NeokaalStudy

Areas Related to Circles: Practice Worksheet

Arc lengths and areas of sectors, segments, and combined circular regions.

14 problems·20–25 min·★★★★☆
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  1. Problem 1
    sector-area·★★☆☆☆
    Find the area of a sector of radius 8 cm and central angle $120^\circ$.
    ▶Answer
    $\frac{64\pi}{3}$ cm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
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  2. Problem 2
    sector-area·★★★☆☆
    A circle has circumference $64\pi$ cm. Find the area of one quadrant.
    ▶Answer
    $256\pi$ cm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  3. Problem 3
    sector-area·★★★★☆
    A clock's minute hand is 21 cm long. Find the area it sweeps in 5 minutes.
    ▶Answer
    $\frac{147\pi}{4}$ cm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  4. Problem 4
    sector-area·★★★★★
    An arc in a circle of radius 24 cm subtends $60^\circ$ at the centre. Find its length, sector area, and minor-segment area.
    ▶Answer
    arc $=24\pi/3$ cm; sector $=576\pi/6$ cm$^2$; segment $=576\pi/6-576\sqrt3/4$ cm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  5. Problem 5
    sector-area·★★☆☆☆
    A grazing rope of length 18 m is tied at a corner of a square field of side 20 m. Find the accessible area.
    ▶Answer
    $\frac{324}4\pi$ m$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  6. Problem 6
    sector-area·★★★☆☆
    A circular wire brooch has radius 25 mm and 4 full diameters made from the same wire. Find total wire length and the area of each of the 8 equal sectors.
    ▶Answer
    length $=50\pi+200$ mm; sector area $=625\pi/8$ mm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  7. Problem 7
    sector-area·★★★★☆
    A flat circular umbrella of radius 54 cm has 8 equally spaced ribs. Find the area between consecutive ribs.
    ▶Answer
    $\frac{2916}{8}\pi$ cm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  8. Problem 8
    sector-area·★★★★★
    Two non-overlapping wipers, each 26 cm long, sweep through $113^\circ$. Find their total cleaned area.
    ▶Answer
    $\frac{19097\pi}{45}$ cm$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  9. Problem 9
    sector-area·★★☆☆☆
    A warning light covers a sector of radius 14 km and angle $90^\circ$. Find the warned area.
    ▶Answer
    $49\pi$ km$^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  10. Problem 10
    sector-area·★★★☆☆
    Write the formula for the area of a sector with radius $R$ and central angle $p^\circ$.
    ▶Answer
    $\frac{p}{360}\pi R^2$
    ▶Step-by-step solution
    1. Identify the central angle as a fraction of $360^\circ$.
    2. Apply the arc or sector formula and simplify.
    Report an issue →
  11. Problem 11
    segment-area·★★★★☆
    A chord subtends $90^\circ$ at the centre of a circle of radius 14 cm. Find the minor segment area.
    ▶Answer
    $\frac{196}4\pi-\frac{196}2$ cm$^2$
    ▶Step-by-step solution
    1. Segment area equals sector area minus the area of the triangle formed by the radii.
    2. Use the complementary part of the circle when a major region is requested.
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  12. Problem 12
    segment-area·★★★★★
    A chord subtends $60^\circ$ at the centre of a circle of radius 6 cm. Find both the minor and major segment areas.
    ▶Answer
    minor $=36\pi/6-36\sqrt3/4$; major $=180\pi/6+36\sqrt3/4$ cm$^2$
    ▶Step-by-step solution
    1. Segment area equals sector area minus the area of the triangle formed by the radii.
    2. Use the complementary part of the circle when a major region is requested.
    Report an issue →
  13. Problem 13
    segment-area·★★☆☆☆
    A chord subtends $120^\circ$ at the centre of a circle of radius 12 cm. Find the corresponding minor segment area.
    ▶Answer
    $\frac{144}3\pi-\frac{144}4\sqrt3$ cm$^2$
    ▶Step-by-step solution
    1. Segment area equals sector area minus the area of the triangle formed by the radii.
    2. Use the complementary part of the circle when a major region is requested.
    Report an issue →
  14. Problem 14
    segment-area·★★★☆☆
    A circular cover of radius 16 cm is divided into six equal segments, each bounded by a $60^\circ$ arc and its chord. Find the total area of the six segments.
    ▶Answer
    $256\pi-\frac{3*radius*radius}2\sqrt3$ cm$^2$
    ▶Step-by-step solution
    1. Segment area equals sector area minus the area of the triangle formed by the radii.
    2. Use the complementary part of the circle when a major region is requested.
    Report an issue →