Segments | Math Coach Amy

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Key Ideas Easy Medium Hard

Segment = 12 r² θ − 12 r² sin θ

c² = r² + r² − 2rr cos θ

What to Remember

Segment

Segment = sector − triangle.

Area of the sector = 12 r² θ

Area of the triangle = 12 r² sin θ

θ is in radians.

Chord

The chord is the third side of the triangle formed by two radii.

c² = r² + r² − 2 r r cos θ

Perimeter of the segment = arc + chord.

Easy

Area and perimeter

θ is given in radians. Subtract the triangle from the sector.

Example

The shaded segment is bounded by a chord and an arc. The radius is 10 cm and the angle at the centre is 1.2 radians. Find the area and the perimeter of the shaded region.

A = 12 r² θ

Area of the sector = 12 × 10² × 1.2

Area of the sector = 60 cm²

Area of the triangle = 12 r² sin θ

Area of the triangle = 12 × 10² × sin 1.2

Area of the triangle = 46.6020 cm²

Segment = sector − triangle

Segment = 60 − 46.6020 = 13.3980

Segment = 13.4 cm²

c² = r² + r² − 2 r r cos θ

c² = 10² + 10² − 2 × 10 × 10 × cos 1.2

c = 11.2928 cm

Perimeter = arc + chord

l = rθ

l = 10 × 1.2 = 12 cm

Perimeter = 12 + 11.2928 = 23.2928

Perimeter = 23.3 cm

Answer: area 13.4 cm², perimeter 23.3 cm

The shaded segment is bounded by a chord and an arc. The radius is 8 cm and the angle at the centre is 0.9 radians. Find the area and the perimeter of the shaded region.

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A = 12 r² θ

Area of the sector = 12 × 8² × 0.9

Area of the sector = 28.8 cm²

Area of the triangle = 12 r² sin θ

Area of the triangle = 12 × 8² × sin 0.9

Area of the triangle = 25.0665 cm²

Segment = sector − triangle

Segment = 28.8 − 25.0665 = 3.7335

Segment = 3.73 cm²

c² = r² + r² − 2 r r cos θ

c² = 8² + 8² − 2 × 8 × 8 × cos 0.9

c = 6.9594 cm

Perimeter = arc + chord

l = rθ

l = 8 × 0.9 = 7.2 cm

Perimeter = 7.2 + 6.9594 = 14.1594

Perimeter = 14.2 cm

Answer: area 3.73 cm², perimeter 14.2 cm

The shaded segment is bounded by a chord and an arc. The radius is 12 cm and the angle at the centre is 1.5 radians. Find the area and the perimeter of the shaded region.

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A = 12 r² θ

Area of the sector = 12 × 12² × 1.5

Area of the sector = 108 cm²

Area of the triangle = 12 r² sin θ

Area of the triangle = 12 × 12² × sin 1.5

Area of the triangle = 71.8196 cm²

Segment = sector − triangle

Segment = 108 − 71.8196 = 36.1804

Segment = 36.2 cm²

c² = r² + r² − 2 r r cos θ

c² = 12² + 12² − 2 × 12 × 12 × cos 1.5

c = 16.3593 cm

Perimeter = arc + chord

l = rθ

l = 12 × 1.5 = 18 cm

Perimeter = 18 + 16.3593 = 34.3593

Perimeter = 34.4 cm

Answer: area 36.2 cm², perimeter 34.4 cm

The shaded segment is bounded by a chord and an arc. The radius is 6 cm and the angle at the centre is 2.1 radians. Find the area and the perimeter of the shaded region.

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A = 12 r² θ

Area of the sector = 12 × 6² × 2.1

Area of the sector = 37.8 cm²

Area of the triangle = 12 r² sin θ

Area of the triangle = 12 × 6² × sin 2.1

Area of the triangle = 15.5378 cm²

Segment = sector − triangle

Segment = 37.8 − 15.5378 = 22.2622

Segment = 22.3 cm²

c² = r² + r² − 2 r r cos θ

c² = 6² + 6² − 2 × 6 × 6 × cos 2.1

c = 10.4091 cm

Perimeter = arc + chord

l = rθ

l = 6 × 2.1 = 12.6 cm

Perimeter = 12.6 + 10.4091 = 23.0091

Perimeter = 23.0 cm

Answer: area 22.3 cm², perimeter 23.0 cm

The shaded segment is bounded by a chord and an arc. The radius is 9 cm and the angle at the centre is 1 radians. Find the area and the perimeter of the shaded region.

