Circles Arcs And Sectors

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टीचर्स के लिए: Circles Arcs And Sectors (Maths [CIE], Core) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स, डायग्राम — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

Circle Basics

A circle is all points equidistant from a single centre point.
The circumference is the perimeter of the circle.
The diameter (d) is twice the radius (r): d = 2r.
π (π) ≈ 3.14159 is the ratio of circumference to diameter.
Answers may be given as 'exact value' or 'in terms of π'.

Area of a Circle

Formula: Area = πr² (given in exam).
Identify the radius (half the diameter).
Square the radius, then multiply by π.
Area is measured in square units (cm², m², etc.).

Circumference of a Circle

Formulae: C = πd or C = 2πr (given in exam).
Identify the diameter (or radius).
Multiply the diameter by π (or 2πr).
Circumference is measured in units of length (cm, m, etc.).

Arcs

An arc is part of the circumference between two points.
The minor arc is the smaller arc; the major arc is the larger.
Arc length formula: Arc length = (θ/360) × 2πr.
θ is the angle at the centre in degrees.

Sectors

A sector is the region enclosed by two radii and an arc.
The minor sector is the smaller sector; the major sector is the larger.
Sector area formula: Area = (θ/360) × πr².
A sector looks like a pizza slice.

Working with Formulae

Write down known values (r, d, θ).
Pick the correct formula (area, circumference, arc length, sector area).
Substitute values and solve.
For arcs and sectors, first find the fraction θ/360 of the full circle.

Semicircles

A semicircle is half a circle (θ = 180°).
Area of semicircle = (1/2)πr².
Perimeter of semicircle = (1/2)×2πr + d = πr + 2r.
Remember to include the diameter (straight edge) in perimeter.

Common Exam Tips

Area is always in square units; circumference/length in linear units.
Leave answers in terms of π if asked for 'exact value'.
For compound shapes (e.g., track), break into circles/rectangles.
Check whether to use radius or diameter in each formula.

स्लाइड्स

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 55 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.A circle has diameter 6 cm. Calculate the area of the circle. (Use π = 3.14)
Easy
- A28.26 cm²
- B18.84 cm²
- C113.04 cm²
- D9.42 cm²
2.Calculate the circumference of a circle with radius 4.5 cm. (Use π = 3.14)
Easy
- A28.26 cm
- B14.13 cm
- C9.42 cm
- D63.585 cm
3.A circle has diameter 7 cm. Show that the circumference is 21.99 cm correct to 2 decimal places. Which calculation gives this value?
Easy
- Aπ × 7
- B2 × π × 7
- Cπ × 3.5
- D2 × π × 3.5
4.A circle has a circumference of 56 mm. Work out the radius of this circle. (Use π = 3.14)
Medium
- A8.92 mm
- B17.83 mm
- C28 mm
- D8.75 mm
5.Rachel has a pond in the shape of a circle. The circumference is 4.25 m. Calculate the diameter in centimetres. (Use π = 3.14)
Medium
- A135.4 cm
- B67.7 cm
- C13.54 cm
- D1.354 cm
6.The diagram shows a semicircle with diameter 9 cm. Calculate the total perimeter of this semicircle. (Use π = 3.14)
Medium
- A23.13 cm
- B14.13 cm
- C28.26 cm
- D18.13 cm
7.A circular pool has a diameter of 8 m. Calculate the circumference of the pool. (Use π = 3.14)
Easy
- A25.12 m
- B50.24 m
- C12.56 m
- D200.96 m
8.A circular pool has a diameter of 8 m. Calculate the area of the pool. (Use π = 3.14)
Easy
- A50.24 m²
- B25.12 m²
- C200.96 m²
- D12.56 m²

