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Worksheet: Area of a Circular Sector

Q1:

Shown is a sector of a circle. If its perimeter is 39 mm, what is its area?

Q2:

The area of a circular sector is 1 790 cm2 and the central angle is 1.5 rad. Find the radius of the circle giving the answer to the nearest centimeter.

  • A 24 cm
  • B 2 387 cm
  • C 597 cm
  • D 49 cm

Q3:

A circular birthday cake with a diameter of 22 cm is divided into eleven equal sectors. Using 3.14 as an approximation for πœ‹ , find the area of one sector.

Q4:

A landscape gardener decides that he wants to design a lawn split into a series of sectors with circular patios laid into the grass, as shown in the given figure. The circular lawn will be split into six equal sectors, each with a radius of eight yards. The lines 𝑂 𝐴 and 𝑂 𝐡 are both tangents to the circle, and the arc 𝐴 𝐡 touches the circle at a single point.

Work out the area of sector 𝑂 𝐴 𝐡 . Give your answer in terms of πœ‹ .

  • A 6 4 πœ‹ 3 sq yards
  • B 3 2 πœ‹ 7 sq yards
  • C 8 πœ‹ 3 sq yards
  • D 3 2 πœ‹ 3 sq yards
  • E 4 πœ‹ 3 sq yards

The gardener needs to calculate the radius of the circular patio. Using trigonometric ratios, calculate the radius of the patio. Give your answer as a fraction.

  • A 8 3 yards
  • B 3 4 yards
  • C 3 8 yards
  • D 4 3 yards
  • E 1 6 3 yards

Calculate the total area of grass in one sector. Give your answer, in terms of πœ‹ , in its simplest form.

  • A 6 4 πœ‹ 9 sq yards
  • B 3 2 πœ‹ 3 sq yards
  • C 3 2 πœ‹ 9 sq yards
  • D 9 πœ‹ 3 2 sq yards
  • E 3 2 9 sq yards

Q5:

An arc has a measure of 6 3 ∘ and a radius of 4.

Work out the length of the arc. Give your answer in terms of πœ‹ and in its simplest form.

  • A 7 πœ‹ 1 0
  • B 5 πœ‹ 7
  • C 1 0 πœ‹ 7
  • D 7 πœ‹ 5
  • E 7 5

Work out the area of the sector. Give your answer in terms of πœ‹ and in its simplest form.

  • A 1 4 πœ‹ 5
  • B 2 8 πœ‹ 5
  • C 7 πœ‹ 5
  • D 7 πœ‹ 1 0
  • E 288

Q6:

In this figure, the diameter of the larger circle is 41 cm and both circles have the same center. Determine, to the nearest tenth, the area of the shaded part.

Q7:

In this figure, the diameter of the larger circle is 61.6 cm and both circles have the same center. Determine, to the nearest tenth, the area of the shaded part.

Q8:

Write an expression for the area of a sector whose arc’s measure is πœƒ radians, knowing that the expression for the area of a sector measuring πœ‹ degrees is πœ‹ π‘Ÿ πœƒ 3 6 0 2 .

  • A 2 π‘Ÿ πœƒ 2
  • B 1 2 π‘Ÿ πœƒ
  • C 2 π‘Ÿ πœƒ
  • D 1 2 π‘Ÿ πœƒ 2
  • E 1 4 π‘Ÿ πœƒ 2

Q9:

Find the area of the shaded part in the figure. Round your answer to the nearest tenth.

Q10:

The circle in the given figure has a radius π‘Ÿ , and the angle of the sector is πœƒ .

Write down an expression for the area of the circle.

  • A πœ‹ π‘Ÿ 2 2
  • B 2 πœ‹ π‘Ÿ
  • C πœ‹ π‘Ÿ 2
  • D πœ‹ π‘Ÿ 2
  • E 2 πœ‹ π‘Ÿ 2

What fraction of the circle is the sector with central angle πœƒ ?

  • A πœƒ 3 6 0
  • B πœƒ 2 7 0
  • C πœƒ 9 0
  • D πœƒ 1 8 0
  • E πœƒ 6 0

Write an expression for the area of the sector.

