Formulas · Mathematics

Circles

8 formulas, each with worked examples. Revision sheet · Practise these formulas

Class 9

Length of a chord from its distance to the centre

l = 2 r^{2} - d^{2}

What each letter means
$l$	the length of the chord (length)
$r$	the radius of the circle (length)
$d$	the distance from the centre to the chord (along the perpendicular) (length)

The perpendicular from the centre cuts a chord exactly in half, making a right triangle with the radius as its longest side. Pythagoras then gives half the chord, and doubling gives the chord.

Why it works

Join the centre to one end of the chord (that line is a radius, r) and drop the perpendicular to the chord (length d). The perpendicular bisects the chord, so the triangle has legs d and l ÷ 2 and hypotenuse r: (l ÷ 2)² + d² = r².

When to use it

Any two of chord length, radius and distance from the centre are known and you want the third; comparing chords (the longer chord is closer to the centre).

How to use it

Square the radius and the distance, and subtract.
Take the square root: that is half the chord.
Double it.

To remember: Half chord, distance, radius: a right triangle.

Other forms

Rearranged $d = r^{2} - \frac{l ^{2}}{4}$ How far a chord of known length is from the centre.
Rearranged $r = d^{2} + \frac{l ^{2}}{4}$ The radius, from a chord and its distance.
Special case $l = 2 r$ When d = 0: the chord passes through the centre, so it is a diameter.

Worked examples

Story

A round pizza of radius 13 cm is cut in a straight line 5 cm from its centre. How long is the cut?

$l = 2 1 3^{2} - 5^{2}$
$= 2 169 - 25 = 2144 = 24$
The cut is 24 cm long.

Answer: 24

Picture

In a circle of radius 5 cm, two parallel chords of lengths 8 cm and 6 cm lie on opposite sides of the centre. How far apart are they?

$d_{1} = 5^{2} - 4^{2} = 9 = 3$
$d_{2} = 5^{2} - 3^{2} = 16 = 4$
On opposite sides of the centre, the distances add: 3 + 4 = 7 cm.

Answer: 7 cm

Direct

A circle has radius 5 cm. A chord is 3 cm from the centre. How long is the chord?

$l = 2 5^{2} - 3^{2}$
$= 2 25 - 9 = 216 = 2 \times 4 = 8$

Answer: 8

Reverse

A chord of length 24 cm is drawn in a circle of radius 13 cm. How far is it from the centre?

Try it first, then show the working

$d = 1 3^{2} - 1 2^{2}$
$= 169 - 144 = 25 = 5$

Answer: 5

Exam

A chord of length 16 cm is 6 cm from the centre of a circle. Find the radius.

Try it first, then show the working

Half the chord is 8 cm.
$r = 6^{2} + 8^{2}$
$= 36 + 64 = 100 = 10$

Answer: 10

Common mistake: Forgetting to double at the end (the right triangle only holds half the chord), or adding r² and d² instead of subtracting.

Class 9

Angle at the centre is twice the angle at the circle

θ = 2 ϕ

What each letter means
$θ$	the angle an arc makes at the centre of the circle (degrees)
$ϕ$	the angle the same arc makes at any point on the rest of the circle (degrees)

An arc makes an angle at the centre that is exactly twice the angle it makes at any point on the remaining part of the circle.

Why it works

Join the point on the circle to the centre and carry the line on. It splits the figure into two isosceles triangles (two sides of each are radii), so their base angles are equal. Each outside angle at the centre is the sum of the two equal base angles, twice one of them; adding the two parts gives θ = 2φ.

When to use it

Finding angles in a circle when the centre is involved; the angle in a semicircle (90°); showing that angles on the same arc are equal.

How to use it

Find the arc both angles stand on.
The angle at the centre is double the angle at the circle; the angle at the circle is half the angle at the centre.
Make sure the point on the circle is on the remaining part of the circle, not on the arc itself.

To remember: The centre sees twice as much.

