Circles: circumference and area

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Apuntes de la lección

Parts of a Circle

Centre: the point that is the same distance from every point on the circle.
Radius (r): a line segment from the centre to any point on the circle; also its length.
Diameter (d): a line segment through the centre with both endpoints on the circle; the longest chord, and d = 2r.
Circumference: the distance all the way around the circle.
Chord: a line segment with both endpoints on the circle; the diameter is the longest possible chord.
Tangent: a straight line that touches the circle at exactly one point.
Arc: any connected part of the circle.
Sector: the region bounded by two radii and an arc; segment: the region bounded by a chord and an arc.

The Number π

π (π) is the ratio of a circle's circumference to its diameter: π = C / d.
This ratio is the same for every circle, whatever its size.
π is approximately 3.14159… and is irrational, so it never ends or repeats.
In calculations, use the π button on your calculator, or 3.14 as a rough approximation.

Circumference

Circumference from diameter: C = πd.
Circumference from radius: C = 2πr, because d = 2r.
Example: a circle with r = 5 cm has C = 2 × π × 5 = 10π cm ≈ 31.4 cm.
Leave answers in terms of π when asked for an exact value; otherwise round as instructed.

Area

Area of a circle: A = πr².
Square the radius first, then multiply by π.
Example: a circle with r = 5 cm has A = π × 5² = 25π cm² ≈ 78.5 cm².
If you are given the diameter, halve it to get the radius before using A = πr².

Semicircles and Quarter Circles

A semicircle is half a circle: its arc length is half the circumference, C/2 = πr.
Area of a semicircle: A = (πr²)/2.
A quarter circle is one quarter of a circle: arc length = (2πr)/4 = (πr)/2.
Area of a quarter circle: A = (πr²)/4.
For a semicircle or quarter circle, remember to add the straight edges when finding the perimeter of the whole shape.

Arc Length and Sector Area

An arc or sector is a fraction of the whole circle, set by the angle at the centre.
Fraction of the circle = angle/360°.
Arc length = (angle/360°) × πd = (angle/360°) × 2πr.
Sector area = (angle/360°) × πr².
Example: a sector with angle 90° is 90/360 = 1/4 of the circle, so its area is (πr²)/4.

Exact Answers and Rounding

An answer in terms of π keeps the symbol π, e.g. 10π cm or 25π cm².
A rounded answer replaces π with a decimal, e.g. 31.4 cm or 78.5 cm².
Check whether the question asks for an exact answer or a decimal before you finish.
Give units with every answer: cm, cm², m, m² and so on.

Key Facts to Remember

d = 2r and r = d/2.
C = πd = 2πr.
A = πr².
A circle is the boundary; the region inside it is called a disc.
π is the same value for every circle, which is why these formulas work for all circles.

Diapositivas

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Preguntas de práctica

Vista previa gratis — 8 de 63 preguntas. Regístrate para verlas todas.

1.Which term means the distance around a circle?
Easy
- ARadius
- BDiameter
- CCircumference
- DArea
2.Which formula gives the circumference of a circle with radius r?
Easy
- AC = \π r²
- BC = 2\π r
- CC = \π d²
- DC = \frac{r}{2}
3.Which formula gives the area of a circle with radius r?
Easy
- AA = 2\π r
- BA = \π d
- CA = \π r²
- DA = \frac{1}{2}\π r²
4.What is π defined as?
Medium
- AThe ratio of area to radius
- BThe ratio of circumference to diameter
- CThe ratio of diameter to radius
- DThe ratio of circumference to radius
5.Which of the following are parts of a circle? (select all that apply)
Medium
- AChord
- BTangent
- CSector
- DHypotenuse
- EDiameter
6.The diameter of a circle is twice the length of its radius.
Easy
True or false?
7.A tangent is a straight line that crosses a circle at two points.
Easy
True or false?
8.Match each circle term to its definition.
Medium
- Radius
- Chord
- Tangent
- Arc
- A straight line touching the circle at exactly one point
- A line segment from the centre to a point on the circle
- A line segment joining two points on the circle
- A connected part of the circle's edge