  • A 3 6 0 πœƒ Γ— πœ‹ π‘Ÿ
  • B πœƒ 1 8 0 Γ— πœ‹ π‘Ÿ 2
  • C πœƒ 3 6 0 Γ— πœ‹ π‘Ÿ 2
  • D πœƒ 3 6 0 Γ— πœ‹ π‘Ÿ
  • E πœƒ 1 8 0 Γ— πœ‹ π‘Ÿ 2

Q11:

Work out the area of the given shape, giving your answer in terms of πœ‹ .

  • A 2 5 πœ‹
  • B 2 5 4 πœ‹
  • C 2 5 2 πœ‹
  • D 7 5 4 πœ‹
  • E 7 5 2 πœ‹

Q12:

Work out the area of the quarter circle, giving your answer in terms of πœ‹ .

  • A 8 1 4 πœ‹
  • B 9 2 πœ‹
  • C 9 πœ‹
  • D 9 4 πœ‹
  • E 8 1 πœ‹

Q13:

The area of a circular sector is 12 cm2 and the perimeter is 16 cm. Find all possible values for the radius.

  • A 12 cm or 3 cm
  • B 12 cm or 4 cm
  • C 7 cm or 2 cm
  • D 6 cm or 2 cm
  • E 7 cm or 4 cm

Q14:

𝐴 𝐡 and 𝐴 𝐢 are two tangents to the circle 𝑀 where 𝐡 and 𝐢 lie on the circumference. 𝑀 𝐴 = 1 4 c m and the radius of the circle is 7 cm. Find the area of the part between the two tangents and the smaller arc 𝐡 𝐢 giving the answer to the nearest square centimeter.

  • A 59 cm2
  • B 118 cm2
  • C 17 cm2
  • D 34 cm2
  • E 67 cm2

Q15:

Find the area of the shaded part of the diagram in terms of πœ‹ .

  • A 1 5 πœ‹ cm2
  • B 7 . 5 πœ‹ cm2
  • C 5 6 . 2 5 πœ‹ cm2
  • D 1 1 2 . 5 πœ‹ cm2

Q16:

𝑋 π‘Œ 𝑍 is an equilateral triangle with a side length of 92 cm. Three circular sectors are drawn in the triangle such that their centers are the vertices 𝑋 , π‘Œ and 𝑍 . The radius of each sector is 46 cm. Find the area of the part of the triangle bounded by the arcs of the circular sectors giving the answer to one decimal place.

  • A 2 667.9 cm2
  • B 2 003.1 cm2
  • C 332.4 cm2
  • D 341.2 cm2

Q17:

The area of a circular sector is 561.3 cm2 and the central angle is 2 7 ∘ . Find the radius of the circle giving the answer to the nearest centimeter.

  • A 98 cm
  • B 23 cm
  • C 26 cm
  • D 49 cm

Q18:

The area of a circular sector is 387.8 cm2 and the central angle is 1 4 3 ∘ . Find the radius of the circle giving the answer to the nearest centimeter.

  • A 35 cm
  • B 44 cm
  • C 26 cm
  • D 18 cm

Q19:

Work out the area of the given shape, giving your answer accurate to two decimal places.

Q20:

The perimeter of a circular sector is 36 cm and the central angle is 0.4 rad. Find the area of the sector giving the answer to the nearest square centimeter.

  • A 22.5 cm2
  • B 90 cm2
  • C 180 cm2
  • D 45 cm2

Q21:

The arc length of a circular sector is 22 cm and the central angle is 7 7 ∘ . Find the area of the sector giving the answer to the nearest square centimeter.

  • A 360 cm2
  • B 90 cm2
  • C 115 cm2
  • D 180 cm2

Q22:

The diameter of a circle is 50 cm and the angle of a sector is 7 0 ∘ . Find the area of the sector giving the answer to the nearest square centimeter.

  • A 122 cm2
  • B 15 cm2
  • C 764 cm2
  • D 382 cm2
  • E 31 cm2

Q23:

Given that the area of the colored part in the figure is 155 ft2, find the value of π‘₯ to the nearest tenth.

Q24:

Given that the area of the colored part in the figure is 58 ft2, find the value of π‘₯ to the nearest tenth.

Q25:

Given that the area of the colored part in the figure is 140 ft2, find the value of π‘₯ to the nearest tenth.