Parts of a circle: radius, chord, diameter, arc
To draw a rangoli circle, you fix a mark in the middle and swing a stick tied to it with a string. The mark is the centre and the length of the string is the radius. Whether the circle is big or small, its parts are the same.
The distance from the centre to any point on the circle is the radius. A line joining two points on the circle is a chord. A chord through the centre is the diameter. Any piece of the circle is an arc.
In the figure: The amber line goes through the centre; it is the diameter 10. The green line goes from the centre to the circle; it is the radius 5. The blue line does not pass through the centre; it is a chord 8.
Rule 1: All radii are equal and the diameter is double the radius — All radii of one circle have the same length. To find the diameter, multiply the radius by 2. To find the radius, divide the diameter by 2.
- Easy: Radius 5 cm. Diameter = 2 × 5 = 10 cm.
- Medium: Diameter 26 cm. Radius = 26 ÷ 2 = 13 cm.
- Exam type: Radius 2.5 metres. Diameter = 2 × 2.5 = 5 metres.
In the figure: In one circle there is a chord 6, a chord 8 and a chord 10 through the centre. The nearer a chord is to the centre, the longer it is. The one through the centre, 10, is the longest. So 6 < 8 < 10.
Rule 2: The diameter is the longest chord of a circle — The farther a chord is from the centre, the shorter it is. No chord can be longer than the diameter.
- Easy: Radius 5 cm. A chord 3 cm from the centre = 8 cm. It is shorter than the diameter 10 cm.
- Medium: Radius 13 cm. A chord 5 cm from the centre = 24 cm. The diameter is 26 cm and 24 < 26.
- Exam type: In a circle of radius 7 cm, a chord of 15 cm is not possible. The longest chord is the diameter 2 × 7 = 14 cm.
In the figure: On the left the chord does not pass through the centre, so it makes 2 unequal arcs: a minor arc (amber) and a major arc (blue). On the right a diameter makes 2 equal arcs. Each semicircle is 180°.
Rule 3: A chord makes two arcs of the circle — a minor arc and a major arc. If the chord is a diameter, both arcs are equal and are called semicircles.
- Easy: A diameter divides the circle into 2 parts. Both are equal, each a semicircle.
- Medium: If a chord does not pass through the centre, the 2 arcs are unequal: one minor, one major.
- Exam type: The minor arc and the major arc together make the whole circle. So they are corresponding arcs of each other.
Parts of a circle: radius, chord, diameter, arc
In the app you can move this figure with your finger.
Open in the appDiameter = 2 × radius. Whatever the chord, the longest chord is always the diameter.
Taking the radius as the diameter. In the figure the radius is 5 and the diameter is 10; 5 is not the diameter.
If one number is given, first spot whether it is a radius or a diameter. If it is a radius, multiply by 2; if it is a diameter, divide by 2.
Spot the difference
| Point | Radius | Diameter |
|---|---|---|
| From where to where | Centre to circle | Circle to circle through the centre |
| Measure in the figure | 5 | 10 |
| Relation | Half of the diameter | Double the radius |
| Is it a chord? | No | Yes, the longest |
Practice questions
- 1What is the diameter of a circle with radius 12 cm?
- 2What is the radius of a circle with diameter 50 cm?
- 3What is the diameter of a circle with radius 7.5 cm?
- 4Can a chord of length 25 cm be drawn in a circle of radius 10 cm? What is the longest chord?
- 5In a circle of radius 13 cm, what is the length of a chord 12 cm from the centre? (Take the square root of 13² − 12².)
- 6A circle has radius 6.5 cm. What are its diameter and its longest chord?
- 7The radii of two circles are 4 cm and 9 cm. What is the difference between their diameters?
- 8A circle has diameter 30 cm. How long is a chord 9 cm from the centre?
Show answers (try them yourself first)
- 1Diameter = 2 × 12 = 24 cm. 2) Radius = 50 ÷ 2 = 25 cm. 3) Diameter = 2 × 7.5 = 15 cm.