Más preguntas

9.Order these circle measures from smallest to largest for a circle with radius 5 cm. Use π ≈ 3.14.
Easy
- Diameter = 10 cm
- Circumference ≈ 31.4 cm
- Area ≈ 78.5 cm²
10.Match each circle term to its correct description.
Easy
- Arc
- Sector
- Segment
- Annulus
- a region bounded by a chord and the arc between the chord's endpoints
- any connected part of the circumference
- a ring-shaped region between two concentric circles
- a region bounded by two radii and the arc between them
11.For a circle, which of the following statements are true? (select all that apply)
Medium
- AThe circumference is the distance around the circle.
- BThe diameter is the longest possible chord.
- CA sector is bounded by a chord and an arc.
- DThe radius is half the diameter.
- EA tangent meets the circle at two points.
12.Order these circle measures from smallest to largest for a circle with radius 10 cm. Use π ≈ 3.14.
Medium
- radius
- diameter
- circumference
- area
13.A circle has diameter 12 cm. What is its circumference? Use π ≈ 3.14.
Medium
- A37.68 cm
- B18.84 cm
- C113.04 cm
- D75.36 cm
14.A circle has radius 5 cm. What is its area? Give your answer in terms of π.
Medium
- A25π cm²
- B10π cm²
- C50π cm²
- D5π cm²
15.A tangent is a straight line that touches a circle at exactly one point.
Easy
True or false?
16.A circle has radius 4 cm. What is its circumference? Use π ≈ 3.14.
Easy
- A25.12 cm
- B12.56 cm
- C50.24 cm
- D8.00 cm
17.A circle has diameter 20 cm. What is its area? Use π ≈ 3.14.
Medium
- A314 cm²
- B1256 cm²
- C62.8 cm²
- D31.4 cm²
18.A circle has circumference 50 cm. Which calculation gives its radius, correct to 1 decimal place? Use π ≈ 3.142.
Medium
- A50 ÷ (2 × 3.142) = 8.0 cm
- B50 × 2 ÷ 3.142 = 31.8 cm
- C50 ÷ 3.142 = 15.9 cm
- D50 × 3.142 ÷ 2 = 78.6 cm
19.The area of a circle is measured in square units, such as cm².
Easy
True or false?
20.If the radius of a circle is doubled, its area also doubles.
Easy
True or false?
21.A quarter circle has radius 6 cm. What is its area? Use π ≈ 3.14.
Medium
- A28.26 cm²
- B113.04 cm²
- C56.52 cm²
- D9.42 cm²
22.A circle has circumference 44 cm. Which calculation gives its diameter, correct to 1 decimal place? Use π ≈ 3.142.
Medium
- Ad = 44 ÷ 3.142 = 14.0 cm
- Bd = 44 × 3.142 = 138.2 cm
- Cd = 44 ÷ (2 × 3.142) = 7.0 cm
- Dd = 44 ÷ 2 = 22.0 cm
23.A circular pond has radius 3 m. Which expression gives the exact distance around its edge?
Medium
- A6π m
- B9π m
- C3π m
- D2π m
24.A circle has radius 4 cm. Which value is its area, correct to 1 decimal place? Use π ≈ 3.142.
Medium
- A50.3 cm²
- B25.1 cm²
- C12.6 cm²
- D201.1 cm²
25.A circle has radius 9 cm. Which of the following statements are true? Use π ≈ 3.142. (select all that apply)
Medium
- AIts diameter is 18 cm.
- BIts circumference is about 56.6 cm.
- CIts area is about 254.5 cm².
- DIts area is about 28.3 cm².
- EIts diameter is 4.5 cm.
26.The circumference of a circle is the distance around its edge.
Easy
True or false?
27.If the diameter of a circle is doubled, its area is doubled.
Medium
True or false?
28.Match each circle measure for a circle of radius 5 cm to its value. Use π ≈ 3.14.
Medium
- Diameter
- Circumference
- Area
- 31.4 cm
- 10 cm
- 78.5 cm²
29.Order these steps for finding the area of a circle from its circumference, starting with the first step.
Medium
- Divide the circumference by 2π to get the radius
- Write down the circumference formula C = 2πr
- Substitute the radius into A = πr²
- Round the area to the required accuracy
30.A circular tabletop has a diameter of 1.4 m. Work out its circumference, correct to 2 decimal places. Use π = 3.142.
Medium
- A4.40 m
- B2.20 m
- C6.16 m
- D1.54 m
31.A running track is made from two straight sections and two semicircular ends. Each semicircular end has a diameter of 60 m. Work out the total length of the two curved ends, correct to 1 decimal place. Use π = 3.142.
Hard
- A188.5 m
- B94.3 m
- C377.0 m
- D60.0 m
32.A circular garden pond has a circumference of 18 m. Work out its area, correct to 2 decimal places. Use π = 3.142.
