposite point is 40 m.

Rule 2: Does it fit? — Compare the diameter of the object with the width of the space.

  • Easy : A 24 cm plate cannot fit on a 20 cm tray.
  • Medium : A coin of radius 1 cm (diameter 2 cm) fits in a slot 3 cm wide.
  • Exam-type : A round table of radius 60 cm needs 120 cm of width; it fits in a corner 125 cm wide but not in one 110 cm wide.
Remember

Diameter = 2 × radius is the key link between the two measures.

Common mistake

Comparing the radius of an object with the width of the space. Compare the diameter.

Shortcut

Memory clue: Fits? Compare the diameter.

Spot the difference

ObjectDiameterFits a 25 cm space?
plate24 cmyes
tray26 cmno
coin2 cmyes

Practice questions

Easy
  1. 1What is the diameter of a wheel whose spokes are 30 cm long?
  2. 2Find the radius of a coin whose diameter is 3 cm.
  3. 3A garden has radius 10 m. How long is a path through its centre?
Medium
  1. 4A 24 cm plate is placed on a 20 cm tray. Does it fit?
  2. 5Which has a longer diameter: a wheel with radius 28 cm or a wheel with diameter 50 cm?
  3. 6A circular pond has a diameter of 18 m. A boat 20 m long travels along its diameter. Will it fit?
Challenging
  1. 7A round table has radius 60 cm. A corner is 125 cm wide. Does the table fit?
  2. 8Concentric circles of radii 5 cm and 8 cm are drawn. What is the difference between their diameters?
Show answers (try them yourself first)
  1. 160 cm
  2. 21.5 cm
  3. 320 m
  4. 4No, 24 cm > 20 cm.
  5. 5Radius 28 cm → diameter 56 cm, which is longer than 50 cm.
  6. 6No, because 20 m is longer than 18 m.
  7. 7Diameter = 120 cm < 125 cm, so it fits.
  8. 816 − 10 = 6 cm.