- 4The longest chord is the diameter = 2 × 10 = 20 cm. 25 > 20, so it is not possible.
- 513² − 12² = 169 − 144 = 25 and √25 = 5. Chord = 2 × 5 = 10 cm. 6) Diameter = 2 × 6.5 = 13 cm. The longest chord is the diameter itself, so 13 cm.
- 7The diameters are 2 × 4 = 8 and 2 × 9 = 18. The difference is 18 − 8 = 10 cm.
- 8Radius 30 ÷ 2 = 15. 15² − 9² = 225 − 81 = 144 and √144 = 12. Chord = 2 × 12 = 24 cm.
Circumference and π
If you wind a thread round a round plate and then straighten and measure it, you get the circumference of the plate. The ratio of the plate's diameter to this circumference comes out nearly the same for any round object.
The length of one trip round a circle is its circumference. Circumference ÷ diameter is the same number for all circles. It is called π, and its value is taken as about 3.14 or 22/7.
In the figure: A circle of diameter 7 has circumference 22 when straightened. The diameter 7, 7, 7 fits 3 times and 1 is left over. So the circumference is a little more than 3 diameters; 22 ÷ 7 is about 3.14.
Rule 1: Circumference ÷ diameter is always about 3.14 — Big or small, if you divide the circumference by the diameter you get about 3.14. This number is π.
- Easy: Diameter 7 cm, circumference 22 cm. 22 ÷ 7 ≈ 3.14.
- Medium: A plate has diameter 35 cm and circumference 110 cm. 110 ÷ 35 ≈ 3.14.
- Exam type: Diameter 50 cm, circumference 157 cm. 157 ÷ 50 = 3.14.
In the figure: The diameter is 42 and the radius is 21. From the diameter, 22/7 × 42 = 132. From the radius, 2 × 22/7 × 21 = 132. Both ways give circumference 132.
Rule 2: Circumference = π × diameter — If the diameter is known, multiply it by π. The answer is the circumference.
- Easy: Diameter 42 cm. Circumference = 22/7 × 42 = 132 cm.
- Medium: Diameter 21 cm. Circumference = 22/7 × 21 = 66 cm.
- Exam type: Diameter 5 cm, π = 3.14. Circumference = 3.14 × 5 = 15.7 cm.
Rule 3: Circumference = 2 × π × radius — If the radius is known, first multiply it by 2 to get the diameter, then multiply by π.
- Easy: Radius 21 cm. Circumference = 2 × 22/7 × 21 = 132 cm.
- Medium: Radius 3.5 cm. Circumference = 2 × 22/7 × 3.5 = 22 cm.
- Exam type: Radius 15 cm, π = 3.14. Circumference = 2 × 3.14 × 15 = 94.2 cm.
In the figure: For diameter 21, 22/7 was used: 22/7 × 21 = 66. For diameter 5, 3.14 was used: 3.14 × 5 = 15.7. Where the number is in the table of 7, 22/7 is easier.
Rule 4: Choose which π to use from the number — If the number divides by 7, take 22/7. Otherwise take 3.14. If the question gives a value of π, use that.
- Easy: Diameter 28 cm. It divides by 7, so 22/7 × 28 = 88 cm.
- Medium: Diameter 10 cm. It does not divide by 7, so 3.14 × 10 = 31.4 cm.
- Exam type: Diameter 42 cm. 22/7 × 42 = 132 and 3.14 × 42 = 131.88. Both values are nearly equal.
Circumference and π
In the app you can move this figure with your finger.
Open in the appπ is one fixed number. However big the circle, circumference ÷ diameter is about 3.14.
Multiplying the radius directly by π. With radius 21, taking 22/7 × 21 = 66 is wrong; the real circumference is 132.
If the radius is given, first make the diameter. Double 21 is 42, then 22/7 × 42 = 132.