Other forms

Rearranged $ϕ = \frac{θ}{2}$ The angle at the circle, from the angle at the centre.
Special case $ϕ = 90$ When the arc is a semicircle (θ = 180°): the angle in a semicircle is a right angle.
Same formula, another formAngles standing on the same arc, at points on the same side, are all equal, since each is half the same angle at the centre.

Worked examples

Story

A statue stands at the centre of a circular path. From the statue, the two ends of a bench on the path are 100° apart. What angle do the bench ends make for someone standing on the far side of the path?

$ϕ = \frac{100}{2} = 50$
The person sees the bench ends 50° apart.

Answer: 50

Picture

AB is a diameter of a circle and C is any other point on the circle. Find angle ACB.

A diameter makes a straight angle, 180°, at the centre.
$ϕ = \frac{180}{2} = 90$

Answer: 90

Direct

An arc makes an angle of 35° at a point on the circle. What angle does it make at the centre?

$θ = 2 \times 35 = 70$

Answer: 70

Reverse

A chord makes an angle of 42° at a point on the major arc. What angle does it make at the centre, and what kind of angle is that?

Try it first, then show the working

$θ = 2 \times 42 = 84$
84° is less than 90°, so it is an acute angle.

Answer: 84

Exam

O is the centre of a circle, and C is a point on the major arc AB. If angle AOB = (3x + 10)° and angle ACB = (x + 20)°, find x and both angles.

Try it first, then show the working

$3 x + 10 = 2 (x + 20)$
$3 x + 10 = 2 x + 40$
$x = 30$
Angle AOB = 3 × 30 + 10 = 100° and angle ACB = 30 + 20 = 50°. Check: 100 = 2 × 50.

Answer: x = 30; angle AOB = 100°, angle ACB = 50°

Common mistake: Halving when you should double (the angle at the centre is the bigger one), or using a point that lies on the arc itself.

Class 9

Circumference of a circle

C = 2 π r

What each letter means
$C$	the circumference: the distance all the way round the circle (length)
$r$	the radius (length)

For every circle, the distance round it divided by the distance across it is the same number, π (about 3.14, or 22/7). So the distance round is π times the diameter, which is 2πr.

Why it works

All circles are the same shape, only scaled: doubling the radius doubles every length, including the distance round. So circumference ÷ diameter never changes; that fixed ratio is called π, and C = π × 2r.

When to use it

Finding the distance round a circular object (a wheel, a track, a plate), how far a wheel rolls in one turn, or the radius from a measured distance round.

How to use it

Find the radius (half the diameter).
Multiply 2 × π × radius.
Leave the answer in terms of π, or use π ≈ 22/7 or 3.14 for a number.

To remember: Round is π times across.

Other forms

Same formula, another formC = πd, where d is the diameter, because d = 2r.
Rearranged $r = \frac{C}{2 π}$ The radius, from a measured distance round.

Worked examples

Story

A bicycle wheel has a radius of 35 cm. How far does the bicycle move when the wheel turns once?

One turn rolls out one circumference.
$C = 2 \times π \times 35 = 70 π$
That is about 220 cm, or 2.2 m, using π ≈ 22/7.

Answer: 70π cm

Picture

A protractor is a semicircle of radius 7 cm. Find the length of its whole edge (the curved part and the straight part).

The curved part is half a circumference: π × 7 = 7π cm.
The straight part is a diameter: 14 cm.
The edge is 7π + 14 cm, about 22 + 14 = 36 cm.

Answer: 7π + 14 cm, about 36 cm

Direct

Find the circumference of a circle of radius 7 cm.

$C = 2 \times π \times 7 = 14 π$
That is about 44 cm, using π ≈ 22/7.

Answer: 14π cm

Reverse

A circular running track is 440 m round. Find its radius, using π ≈ 22/7.

