AFCAT Numerical Ability · Area and Perimeter

Circles, Rings, Sectors and Grazing

Circumference 2πr and area πr², and the fraction θ/360 of each for an arc or a sector; rings, wheels, clock hands and tethered animals are those formulas in a story.

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Why this matters

This is the chapter's biggest page and only one question on it is HARD. The marks go on small slips: a radius read as a diameter, a semicircle's perimeter without its straight edge, a ring worked as the square of the difference, or π taken as 3.14 when the options were worked with 22/7.

Concept 1 of 4: Circumference and area

The distance round any circle is a little over three times its diameter: that ratio is π. The area is π times the square of the radius. Know one of radius, diameter, circumference or area and the other three follow.

Definition

Circumference $C = 2\pi r = \pi d$ . Area $A = \pi r^2$ .
From the circumference: $r = \dfrac{C}{2\pi}$ . From the area: $r = \sqrt{\dfrac{A}{\pi}}$ .
A wheel covers one circumference per turn: distance = turns × $2\pi r$ .
A wire bent into a new shape keeps its length: a circle's circumference becomes a square's perimeter, or the other way round.
Use the value of π the question states. With 22/7, radii that are multiples of 7 come out whole.

Circle

C = 2\pi r = \pi d,\qquad A = \pi r^2

Worked example

A circle has a circumference of 88 cm. Find its radius and its area. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Area and Perimeter · Circles and Sectors

A bicycle wheel makes 5000 revolutions in moving 11 km. The diameter of the wheel is

Radius or diameter

A circumference of 88 gives a radius of 14 and a diameter of 28. Options usually carry both; read which one the question asks for.

Convert the distance first

A wheel's circumference is in cm and the road is often in km or m. 660 m is 66,000 cm; dividing 660 by 132 gives 5 turns instead of 500.

Drill 8 more on circumference and area

Concept 2 of 4: Rings, tracks and two circles

A ring is a big circle with a small one taken out of its middle. Its area is the difference of the two areas, and its width is the difference of the two radii. Two circles that touch have their centres a sum of radii apart, or a difference.

Definition

Ring with outer radius $R$ , inner $r$ : area $\pi(R^2 - r^2) = \pi(R + r)(R - r)$ .
A track of width $w$ round a circle of radius $r$ : $R = r + w$ , and the outer diameter is the inner diameter plus $2w$ .
Touching externally: distance between centres $= r_1 + r_2$ . Internally: $R - r$ .
Given the sum of two radii and the sum of their squares: $2r_1 r_2 = (r_1 + r_2)^2 - (r_1^2 + r_2^2)$ .

Area of a ring

A = \pi(R^2 - r^2) = \pi(R + r)(R - r)

Worked example

A circular park of radius 35 m has a track 7 m wide all round it, outside. Find the area of the track. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Area and Perimeter · Circles and Sectors

The area of ring is

418 \text{ cm}^2

. If the radius of the smaller circle is 6 cm, the radius of the bigger circle is

Difference of squares, not square of the difference

For radii 42 and 35 the ring is

\pi(42^2 - 35^2)

, which is 1694 with 22/7. Squaring the width,

\pi \times 7^2 = 154

, is far too small.

Width adds to the radius

A 7 m track round a circle of radius 14 m makes the outer radius 21 m and the outer diameter 42 m. Adding 7 to the diameter gives 35, which is wrong.

Drill 3 more on rings, tracks and two circles

Concept 3 of 4: Semicircles, arcs, sectors and clock hands

A sector is a slice of the circle, like a slice of a round cake. If its angle is θ, it is

\tfrac{\theta}{360}

of the whole, so its arc is that fraction of the circumference and its area that fraction of the area. A semicircle is the 180° slice; a quadrant the 90° one.