Spot the difference
| Point | Diameter given | Radius given |
|---|---|---|
| Circumference formula | π × diameter | 2 × π × radius |
| Measure in the figure | 42 | 21 |
| Calculation | 22/7 × 42 | 2 × 22/7 × 21 |
| Circumference | 132 | 132 |
Practice questions
- 1Find the circumference of a circle of diameter 49 cm. (π = 22/7)
- 2Find the circumference of a circle of radius 10.5 cm. (π = 22/7)
- 3Find the circumference of a circle of diameter 40 cm. (π = 3.14)
- 4A round table has diameter 84 cm. What is its circumference? (π = 22/7)
- 5Find the circumference of a circle of radius 5 cm. (π = 3.14)
- 6A circle has diameter 28 cm and circumference 88 cm. What does circumference ÷ diameter come to?
- 7A circle has diameter 91 cm. Find its circumference and half the circumference. (π = 22/7)
- 8The radii of two circles are 7 cm and 21 cm. Find the difference between their circumferences. (π = 22/7)
Show answers (try them yourself first)
- 1Circumference = 22/7 × 49 = 154 cm. 2) Circumference = 2 × 22/7 × 10.5 = 66 cm. 3) Circumference = 3.14 × 40 = 125.6 cm.
- 4Circumference = 22/7 × 84 = 264 cm. 5) Circumference = 2 × 3.14 × 5 = 31.4 cm. 6) 88 ÷ 28 ≈ 3.14, that is about 3.14.
- 7Circumference = 22/7 × 91 = 286 cm. Half the circumference = 286 ÷ 2 = 143 cm.
- 8The circumferences are 2 × 22/7 × 7 = 44 and 2 × 22/7 × 21 = 132. The difference is 132 − 44 = 88 cm.
Finding the radius and diameter from the circumference
If you wind a thread round a tree trunk and measure it, you know the circumference. How thick the trunk is, that is its diameter, can be found from the length of the thread without a ruler across it.
If the circumference is known, divide it by π to find the diameter. Then divide the diameter by 2 to find the radius. If π = 22/7, dividing by 22/7 means multiplying by 7 and dividing by 22.
In the figure: The circumference is 110 and the diameter is unknown (?). 110 = 22/7 × ?, so ? = 110 ÷ 22/7 = 110 × 7 ÷ 22 = 35. The radius is 35 ÷ 2 = 17.5.
Rule 1: Diameter = circumference × 7 ÷ 22 — Multiply the circumference by 7 and divide by 22. The answer is the diameter.
- Easy: Circumference 110 cm. Diameter = 110 × 7 ÷ 22 = 35 cm.
- Medium: Circumference 66 cm. Diameter = 66 × 7 ÷ 22 = 21 cm.
- Exam type: Circumference 154 cm. Diameter = 154 × 7 ÷ 22 = 49 cm.
Rule 2: Radius = diameter ÷ 2 — From the circumference, first find the diameter. After that, divide it by 2.
- Easy: Diameter 35 cm. Radius = 35 ÷ 2 = 17.5 cm.
- Medium: Circumference 88 cm. Diameter = 88 × 7 ÷ 22 = 28 cm, radius = 28 ÷ 2 = 14 cm.
- Exam type: Circumference 308 cm. Diameter = 308 × 7 ÷ 22 = 98 cm, radius = 98 ÷ 2 = 49 cm.
Rule 3: If π = 3.14 is given, divide the circumference by 3.14 — Do the decimal divisions carefully. The answer can be checked.
- Easy: Circumference 94.2 cm. Diameter = 94.2 ÷ 3.14 = 30 cm.
- Medium: Circumference 125.6 cm. Diameter = 125.6 ÷ 3.14 = 40 cm.
- Exam type: Circumference 18.84 cm. Diameter = 18.84 ÷ 3.14 = 6 cm, radius = 6 ÷ 2 = 3 cm.
Rule 4: Check the answer by multiplying back — Multiply the diameter you found by π. The original circumference must come out.
- Easy: Diameter 35 cm. 22/7 × 35 = 110. The circumference 110 came out, so the answer is right.
- Medium: Diameter 21 cm. 22/7 × 21 = 66. The circumference 66 came out, so the answer is right.