Try it first, then show the working

r = C ÷ 2π
$\frac{440}{2 \times \frac{22}{7}} = \frac{440 \times 7}{44} = 70$

Answer: 70 m

Exam

A wire bent into a square of side 11 cm is straightened and bent into a circle. Find the radius of the circle (π ≈ 22/7).

Try it first, then show the working

The wire is 4 × 11 = 44 cm long, and that becomes the circumference.
$2 \times \frac{22}{7} \times r = 44$
$r = \frac{44 \times 7}{44} = 7$

Answer: 7 cm

Common mistake: Using the diameter in place of the radius in 2πr (that doubles the answer), or mixing up circumference (a length) with area (πr², a square measure).

Class 9

Length of an arc

l = 2 π r \times \frac{θ}{360}

What each letter means
$l$	the length of the arc (length)
$r$	the radius (length)
$θ$	the angle the arc makes at the centre (degrees)

An arc is a fraction of the whole circle, and that fraction is its angle out of 360°. So its length is the same fraction of the circumference.

Why it works

Turning the circle about its centre moves any arc onto another arc with the same angle, so equal angles give equal arcs. The full 360° gives the whole circumference 2πr, so each degree gives 2πr ÷ 360, and θ degrees give 2πr × θ ÷ 360.

When to use it

The length of a curved edge: a semicircle (180°), a quarter circle (90°), the path of the tip of a clock hand, a curved fence or road.

How to use it

Find the circumference 2πr.
Write the angle as a fraction of 360°.
Multiply the two.

To remember: The arc is its angle's share of the whole circle.

Other forms

Special case $l = π r$ When a semicircle (θ = 180°).
Special case $l = \frac{π r}{2}$ When a quarter circle (θ = 90°).
Rearranged $θ = \frac{360 l}{2 π r}$ The angle, from the arc length and the radius.

Worked examples

Story

The minute hand of a clock is 14 cm long. How far does its tip travel in 15 minutes?

In 15 minutes the hand turns a quarter of the way round: 90°.
$l = 2 π \times 14 \times \frac{90}{360} = 7 π$
That is about 22 cm, using π ≈ 22/7.

Answer: 7π cm

Picture

A flower bed is a sector of a circle of radius 21 m with an angle of 120° at the centre. How long is the fence all round it?

The curved edge: 2π × 21 × 120 ÷ 360 = 14π m.
The two straight edges are radii: 21 + 21 = 42 m.
The fence is 14π + 42 m, about 44 + 42 = 86 m.

Answer: 14π + 42 m, about 86 m

Direct

Find the length of an arc of a circle of radius 6 cm that makes an angle of 60° at the centre.

$l = 2 π \times 6 \times \frac{60}{360}$
The fraction is 60/360 = 1/6 of 12π, which is 2π cm, about 6.3 cm.

Answer: 2π cm

Reverse

An arc of a circle of radius 10 cm is 5π cm long. What angle does it make at the centre?

Try it first, then show the working

$2 π \times 10 \times \frac{θ}{360} = 5 π$
$\frac{20 θ}{360} = 5$
$θ = 90$

Answer: 90°

Exam

The minute hand of a clock is 7 cm long. How far does its tip travel in 20 minutes? Use π ≈ 22/7.

Try it first, then show the working

In 20 minutes the hand turns 20/60 of 360°, which is 120°.
$2 \times \frac{22}{7} \times 7 \times \frac{120}{360} = \frac{44}{3}$
That is 14 2/3 cm, about 14.7 cm.

Answer: 44/3 cm, about 14.7 cm

Common mistake: Dividing by 180 instead of 360, or using the angle in the wrong place (it is θ out of 360, so a small angle gives a short arc).

Class 9

Area of a circle

A = π r^{2}

What each letter means
$A$	the area of the circle (area)
$r$	the radius (length)

The area of a circle is π times the square of its radius.

Why it works

Cut the circle into many thin slices, like a pizza, and lay them in a row, pointing up and down in turn. The row is almost a rectangle: its long side is half the circumference, πr, and its short side is the radius r. The more slices, the closer it gets, so the area is πr × r = πr².