और सवाल

9.The diagram shows a square with vertices on the circumference of a circle, centre O. The radius of the circle is 6 cm. Work out the shaded area (area of circle minus area of square). (Use π = 3.14)
Hard
- A41.04 cm²
- B72 cm²
- C113.04 cm²
10.Points P, Q and R are on the circumference of a semicircle, centre O and radius 8 cm. Angle POQ = 90°. Calculate the shaded area (area of semicircle minus area of triangle POQ). (Use π = 3.14)
Hard
- A68.48 cm²
- B32 cm²
- C100.48 cm²
- D36.48 cm²
11.A shape is drawn using two semicircles that have the same centre. The large semicircle has radius 7 cm. The small semicircle has radius 3 cm. Calculate the area of the shape (area of large semicircle minus area of small semicircle). (Use π = 3.14)
Medium
- A62.8 cm²
- B87.92 cm²
- C50.24 cm²
- D100.48 cm²
12.Points A and B lie on a circle. A region has been shaded between line AB and the circumference. The area of the circle is 76.4 cm². Calculate the radius of the circle. (Use π = 3.14)
Medium
- A4.93 cm
- B4.87 cm
- C12.17 cm
- D24.33 cm
13.The diameter of a different circle is 38 mm. Calculate the circumference of this circle. Give your answer in centimetres. (Use π = 3.14)
Medium
- A11.932 cm
- B119.32 cm
- C1.1932 cm
- D1193.2 cm
14.A circle has radius 5 cm. What is its circumference? (Use π = 3.14)
Easy
- A15.7 cm
- B31.4 cm
- C78.5 cm
- D25 cm
15.The diameter of a circle is 12 cm. What is its area? (Use π = 3.14)
Easy
- A37.68 cm²
- B75.36 cm²
- C113.04 cm²
- D452.16 cm²
16.A circle has radius 7 cm. What is its area in terms of π?
Easy
- A7π cm²
- B14π cm²
- C49π cm²
17.A circle has circumference 44 cm. What is its radius? (Use π = 22/7)
Easy
- A7 cm
- B14 cm
- C22 cm
- D28 cm
18.A sector of a circle has radius 6 cm and angle 60°. What is the area of the sector in terms of π?
Medium
- A6π cm²
- B12π cm²
- C36π cm²
- D2π cm²
19.A sector has radius 9 cm and angle 40°. What is the length of its arc in terms of π?
Medium
- Aπ cm
- B2π cm
- C3π cm
- D4π cm
20.A semicircle has diameter 10 cm. What is its perimeter? (Use π = 3.14)
Medium
- A15.7 cm
- B25.7 cm
- C31.4 cm
- D20.7 cm
21.A circle has area 154 cm². What is its radius? (Use π = 22/7)
Medium
- A3.5 cm
- B7 cm
- C14 cm
- D49 cm
22.A sector has area 30π cm² and radius 10 cm. What is the angle at the centre?
Hard
- A108°
- B120°
- C90°
- D72°
23.An arc of a circle has length 4π cm and radius 8 cm. What is the angle subtended at the centre?
Medium
- A90°
- B120°
- C60°
- D45°
24.The diagram shows a circle with centre O and radius 5 cm. Points A and B are on the circumference such that angle AOB = 72°. What is the area of the minor sector OAB? (Use π = 3.14)
Medium
- A15.7 cm²
- B31.4 cm²
- C7.85 cm²
- D3.14 cm²
25.What is the value of π (π) approximately?
Easy
- A3.14
- B3.41
- C2.71
- D1.62
26.A circle has radius 4 cm. What is the area of a sector with angle 90°? Give your answer in terms of π.
Medium
- A4π cm²
- B8π cm²
- C16π cm²
- D2π cm²
27.The sector of a circle has area 12π cm² and angle 120°. What is the radius of the circle?
Medium
- A6 cm
- B3 cm
- C9 cm
- D12 cm
28.A circle has diameter 14 cm. What is its circumference in terms of π?
Easy
- A7π cm
- B14π cm
- C28π cm
- D49π cm
29.The diagram shows a semicircle with diameter 12 cm. What is the area of the semicircle? (Use π = 3.14)
Medium
- A56.52 cm²
- B113.04 cm²
- C28.26 cm²
- D37.68 cm²
30.OAB is a sector of a circle with radius 9 cm and centre O. The angle at O is 30°. Calculate the area of the sector in terms of π.
Medium
- A(27/4)π cm²
- B(27/2)π cm²
- C27π cm²
- D54π cm²
31.What is the mathematical name for the straight line that touches a circle at exactly one point?
Easy
- ATangent
- BChord
- CDiameter
- DRadius
32.Which of the following is the correct formula for the area of a circle?
Easy
- AA = πr²
- BA = 2πr
- CA = πd
- DA = πr
33.π (π) is approximately equal to which of the following?
Easy
- A3.14
- B2.14
- C4.14
- D1.14
34.What is the value of π (π) approximately equal to?
Easy
- A3.14
- B2.14
- C4.14
- D1.14
35.Which of the following is the correct formula for the circumference of a circle?
Easy
- AC = πd
- BC = πr²
- CC = 2πd
- DC = πr
36.The diameter of a circle is half of its radius.
Easy
True or false?
37.A circle has a diameter of 14 cm. Using π = 22/7, what is its circumference?
Medium
- A44 cm
- B22 cm
- C88 cm
- D11 cm
38.A circle has a radius of 6 cm. What is its area in terms of π?
Easy
- A36π cm²
- B12π cm²
- C6π cm²
- D72π cm²
39.A sector of a circle has radius 8 cm and angle 45°. What is the area of the sector in terms of π?
Medium
- A8π cm²
- B4π cm²
- C16π cm²
- D32π cm²
40.The area of a sector is 12π cm² and the radius is 6 cm. What is the angle at the centre?
Medium
- A120°
- B60°
- C90°
- D180°
41.What is the formula for the area of a circle with radius r?
Easy
- AA = πr²
- BA = 2πr
- CA = πd
- DA = πr
42.Which of the following are correct formulas for the circumference of a circle? (Select all that apply)
Medium
- AC = πd
- BC = 2πr
- CC = πr²
- DC = 2πd
43.Match each quantity to its correct formula for a circle of radius r and diameter d.
Medium
- Area
- Circumference
- Arc length for angle θ
- πr²
- 2πr
- θ/360 × 2πr
44.A circle has a radius of 5 cm. What is its diameter in cm?
Easy
45.Complete the sentence: The distance around the edge of a circle is called its ______.
Easy
The distance around the edge of a circle is called its ____.
46.A circle has an area of 154 cm². Using π = 22/7, find its radius in cm.
Easy
47.A sector has radius 10 cm and angle 36°. Find the length of its arc. Give your answer in terms of π.
Medium
48.Calculate the area of the sector shown in the diagram. Give your answer in terms of π.
Medium
49.A sector has an arc length of 6π cm and a radius of 9 cm. Find the angle at the centre in degrees.
Medium
50.What is the name of the line that touches a circle at exactly one point?
Easy
51.A circle has a radius of 5 cm. What is its diameter in cm?
Easy
- A11 cm
- B10 cm
- C9 cm
- D12 cm
52.A circle has an area of 154 cm². Using π = 22/7, find its radius in cm.
Easy
- A8 cm
- B6 cm
- C7 cm
- D9 cm
53.A sector has radius 10 cm and angle 36°. Find the length of its arc. Give your answer in terms of π.
Medium
- A1 π cm
- B4 π cm
- C3 π cm
- D2 π cm
54.Calculate the area of the sector shown in the diagram. Give your answer in terms of π.
Medium
- A7 π cm²
- B5 π cm²
- C4 π cm²
- D6 π cm²
55.A sector has an arc length of 6π cm and a radius of 9 cm. Find the angle at the centre in degrees.
Medium
- A132 °
- B120 °
- C144 °
- D108 °