- Exam type: Diameter 30 cm. 3.14 × 30 = 94.2. The circumference 94.2 came out, so the answer is right.
Finding the radius and diameter from the circumference
In the app you can move this figure with your finger.
Open in the appIf the diameter is given, multiply to get the circumference. If the circumference is given, divide to get the diameter.
Multiplying the circumference by 22/7. With circumference 110, doing so gives an answer far bigger than 110, but a diameter is always smaller than the circumference. Correct: 110 × 7 ÷ 22 = 35.
Dividing by 22/7 means multiplying by 7 and dividing by 22. 110 × 7 = 770 and 770 ÷ 22 = 35.
Spot the difference
| Point | Finding the circumference | Finding the diameter |
|---|---|---|
| Given | Diameter | Circumference |
| Operation | Multiplication | Division |
| Calculation in the figure | 22/7 × 42 = 132 | 110 × 7 ÷ 22 = 35 |
| Answer | 132 | 35 |
Practice questions
- 1Find the diameter of a circle of circumference 132 cm. (π = 22/7)
- 2Find the diameter of a circle of circumference 264 cm. (π = 22/7)
- 3Find the radius of a circle of circumference 66 cm. (π = 22/7)
- 4Find the diameter of a circle of circumference 31.4 cm. (π = 3.14)
- 5Find the radius of a circle of circumference 330 cm. (π = 22/7)
- 6Find the diameter of a circle of circumference 43.96 cm. (π = 3.14)
- 7The circumferences of two circles are 88 cm and 132 cm. Find the difference between their radii. (π = 22/7)
- 8A thread 242 cm long was wound once round a tree trunk. Find the diameter and the radius of the trunk. (π = 22/7)
Show answers (try them yourself first)
- 1Diameter = 132 × 7 ÷ 22 = 42 cm. 2) Diameter = 264 × 7 ÷ 22 = 84 cm. 3) Diameter = 66 × 7 ÷ 22 = 21 cm, radius = 21 ÷ 2 = 10.5 cm.
- 4Diameter = 31.4 ÷ 3.14 = 10 cm. 5) Diameter = 330 × 7 ÷ 22 = 105 cm, radius = 105 ÷ 2 = 52.5 cm. 6) Diameter = 43.96 ÷ 3.14 = 14 cm.
- 7First diameter 88 × 7 ÷ 22 = 28, radius 14. Second diameter 132 × 7 ÷ 22 = 42, radius 21. The difference is 21 − 14 = 7 cm.
- 8Diameter = 242 × 7 ÷ 22 = 77 cm, radius = 77 ÷ 2 = 38.5 cm.
Uses of the circumference: fencing and wheel turns
To put a wire fence round a round garden, the wire must be as long as the garden's circumference. Each time a wheel turns once, it moves forward on the road by a distance equal to its circumference.
Length of fence = rounds × circumference, and cost = length × rate per metre. Turns of a wheel = total distance ÷ circumference. The distance and the circumference must be in the same unit.
In the figure: The garden radius is 21 metres. The circumference is 2 × 22/7 × 21 = 132 metres. 3 rounds of wire are needed, that is 3 × 132 = 396 metres. At 25 rupees per metre the cost is 396 × 25 = 9900 rupees.
Rule 1: Length of fence = rounds × circumference, cost = length × rate — First find the circumference. Then multiply by the number of rounds. Finally multiply by the rate.
- Easy: Radius 21 metres, 3 rounds, rate 25 rupees. Circumference 2 × 22/7 × 21 = 132, length 3 × 132 = 396 metres, cost 396 × 25 = 9900 rupees.
- Medium: Radius 14 metres, 3 rounds, rate 30 rupees. Circumference 2 × 22/7 × 14 = 88, length 3 × 88 = 264 metres, cost 264 × 30 = 7920 rupees.
- Exam type: Diameter 28 metres, 4 rounds, rate 20 rupees. Circumference 22/7 × 28 = 88, length 4 × 88 = 352 metres, cost 352 × 20 = 7040 rupees.