When to use it

The space inside a circle: a round field, a coin, the region a tethered animal can graze, a ring between two circles.

How to use it

Find the radius (half the diameter).
Square it.
Multiply by π, and give the answer in square units.

To remember: Area is π r squared: square units need a square.

Other forms

Same formula, another formA = πd²/4, where d is the diameter, because r = d/2.
Rearranged $r = \frac{A}{π}$ The radius of a circle with a given area.

Worked examples

Story

A cow is tied to a peg with a rope 14 m long. How much ground can it graze?

It can reach every point within 14 m of the peg: a circle of radius 14 m.
$A = π \times 1 4^{2} = 196 π$
That is about 616 m², using π ≈ 22/7.

Answer: 196π m²

Picture

A circular path runs between two circles with the same centre, of radii 10 m and 6 m. Find the area of the path.

Path area = big circle − small circle.
$π \times 1 0^{2} - π \times 6^{2} = 100 π - 36 π = 64 π$
That is about 201 m².

Answer: 64π m²

Direct

Find the area of a circle of radius 7 cm.

$A = π \times 7^{2} = 49 π$
That is about 154 cm², using π ≈ 22/7.

Answer: 49π cm²

Reverse

A circle has an area of 154 cm². Find its radius, using π ≈ 22/7.

Try it first, then show the working

r² = 154 ÷ (22/7) = 154 × 7 ÷ 22 = 49
$r = 49 = 7$

Answer: 7 cm

Exam

The circumference of a circle is 88 cm. Find its area, using π ≈ 22/7.

Try it first, then show the working

From 2πr = 88: r = 88 ÷ (2 × 22/7) = 14 cm.
$\frac{22}{7} \times 1 4^{2} = \frac{22}{7} \times 196 = 616$

Answer: 616 cm²

Common mistake: Using the diameter instead of the radius (that makes the answer 4 times too big), or computing 2πr (a length) instead of πr².

Class 9

Area of a sector

A = π r^{2} \times \frac{θ}{360}

What each letter means
$A$	the area of the sector (a slice of the circle) (area)
$r$	the radius (length)
$θ$	the angle of the sector at the centre (degrees)

A sector is a slice of a circle, and its share of the whole area is its angle out of 360°.

Why it works

Rotating the circle carries any sector onto another with the same angle, so equal angles give equal areas. The full 360° is the whole area πr², so θ degrees give πr² × θ ÷ 360.

When to use it

The area of a slice: a pizza slice, the region a wiper or a sprinkler covers, a fan-shaped plot; and, by taking away a triangle, a segment of a circle.

How to use it

Find the area of the whole circle, πr².
Write the angle as a fraction of 360°.
Multiply the two.

To remember: The slice is its angle's share of the whole circle.

Other forms

Special case $A = \frac{π r ^{2}}{2}$ When a semicircle (θ = 180°).
Same formula, another formA = l × r ÷ 2, where l is the length of the sector's arc: like a triangle with the arc as its base and the radius as its height.

Worked examples

Story

A pizza of radius 12 cm is cut into 8 equal slices. What is the area of one slice?

8 equal slices means each angle is 360° ÷ 8 = 45°.
$A = π \times 1 2^{2} \times \frac{45}{360} = 18 π$
That is about 56.5 cm².

Answer: 18π cm²

Picture

In a circle of radius 10 cm, a 90° sector is cut by the straight line joining the ends of its arc. Find the area of the piece between the chord and the arc (the segment).

Sector: π × 10² × 90 ÷ 360 = 25π cm².
The triangle made by the two radii and the chord is right-angled: its area is 1/2 × 10 × 10 = 50 cm².
Segment = sector − triangle = 25π − 50 cm², about 28.5 cm².

Answer: 25π − 50 cm², about 28.5 cm²

Direct

Find the area of a sector of radius 6 cm with an angle of 60°.