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A = 12 r² θ

Area of the sector = 12 × 9² × 1

Area of the sector = 40.5 cm²

Area of the triangle = 12 r² sin θ

Area of the triangle = 12 × 9² × sin 1

Area of the triangle = 34.0796 cm²

Segment = sector − triangle

Segment = 40.5 − 34.0796 = 6.4204

Segment = 6.42 cm²

c² = r² + r² − 2 r r cos θ

c² = 9² + 9² − 2 × 9 × 9 × cos 1

c = 8.6297 cm

Perimeter = arc + chord

l = rθ

l = 9 × 1 = 9 cm

Perimeter = 9 + 8.6297 = 17.6297

Perimeter = 17.6 cm

Answer: area 6.42 cm², perimeter 17.6 cm

The shaded segment is bounded by a chord and an arc. The radius is 7 cm and the angle at the centre is 1.6 radians. Find the area and the perimeter of the shaded region.

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A = 12 r² θ

Area of the sector = 12 × 7² × 1.6

Area of the sector = 39.2 cm²

Area of the triangle = 12 r² sin θ

Area of the triangle = 12 × 7² × sin 1.6

Area of the triangle = 24.4896 cm²

Segment = sector − triangle

Segment = 39.2 − 24.4896 = 14.7104

Segment = 14.7 cm²

c² = r² + r² − 2 r r cos θ

c² = 7² + 7² − 2 × 7 × 7 × cos 1.6

c = 10.0430 cm

Perimeter = arc + chord

l = rθ

l = 7 × 1.6 = 11.2 cm

Perimeter = 11.2 + 10.0430 = 21.2430

Perimeter = 21.2 cm

Answer: area 14.7 cm², perimeter 21.2 cm

Medium

Arc minus chord

Find each length, then subtract.

Example

A circle has radius 8 cm. A chord subtends an angle of 0.9 radians at the centre. Find the difference between the length of the arc and the length of the chord.

l = rθ

l = 8 × 0.9

l = 7.2 cm

c² = r² + r² − 2 r r cos θ

c² = 2 × 8² × (1 − cos 0.9)

c = 6.9594 cm

Difference = arc − chord

Difference = 7.2 − 6.9594 = 0.2406

Difference = 0.241 cm

Answer: 0.241 cm

A circle has radius 10 cm. A chord subtends an angle of 1.2 radians at the centre. Find the difference between the length of the arc and the length of the chord.

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l = rθ

l = 10 × 1.2

l = 12 cm

c² = r² + r² − 2 r r cos θ

c² = 2 × 10² × (1 − cos 1.2)

c = 11.2928 cm

Difference = arc − chord

Difference = 12 − 11.2928 = 0.7072

Difference = 0.707 cm

Answer: 0.707 cm

A circle has radius 6 cm. A chord subtends an angle of 1.4 radians at the centre. Find the difference between the length of the arc and the length of the chord.

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l = rθ

l = 6 × 1.4

l = 8.4 cm

c² = r² + r² − 2 r r cos θ

c² = 2 × 6² × (1 − cos 1.4)

c = 7.7306 cm

Difference = arc − chord

Difference = 8.4 − 7.7306 = 0.6694

Difference = 0.669 cm

Answer: 0.669 cm

A circle has radius 5 cm. A chord subtends an angle of 2 radians at the centre. Find the difference between the length of the arc and the length of the chord.

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l = rθ

l = 5 × 2

l = 10 cm

c² = r² + r² − 2 r r cos θ

c² = 2 × 5² × (1 − cos 2)

c = 8.4147 cm

Difference = arc − chord

Difference = 10 − 8.4147 = 1.5853

Difference = 1.59 cm

Answer: 1.59 cm

A circle has radius 12 cm. A chord subtends an angle of 0.8 radians at the centre. Find the difference between the length of the arc and the length of the chord.

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l = rθ

l = 12 × 0.8

l = 9.6 cm

c² = r² + r² − 2 r r cos θ

c² = 2 × 12² × (1 − cos 0.8)

c = 9.3460 cm

Difference = arc − chord

Difference = 9.6 − 9.3460 = 0.2540

Difference = 0.254 cm

Answer: 0.254 cm

A circle has radius 9 cm. A chord subtends an angle of 1.3 radians at the centre. Find the difference between the length of the arc and the length of the chord.

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l = rθ

l = 9 × 1.3

l = 11.7 cm

c² = r² + r² − 2 r r cos θ

c² = 2 × 9² × (1 − cos 1.3)

c = 10.8934 cm

Difference = arc − chord

Difference = 11.7 − 10.8934 = 0.8066

Difference = 0.807 cm

Answer: 0.807 cm

Hard

Find θ first

Turn the area into θ − sin θ = a number, then solve on a calculator.

Example

A segment of a circle of radius 5 cm has area 10 cm². The angle at the centre is θ radians. Show that θ − sin θ = 0.8, find θ to 3 significant figures, and find the perimeter of the segment.

Area of a segment = 12 r² (θ − sin θ)

10 = 12 × 5² × (θ − sin θ)

10 = 12.5 (θ − sin θ)

θ − sin θ = 1012.5

θ − sin θ = 0.8

Solve θ − sin θ = 0.8 on a calculator. Keep extra digits until the last line.