In the figure: The wheel's diameter is 0.35 metres. In one turn the wheel moves its circumference, 22/7 × 0.35 = 1.1 metres. A distance of 5.5 km is 5500 metres. Turns = 5500 ÷ 1.1 = 5000.
Rule 2: Turns of a wheel = total distance ÷ circumference — Put the distance and the circumference in the same unit. Then divide.
- Easy: Diameter 0.35 metres, distance 5.5 km. Circumference 22/7 × 0.35 = 1.1 metres, distance 5.5 km = 5500 metres. Turns 5500 ÷ 1.1 = 5000.
- Medium: Diameter 56 cm, distance 3.52 km. Circumference 22/7 × 56 = 176 cm = 1.76 metres, distance 3.52 km = 3520 metres. Turns 3520 ÷ 1.76 = 2000.
- Exam type: Diameter 42 cm, distance 660 metres = 66000 cm. Circumference 22/7 × 42 = 132 cm. Turns 66000 ÷ 132 = 500.
Rule 3: Distance = turns × circumference — If the turns are known, multiply the other way. The answer comes in the same unit as the circumference.
- Easy: Circumference 2 metres, 300 turns. Distance 300 × 2 = 600 metres.
- Medium: Circumference 1.76 metres, 500 turns. Distance 500 × 1.76 = 880 metres.
- Exam type: Diameter 28 cm. Circumference 22/7 × 28 = 88 cm = 0.88 metres. In 1500 turns the distance is 1500 × 0.88 = 1320 metres, that is 1.32 km.
Uses of the circumference: fencing and wheel turns
In the app you can move this figure with your finger.
Open in the appOne turn of a wheel is one circumference of distance. So to find the turns, always divide the distance by the circumference.
Mixing km and metres. 5.5 ÷ 1.1 = 5 is wrong. Changing 5.5 km to 5500 metres and doing 5500 ÷ 1.1 = 5000 is right.
Put the circumference in metres and the distance in metres. Then estimate: if one turn is 1.1 metres, 5500 metres takes a few thousand turns.
Spot the difference
| Point | Fence | Wheel turns |
|---|---|---|
| Calculation in the figure | 2 × 22/7 × 21 = 132 | 22/7 × 0.35 = 1.1 |
| Next step | 3 × 132 = 396 | 5500 ÷ 1.1 = 5000 |
| Operation | Multiplication | Division |
| Final answer | 9900 rupees | 5000 turns |
Practice questions
- 1The diameter of a garden is 42 metres. What is the circumference round it? (π = 22/7)
- 2The radius of a garden is 63 metres. Wire is to be put round it for 2 rounds. How long is the wire? (π = 22/7)
- 3The circumference of a garden is 88 metres. If wire for 3 rounds is bought at 10 rupees per metre, what is the cost?
- 4The diameter of a wheel is 77 cm. How many turns will it make to cover 4.84 km? (π = 22/7)
- 5The diameter of a wheel is 21 cm. How many turns are needed to cover 990 metres? (π = 22/7)
- 6The circumference of a wheel is 1.5 metres. What distance will it cover in 400 turns?
- 7A round ground has diameter 70 metres. A fence is to be put round it for 2 rounds. If the rate is 15 rupees per metre, what is the cost? (π = 22/7)
- 8The diameter of a wheel is 49 cm. How many turns are needed to cover 3.08 km? (π = 22/7)
Show answers (try them yourself first)
- 1Circumference = 22/7 × 42 = 132 metres. 2) Circumference = 2 × 22/7 × 63 = 396 metres. For 2 rounds, 2 × 396 = 792 metres. 3) Length 3 × 88 = 264 metres. Cost 264 × 10 = 2640 rupees.
- 4Circumference 22/7 × 77 = 242 cm = 2.42 metres. Distance 4.84 km = 4840 metres. Turns 4840 ÷ 2.42 = 2000.
- 5Circumference 22/7 × 21 = 66 cm = 0.66 metres. Turns 990 ÷ 0.66 = 1500. 6) Distance 400 × 1.5 = 600 metres.