Hard
- A25.78 m²
- B5.73 m²
- C103.13 m²
- D32.17 m²
33.A semicircular window has a straight edge (diameter) of 80 cm. Work out the area of the window, correct to 1 decimal place. Use π = 3.142.
Medium
- A2513.6 cm²
- B5027.2 cm²
- C1256.8 cm²
- D628.4 cm²
34.A circular pizza of diameter 30 cm is cut into 8 equal slices. Work out the area of one slice, correct to 1 decimal place. Use π = 3.142.
Hard
- A88.4 cm²
- B707.0 cm²
- C176.7 cm²
- D353.5 cm²
35.A circular pond has a diameter of 8 m. A fence is built around the edge. Fence panels are 2 m long and can only be bought as whole panels. How many panels are needed? Use π = 3.142.
Hard
- A13
- B12
- C14
- D7
36.The minute hand of a clock is 12 cm long. Work out the distance travelled by the tip of the hand in 20 minutes, correct to 1 decimal place. Use π = 3.142.
Hard
- A50.3 cm
- B75.4 cm
- C25.1 cm
- D12.0 cm
37.A circle with a larger radius always has a larger area than a circle with a smaller radius.
Medium
True or false?
38.Which term describes the point that is the same distance from every point on a circle?
Easy
- ACentre
- BRadius
- CDiameter
- DTangent
39.Which term describes a straight line that meets a circle at exactly two points?
Easy
- ASecant
- BTangent
- CChord
- DArc
40.A circular disc has radius 6 cm. Which value is its diameter?
Medium
- A3 cm
- B6 cm
- C12 cm
- D36 cm
41.Name the line segment whose endpoints lie on the circle and which passes through the centre.
Easy
42.A circle has radius 7 cm. Find its diameter in cm.
Medium
43.Complete the formula for the area of a circle.
Medium
Area = π × ____²
44.A circle has radius 7 cm. Find its circumference in cm, correct to 1 decimal place. Use π ≈ 3.142.
Medium
45.A semicircle has diameter 16 cm. Find its area in cm², correct to 1 decimal place. Use π ≈ 3.142.
Medium
46.Complete the statement about a quarter circle of radius 20 cm. Use π ≈ 3.14.
Easy
The arc length of a quarter circle of radius 20 cm is ____ cm, and its area is ____ cm².
47.The circumference of a circle is 100 cm. Find its radius in cm, correct to 1 decimal place. Use π ≈ 3.142.
Hard
48.Name the circle term for the region bounded by two radii and the arc between them.
Medium
49.A circle has radius 9 cm. Find its circumference in cm, correct to 1 decimal place. Use π ≈ 3.142.
Medium
50.A circle has diameter 25 cm. Find its area in cm², correct to 1 decimal place. Use π ≈ 3.142.
Medium
51.Complete the formula for the circumference of a circle, using d for the diameter.
Easy
C = ____ × d
52.Complete the formula for the area of a circle, using r for the radius.
Easy
A = ____
53.Explain why the diameter is the longest chord of a circle.
Medium
54.Explain why giving an answer in terms of π can be more exact than giving a rounded decimal value. Use the circumference of a circle with radius 3 cm as an example.
Hard
55.Explain why the circumference of a circle is always less than 4 times its diameter, but more than 3 times its diameter.
Hard
56.A student says the area of a circle with radius 5 cm is 31.4 cm². Explain the mistake and give the correct area in terms of π.
Hard
57.Explain why π is described as a constant and why its decimal never terminates or repeats.
Hard
58.A circle has radius 7 cm. Find its diameter in cm.
Medium
- A15 cm
- B13 cm
- C14 cm
- D16 cm
59.A circle has radius 7 cm. Find its circumference in cm, correct to 1 decimal place. Use π ≈ 3.142.
Medium
- A44 cm
- B40 cm
- C48 cm
- D52 cm
60.A semicircle has diameter 16 cm. Find its area in cm², correct to 1 decimal place. Use π ≈ 3.142.
Medium
- A90.4 cm²
- B120.7 cm²
- C100.5 cm²
- D110.6 cm²
61.The circumference of a circle is 100 cm. Find its radius in cm, correct to 1 decimal place. Use π ≈ 3.142.
Hard
- A16.9 cm
- B14.9 cm
- C17.9 cm
- D15.9 cm
62.A circle has radius 9 cm. Find its circumference in cm, correct to 1 decimal place. Use π ≈ 3.142.
Medium
- A62.3 cm
- B56.6 cm
- C68 cm
- D50.9 cm
63.A circle has diameter 25 cm. Find its area in cm², correct to 1 decimal place. Use π ≈ 3.142.
Medium
- A589.1 cm²
- B441.8 cm²
- C490.9 cm²
- D540 cm²