$A = π \times 6^{2} \times \frac{60}{360}$
That is 1/6 of 36π, which is 6π cm², about 18.8 cm².

Answer: 6π cm²

Reverse

A sector of a circle of radius 6 cm has an area of 12π cm². What is its angle?

Try it first, then show the working

$π \times 6^{2} \times \frac{θ}{360} = 12 π$
$\frac{36 θ}{360} = 12$
$θ = 120$

Answer: 120°

Exam

The minute hand of a clock is 14 cm long. Find the area it sweeps in 5 minutes, using π ≈ 22/7.

Try it first, then show the working

In 5 minutes the hand turns 5/60 of 360°, which is 30°.
$\frac{22}{7} \times 1 4^{2} \times \frac{30}{360} = \frac{154}{3}$
That is 51 1/3 cm², about 51.3 cm².

Answer: 154/3 cm², about 51.3 cm²

Common mistake: Using the arc-length formula (2πr) in place of the area formula (πr²), or forgetting to square the radius.

Class 10

Length of a tangent from a point outside a circle

t = d^{2} - r^{2}

What each letter means
$t$	the length of the tangent from the point to where it touches the circle (length)
$d$	the distance from the point to the centre (length)
$r$	the radius (length)

A tangent meets the radius at a right angle, so the tangent, the radius and the line to the centre make a right triangle; the tangent length comes from Pythagoras.

Why it works

At the point of contact the tangent is perpendicular to the radius (Theorem 10.1). So the line from the point to the centre (d) is the hypotenuse, with legs r and t: t² + r² = d². Both tangents from the same point give the same triangle, so they are equal in length (Theorem 10.2).

When to use it

Tangents from an outside point: a thread or a belt touching a round object, a road just touching a circular pond, kites and triangles made of tangents and radii.

How to use it

Find the distance from the point to the centre and the radius.
Square both, subtract the radius squared, and take the square root.
Remember both tangents from that point have this same length.

To remember: Tangent meets radius at a right angle.

Other forms

Rearranged $d = t^{2} + r^{2}$ How far the point is from the centre.
Special case $t = 0$ When d = r: the point is on the circle itself.
Same formula, another formThe two tangents drawn from the same outside point are equal in length.

Worked examples

Story

A thread is pulled tight from a nail 25 cm from the centre of a round plate of radius 7 cm, so that it just touches the edge of the plate. How long is the thread from the nail to the edge?

$t = 2 5^{2} - 7^{2}$
$= 625 - 49 = 576 = 24$

Answer: 24

Picture

From a point T 17 cm from the centre O of a circle of radius 8 cm, tangents TP and TQ are drawn. Find their length and the area of the kite OPTQ.

$t = 1 7^{2} - 8^{2} = 225 = 15$
The kite is two right triangles, each with legs 8 cm and 15 cm.
$2 \times \frac{1}{2} \times 8 \times 15 = 120$

Answer: each tangent 15 cm; kite 120 cm²

Direct

A point is 13 cm from the centre of a circle of radius 5 cm. How long is a tangent from the point to the circle?

$t = 1 3^{2} - 5^{2}$
$= 169 - 25 = 144 = 12$

Answer: 12

Reverse

A tangent from a point 5 cm from the centre of a circle is 4 cm long. What is the radius?

Try it first, then show the working

$r = 5^{2} - 4^{2} = 9 = 3$

Answer: 3

Exam

PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q meet at T. Find the length TP.

Try it first, then show the working

OT is perpendicular to PQ at its midpoint R, so PR = 4 cm and OR = √(25 − 16) = 3 cm.
Let TR = y. In triangle TRP: TP² = y² + 16. In triangle OPT (right-angled at P): TP² = (y + 3)² − 25.
So y² + 16 = y² + 6y − 16, which gives 6y = 32 and y = 16/3.
TP = √(256/9 + 16) = √(400/9) = 20/3 cm.