रिवीज़न नोट्स

## Circles, Arcs & Sectors ### Key Formulas - **Area of a circle**: \(A = \pi r^2\) - **Circumference**: \(C = \pi d\) or \(C = 2\pi r\) - **Arc length**: \(\frac{\theta}{360} \times 2\pi r\) - **Area of a sector**: \(\frac{\theta}{360} \times \pi r^2\) ### Definitions - **Radius** (\(r\)): distance from centre to circumference. - **Diameter** (\(d\)): \(d = 2r\). - **Circumference**: perimeter of a circle. - **Arc**: part of the circumference (minor/major). - **Sector**: region between two radii and an arc (minor/major). - **\(\pi\)**: constant ≈ 3.14159; ratio of circumference to diameter. ### Tips - Area is in square units (e.g., cm²), circumference in linear units (e.g., cm). - For semicircles: area = \(\frac{1}{2}\pi r^2\), perimeter = \(\pi r + 2r\). - To find radius from circumference: \(r = \frac{C}{2\pi}\). - To find diameter from circumference: \(d = \frac{C}{\pi}\). - For arcs/sectors: fraction = \(\frac{\theta}{360}\). - "In terms of \(\pi\)" means leave \(\pi\) in the answer (do not multiply). - Always check units (e.g., convert cm to m if needed). ### Common Question Types - Calculate area/circumference given radius or diameter. - Find radius/diameter given area or circumference. - Find arc length or sector area given angle and radius. - Perimeter of compound shapes (e.g., semicircle + diameter). - Shaded area problems (subtract areas).