Definition

Arc $= \dfrac{\theta}{360} \times 2\pi r$ . Sector area $= \dfrac{\theta}{360} \times \pi r^2 = \tfrac12 \times \text{arc} \times r$ .
Perimeter of a sector = arc + $2r$ . For a semicircle that is $\pi r + 2r$ ; its area is $\tfrac12 \pi r^2$ .
Clock hands: the minute hand turns 6° a minute and sweeps a sector whose radius is its length. The hour hand turns half a degree a minute.
A sector of a ring (between two radii of the same angle) is the big sector less the small one.

Arc and sector

\text{arc} = \dfrac{\theta}{360} \times 2\pi r,\qquad \text{sector} = \dfrac{\theta}{360} \times \pi r^2

Worked example

A sector of a circle of radius 21 cm has an angle of 120°. Find its arc, area and perimeter. (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Area and Perimeter · Circles and Sectors

The minute hand of a clock is 12 cm long. The area of the face of the clock described by the minute hand in 35 min is

A semicircle's edge includes its diameter

The perimeter of a semicircle of radius 14 is

44 + 28 = 72

. The curved part alone, 44, is the arc, not the perimeter.

Minutes into degrees

In 20 minutes the minute hand turns

20 \times 6° = 120°

, a third of the circle. Using

\tfrac{20}{360}

of the circle treats minutes as degrees.

Drill 6 more on semicircles, arcs, sectors and clock hands

Concept 4 of 4: Animals tied with a rope

A tethered animal can reach every point within a rope's length of the peg, so it grazes a circle. Walls and fences cut that circle down to the sector that lies on the animal's side of them.

Definition

Tied to a pole in the open: a full circle, radius = rope.
Tied at a corner inside a field whose angle there is θ: a sector of angle θ.
Tied at a corner outside a building or a fenced triangle with angle θ there: the animal cannot go in, so it grazes a sector of $360° - \theta$ .
Animals at the three corners of a triangle with equal ropes graze sectors whose angles add to 180°: half a circle in all. At the four corners of a rectangle the quadrants make one full circle.
These hold only while the rope is shorter than the walls beside the peg. A longer rope wraps round the next corner and adds more.

Grazed area

\text{area} = \dfrac{\theta}{360} \times \pi\,(\text{rope})^2

Worked example

A goat is tied at one corner of a square field of side 20 m with a rope 14 m long. What area can it graze, and what percentage of the field is that? (Use

\pi = \tfrac{22}{7}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Area and Perimeter · Circles and Sectors

A grassy field has the shape of an equilateral triangle of side 6 m. A horse is tied to one of its vertices with a rope of length 4.2 m. The percentage of the total area of the field which is available for grazing is best approximated by

Only the angle at the peg

At the corner of a square field the goat grazes a quadrant,

\tfrac14

of the circle. Taking the whole circle counts grass on the far side of the fence.

Check the rope against the walls

The sector rule needs a rope shorter than the sides beside the peg. If the rope is longer than a side, the animal walks round that corner and grazes an extra sector there.

Drill 2 more on animals tied with a rope

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Circumference and area
Circle
$C = 2\pi r = \pi d,\qquad A = \pi r^2$
Rings, tracks and two circles
Area of a ring
$A = \pi(R^2 - r^2) = \pi(R + r)(R - r)$
Semicircles, arcs, sectors and clock hands
Arc and sector
$\text{arc} = \dfrac{\theta}{360} \times 2\pi r,\qquad \text{sector} = \dfrac{\theta}{360} \times \pi r^2$
Animals tied with a rope
Grazed area
$\text{area} = \dfrac{\theta}{360} \times \pi\,(\text{rope})^2$

Watch out for (8)

Radius or diameter
→ Circumference and area
Convert the distance first
→ Circumference and area
Difference of squares, not square of the difference
→ Rings, tracks and two circles
Width adds to the radius
→ Rings, tracks and two circles
A semicircle's edge includes its diameter
→ Semicircles, arcs, sectors and clock hands
Minutes into degrees
→ Semicircles, arcs, sectors and clock hands
Only the angle at the peg
→ Animals tied with a rope
Check the rope against the walls
→ Animals tied with a rope

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23 questions from the bank, with cart and Word export.

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