Answer: 20/3 cm

Common mistake: Adding the squares (the line to the centre is the longest side, so you subtract), or using a chord instead of a tangent.

Class 10

Area of a segment of a circle

A = π r^{2} \times \frac{θ}{360} - \frac{1}{2} r^{2} sin θ

What each letter means
$A$	the area of the segment: the part between a chord and its arc (area)
$r$	the radius (length)
$θ$	the angle the chord makes at the centre (degrees)

A segment is a sector with the triangle (two radii and the chord) taken away, so its area is the sector's area minus the triangle's.

Why it works

The sector is the fraction θ/360 of the whole circle, πr² × θ/360. The triangle has two sides r with the angle θ between them, so its area is ½ × r × r × sin θ (half base times height, where the height is r sin θ). Taking one from the other leaves the segment.

When to use it

The area between a chord and its arc: a slice cut off by a straight line, window designs, the part of a round table beyond a straight edge, flower beds.

How to use it

Work out the sector area, πr² × θ/360.
Work out the triangle, ½r² sin θ (for 90° it is ½r²; for 60° it is (√3/4)r²; for 120° it is also (√3/4)r²).
Subtract the triangle from the sector. For the major segment, subtract the minor segment from the whole circle.

To remember: Segment = sector − triangle.

Other forms

Special caseFor a right angle (θ = 90°) the triangle is half a square: A = r²(π/4 − 1/2).When θ = 90°.
Special caseFor θ = 180° the chord is a diameter, the triangle is flat (sin 180° = 0), and the segment is a semicircle, πr²/2.When θ = 180°.
Same formula, another formMajor segment = the whole circle πr² − the minor segment.
Rearranged $A = \frac{r ^{2}}{2} (\frac{π θ}{180} - sin θ)$ The same area with r²/2 taken out: handy when θ is not a friendly angle.

Worked examples

Story

A round table top of radius 21 cm has a straight cut along a chord that makes 120° at the centre. Find the area of the piece cut off (the segment), using π = 22/7.

Sector: 22/7 × 21 × 21 × 120/360 = 462 cm².
Triangle: 1/2 × 21 × 21 × sin 120° = 441√3/4 cm², about 191 cm².
Segment: 462 − 441√3/4 cm², about 271 cm².

Answer: 462 − 441√3/4 cm², about 271 cm²

Picture

A chord of a circle of radius 6 cm makes 60° at the centre, so the triangle is equilateral. Find the area of the minor segment.

Sector: π × 36 × 60/360 = 6π cm².
Triangle: equilateral with side 6, (√3/4) × 36 = 9√3 cm².
Segment: 6π − 9√3 cm², about 3.26 cm².

Answer: 6π − 9√3 cm², about 3.26 cm²

Direct

A chord of a circle of radius 10 cm makes a right angle at the centre. Find the area of the minor segment.

Sector: π × 10² × 90/360 = 25π cm².
Triangle: 1/2 × 10 × 10 = 50 cm².
Segment: 25π − 50 cm², about 28.5 cm².

Answer: 25π − 50 cm², about 28.5 cm²

Reverse

A chord of a circle of radius 14 cm makes a right angle at the centre. Find the area of the major segment.

Try it first, then show the working

Minor segment: π × 196 × 90/360 − 1/2 × 14 × 14 = 49π − 98 cm².
Major segment = whole circle − minor segment = 196π − (49π − 98) = 147π + 98 cm², about 560 cm².

Answer: 147π + 98 cm², about 560 cm²

Exam

A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the minor segment (use π = 3.14 and √3 = 1.73).

Try it first, then show the working

Sector: 3.14 × 144 × 120/360 = 150.72 cm².
Triangle: 1/2 × 144 × sin 120° = 36√3, about 36 × 1.73 = 62.28 cm².
$150.72 - 62.28 = 88.44$

Answer: 88.44 cm²

Common mistake: Subtracting the triangle from the whole circle instead of from the sector, or adding the triangle when you want the segment.