Circle: Set of points equidistant from a centre.
Radius: Distance from centre to circumference; half the diameter.
Diameter: Distance across a circle through centre; twice the radius.
Circumference: Perimeter of a circle; \(C = \pi d\).
Pi (π): Constant ≈ 3.14159; ratio of circumference to diameter.
Arc: Part of the circumference; minor or major.
Sector: Region between two radii and an arc; minor or major.
Arc length: Length of an arc; \(\frac{\theta}{360} \times 2\pi r\).
Area of a sector: Area of a sector; \(\frac{\theta}{360} \times \pi r^2\).

फ्लैशकार्ड

What is the formula for the area of a circle?

A = πr²

What is the formula for the circumference of a circle?

C = πd or C = 2πr

How do you find the radius from the diameter?

Radius = diameter ÷ 2

How do you find the diameter from the radius?

Diameter = 2 × radius

A circle has diameter 6 cm. Calculate its area.

Area = π × (3)² = 9π cm² (or 28.27 cm²)

Calculate the circumference of a circle with radius 4.5 cm.

C = 2π × 4.5 = 9π cm (or 28.27 cm)

A circle has diameter 7 cm. Show that its circumference is 21.99 cm correct to 2 decimal places.

C = πd = π × 7 = 21.9911... ≈ 21.99 cm

A circle has circumference 56 mm. Find its radius.

r = C ÷ (2π) = 56 ÷ (2π) = 28/π ≈ 8.91 mm

The circumference of a pond is 4.25 m. Calculate its diameter in centimetres.

d = C ÷ π = 4.25 ÷ π ≈ 1.3528 m = 135.28 cm

What is the formula for the area of a sector?

Area = (θ/360) × πr²

What is the formula for the arc length of a sector?

Arc length = (θ/360) × 2πr

OAB is a sector with radius 9 cm and angle 30°. Calculate its area in terms of π.

Area = (30/360) × π × 9² = (1/12) × 81π = (27/4)π cm²

A semicircle has diameter 9 cm. Calculate its total perimeter.

Perimeter = (π × 9)/2 + 9 = 4.5π + 9 cm (or 23.14 cm)

A circular pool has diameter 8 m. Find its circumference and area.

Circumference = π × 8 = 8π m (≈ 25.13 m); Area = π × 4² = 16π m² (≈ 50.27 m²)

A circle has radius 11 cm. Find its circumference in terms of π.

C = 2π × 11 = 22π cm

In a circle of radius 11 cm, minor arc FH subtends angle 140° at centre. Calculate the length of minor arc FH.

Arc length = (140/360) × 2π × 11 = (7/18) × 22π = (77/9)π cm (≈ 26.88 cm)

A square is inscribed in a circle of radius 6 cm. Find the shaded area (area of circle minus area of square).

Area of circle = 36π cm²; side of square = 6√2 cm, area = 72 cm²; shaded area = 36π - 72 cm² (≈ 41.10 cm²)

Points P, Q, R on a semicircle of radius 8 cm, centre O, with angle POQ = 90°. Calculate the shaded area.

Shaded area = area of sector POQ - area of triangle POQ = (90/360)×π×8² - (1/2)×8×8 = 16π - 32 cm² (≈ 18.27 cm²)

A running track consists of a rectangle 120m by 60m with two semicircles on the shorter sides. Find the total length of the track.

Length = 2 × 120 + π × 60 = 240 + 60π m (≈ 428.50 m)

Shreva walks at 1.4 m/s around a track of length 428.50 m. How long does it take in minutes and seconds?

Time = 428.50 ÷ 1.4 ≈ 306.07 s = 5 minutes 6 seconds

A shape is made of two semicircles with same centre: large radius 7 cm, small radius 3 cm. Calculate the area of the shape.

Area = (1/2)π×7² - (1/2)π×3² = (1/2)π(49-9) = 20π cm² (≈ 62.83 cm²)

The area of a circle is 76.4 cm². Find its radius.

r = √(76.4/π) ≈ √24.32 ≈ 4.93 cm

A circle has diameter 38 mm. Calculate its circumference in centimetres.

C = π × 38 mm = 38π mm ≈ 119.38 mm = 11.938 cm

What is π (π)?

π is the ratio of a circle's circumference to its diameter, approximately 3.14159.

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