Apuntes de repaso

## Circles: Circumference and Area ### Parts of a circle - **Centre**: the point equidistant from all points on the circle. - **Radius** (r): a line segment from the centre to any point on the circle; its length is half the diameter. - **Diameter** (d): a chord that passes through the centre; the longest chord, and d = 2r. - **Circumference** (C): the distance around the circle (one full circuit). - **Chord**: a line segment whose endpoints lie on the circle. - **Tangent**: a straight line that touches the circle at exactly one point. - **Arc**: any connected part of the circle. - **Sector**: a region bounded by two radii and an arc. - **Segment**: a region bounded by a chord and an arc. - **Semicircle**: an arc from the endpoints of a diameter; the region is a half-disc. ### π (pi) π is the ratio of a circle's circumference to its diameter: π = C / d. It is the same for every circle. Use π ≈ 3.14 or the π button on your calculator unless asked for an exact answer. ### Key formulas - Circumference: C = πd = 2πr - Area: A = πr^2 ### Worked examples **Example 1** — r = 5 cm. Find C and A. C = 2πr = 2 × π × 5 = 10π ≈ 31.4 cm A = πr^2 = π × 5^2 = 25π ≈ 78.5 cm^2 **Example 2** — d = 12 m. Find the area. First r = d / 2 = 6 m, then A = πr^2 = π × 6^2 = 36π ≈ 113.1 m^2 ### Semicircles and quarter circles Treat them as fractions of a full circle. - Semicircle: arc length = (1/2)πd; area = (1/2)πr^2 - Quarter circle: arc length = (1/4)πd; area = (1/4)πr^2 ### Arc length and sector area An arc or sector is a fraction of the whole circle, set by the angle θ at the centre (out of 360°). - Arc length = (θ / 360) × πd - Sector area = (θ / 360) × πr^2 ### Exam tips - Check whether you are given r or d — halve d before using r^2. - Area is in square units (cm^2, m^2); circumference is in units (cm, m). - Leave answers in terms of π when asked for an exact value; otherwise round as instructed. - Never mix up the formulas: circumference uses r (or d) once; area uses r squared.

Centre: The point equidistant from all points on the circle.
Radius (r): A line segment joining the centre of a circle to any point on the circle; its length is half the diameter.
Diameter (d): A line segment whose endpoints lie on the circle and which passes through the centre; the longest chord, with length d = 2r.
Circumference (C): The length of one circuit along the circle, or the distance around the circle; C = πd = 2πr.
Chord: A line segment whose endpoints lie on the circle, dividing the circle into two segments.
Tangent: A straight line that has exactly one point in common with the circle (it touches the circle at that point).
Arc: Any connected part of a circle.
Sector: A region bounded by two radii of equal length with a common centre and either of the two possible arcs.
Segment: A region bounded by a chord and one of the arcs connecting the chord's endpoints.
Semicircle: One of the two arcs determined by the endpoints of a diameter; the region bounded by a diameter and one arc is a half-disc.
π (pi): The ratio of a circle's circumference to its diameter, π = C / d; approximately 3.14.
Area of a circle: A = πr^2, where r is the radius.

Tarjetas

What is the definition of a circle?

A circle is the set of all points in a plane that are a given distance from a fixed point called the centre.

What is the radius of a circle?

The radius is the distance from the centre to any point on the circle. It is half the diameter.

What is the diameter of a circle?

The diameter is the distance across the circle through the centre. It is twice the radius.

What is the circumference of a circle?

The circumference is the distance around the circle (the perimeter).

What is π?

π is the ratio of a circle's circumference to its diameter. It is the same for every circle.

Write the formula for the circumference of a circle in terms of its diameter.

C = πd, where C is the circumference and d is the diameter.

Write the formula for the circumference of a circle in terms of its radius.

C = 2πr, where C is the circumference and r is the radius.

Write the formula for the area of a circle.

A = πr², where A is the area and r is the radius.

What is a chord of a circle?

A chord is a line segment whose endpoints lie on the circle.

What is a tangent to a circle?

A tangent is a straight line that touches the circle at exactly one point.

What is an arc of a circle?

An arc is any connected part of the circumference of a circle.

What is a sector of a circle?

A sector is the region bounded by two radii and the arc between them.

What is a segment of a circle?

A segment is the region bounded by a chord and the arc connecting the chord's endpoints.

How do you find the area of a semicircle?

A semicircle is half a circle, so its area is half of πr²: A = (1/2)πr².

How do you find the perimeter of a semicircle?

The perimeter is half the circumference plus the diameter: P = πr + 2r (or P = (1/2)πd + d).

How do you find the area of a quarter circle?

A quarter circle is one quarter of a circle, so its area is A = (1/4)πr².

How do you find the arc length of a sector with angle θ°?

The arc length is the fraction θ/360 of the circumference: arc length = (θ/360) × 2πr.

How do you find the area of a sector with angle θ°?

The sector area is the fraction θ/360 of the circle's area: sector area = (θ/360) × πr².

What is the relationship between the diameter and the radius?

The diameter is twice the radius: d = 2r, so r = d/2.

What is the difference between a circle and a disc?

A circle is just the boundary (the set of points at a fixed distance from the centre); a disc is the region inside the circle.

What is an annulus?

An annulus is the ring-shaped region between two concentric circles.

What is a secant line?

A secant is a straight line that intersects a circle at two points.

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