θ = 1.7785..., which is 1.78 to 3 significant figures.

l = rθ

l = 5 × 1.7785 = 8.8925 cm

c² = r² + r² − 2 r r cos θ

c = 7.7660 cm

Perimeter = arc + chord

Perimeter = 8.8925 + 7.7660 = 16.6585

Perimeter = 16.7 cm

Answer: θ = 1.78, perimeter 16.7 cm

A segment of a circle of radius 10 cm has area 20 cm². The angle at the centre is θ radians. Show that θ − sin θ = 0.4, find θ to 3 significant figures, and find the perimeter of the segment.

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Area of a segment = 12 r² (θ − sin θ)

20 = 12 × 10² × (θ − sin θ)

20 = 50 (θ − sin θ)

θ − sin θ = 2050

θ − sin θ = 0.4

Solve θ − sin θ = 0.4 on a calculator. Keep extra digits until the last line.

θ = 1.3823..., which is 1.38 to 3 significant figures.

l = rθ

l = 10 × 1.3823 = 13.8228 cm

c² = r² + r² − 2 r r cos θ

c = 12.7484 cm

Perimeter = arc + chord

Perimeter = 13.8228 + 12.7484 = 26.5712

Perimeter = 26.6 cm

Answer: θ = 1.38, perimeter 26.6 cm

A segment of a circle of radius 6 cm has area 10.8 cm². The angle at the centre is θ radians. Show that θ − sin θ = 0.6, find θ to 3 significant figures, and find the perimeter of the segment.

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Area of a segment = 12 r² (θ − sin θ)

10.8 = 12 × 6² × (θ − sin θ)

10.8 = 18 (θ − sin θ)

θ − sin θ = 10.818

θ − sin θ = 0.6

Solve θ − sin θ = 0.6 on a calculator. Keep extra digits until the last line.

θ = 1.5996..., which is 1.60 to 3 significant figures.

l = rθ

l = 6 × 1.5996 = 9.5975 cm

c² = r² + r² − 2 r r cos θ

c = 8.6065 cm

Perimeter = arc + chord

Perimeter = 9.5975 + 8.6065 = 18.2041

Perimeter = 18.2 cm

Answer: θ = 1.60, perimeter 18.2 cm

A segment of a circle of radius 8 cm has area 8 cm². The angle at the centre is θ radians. Show that θ − sin θ = 0.25, find θ to 3 significant figures, and find the perimeter of the segment.

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Area of a segment = 12 r² (θ − sin θ)

8 = 12 × 8² × (θ − sin θ)

8 = 32 (θ − sin θ)

θ − sin θ = 832

θ − sin θ = 0.25

Solve θ − sin θ = 0.25 on a calculator. Keep extra digits until the last line.

θ = 1.1712..., which is 1.17 to 3 significant figures.

l = rθ

l = 8 × 1.1712 = 9.3698 cm

c² = r² + r² − 2 r r cos θ

c = 8.8434 cm

Perimeter = arc + chord

Perimeter = 9.3698 + 8.8434 = 18.2132

Perimeter = 18.2 cm

Answer: θ = 1.17, perimeter 18.2 cm

A segment of a circle of radius 4 cm has area 8 cm². The angle at the centre is θ radians. Show that θ − sin θ = 1, find θ to 3 significant figures, and find the perimeter of the segment.

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Area of a segment = 12 r² (θ − sin θ)

8 = 12 × 4² × (θ − sin θ)

8 = 8 (θ − sin θ)

θ − sin θ = 88

θ − sin θ = 1

Solve θ − sin θ = 1 on a calculator. Keep extra digits until the last line.

θ = 1.9346..., which is 1.93 to 3 significant figures.

l = rθ

l = 4 × 1.9346 = 7.7383 cm

c² = r² + r² − 2 r r cos θ

c = 6.5868 cm

Perimeter = arc + chord

Perimeter = 7.7383 + 6.5868 = 14.3250

Perimeter = 14.3 cm

Answer: θ = 1.93, perimeter 14.3 cm

A segment of a circle of radius 8 cm has area 16 cm². The angle at the centre is θ radians. Show that θ − sin θ = 0.5, find θ to 3 significant figures, and find the perimeter of the segment.

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Area of a segment = 12 r² (θ − sin θ)

16 = 12 × 8² × (θ − sin θ)

16 = 32 (θ − sin θ)

θ − sin θ = 1632

θ − sin θ = 0.5

Solve θ − sin θ = 0.5 on a calculator. Keep extra digits until the last line.

θ = 1.4973..., which is 1.50 to 3 significant figures.

l = rθ

l = 8 × 1.4973 = 11.9784 cm

c² = r² + r² − 2 r r cos θ

c = 10.8904 cm

Perimeter = arc + chord

Perimeter = 11.9784 + 10.8904 = 22.8688

Perimeter = 22.9 cm

Answer: θ = 1.50, perimeter 22.9 cm

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