- 7Circumference 22/7 × 70 = 220 metres. Length 2 × 220 = 440 metres. Cost 440 × 15 = 6600 rupees.
- 8Circumference 22/7 × 49 = 154 cm = 1.54 metres. Distance 3.08 km = 3080 metres. Turns 3080 ÷ 1.54 = 2000.
Measure of an arc and the central angle
When you cut a pizza with two cuts from the centre, the angle between them tells how big the slice is. The edge of that slice is the arc and the angle near the centre is the central angle.
An angle drawn with its corner at the centre of the circle is a central angle. The measure of an arc is the measure of the central angle facing that arc. The measure of the whole circle is 360°.
In the figure: The central angle is 100°, so the minor arc facing it is 100°. The remaining major arc is 360° − 100° = 260°. The 100° and 260° together make the whole circle, 360°.
Rule 1: Measure of a minor arc = its central angle — The measure of the angle and the measure of the arc are the same.
- Easy: Central angle 100°. Minor arc = 100°.
- Medium: Central angle 55°. Minor arc = 55°.
- Exam type: Central angle 145°. Minor arc = 145°.
Rule 2: Major arc = 360° − minor arc — Corresponding arcs together make the whole circle, so their sum is 360°.
- Easy: Minor arc 100°. Major arc = 360° − 100° = 260°.
- Medium: Minor arc 85°. Major arc = 360° − 85° = 275°.
- Exam type: Minor arc 130°. Major arc = 360° − 130° = 230°.
In the figure: The whole circle is 360°. Half of it, a semicircle, is 180°. A quarter of it is 90°.
Rule 3: Whole circle 360°, semicircle 180° — A diameter divides the circle into two equal parts. Each part measures half of 360°.
- Easy: A diameter makes 2 equal arcs. Each = 360° ÷ 2 = 180°.
- Medium: If a circle is divided into 4 equal parts, each arc = 360° ÷ 4 = 90°.
- Exam type: If one of two corresponding arcs is 180°, the other is 360° − 180° = 180°.
Measure of an arc and the central angle
In the app you can move this figure with your finger.
Open in the appThe measure of an arc is in degrees, not in length. A minor arc is less than 180° and a major arc is more than 180°.
Subtracting from 180° for the major arc. With minor arc 100°, a major arc of 180° − 100° = 80° is wrong. Correct: 360° − 100° = 260°.
Two corresponding arcs always add up to 360°. If you know one, subtract it from 360°. 360° − 100° = 260°.
Spot the difference
| Point | Minor arc | Major arc |
|---|---|---|
| Measure | Less than 180° | More than 180° |
| In the figure | 100° | 260° |
| How to find it | Same as the angle | 360° − minor arc |
| Sum of the two | 360° | 360° |
Practice questions
- 1If the central angle is 40°, what is the measure of the minor arc?
- 2If the minor arc is 150°, what is the major arc?
- 3What is the measure of one arc made by a diameter?
- 4If the minor arc is 25°, what is the major arc?
- 5If the major arc is 300°, what is the minor arc?
- 6Of two corresponding arcs, the major arc is 3 times the minor arc. Find both arcs.
- 7The central angle is 72°. Find the measure of the major arc and say how many times the minor arc it is.
- 8A circle is divided into 5 equal arcs. What is the measure of one arc? And of 2 arcs together?
Show answers (try them yourself first)
- 1Minor arc = 40°. 2) Major arc = 360° − 150° = 210°. 3) A diameter makes a semicircle, so 180°.
- 4Major arc = 360° − 25° = 335°. 5) Minor arc = 360° − 300° = 60°.
- 6The minor and major arcs together make 1 + 3 = 4 parts, 360° ÷ 4 = 90°. The minor arc is 90° and the major arc is 3 × 90° = 270°.
- 7Major arc = 360° − 72° = 288°. 288° ÷ 72° = 4, that is 4 times.
- 8One arc = 360° ÷ 5 = 72°. 2 arcs together make 2 × 72° = 144°.