Practice Worksheet

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

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Find the area of a sector of radius 8 cm and central angle $120^\circ$.
A circle has circumference $64\pi$ cm. Find the area of one quadrant.
A clock's minute hand is 21 cm long. Find the area it sweeps in 5 minutes.
An arc in a circle of radius 24 cm subtends $60^\circ$ at the centre. Find its length, sector area, and minor-segment 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.
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.
A flat circular umbrella of radius 54 cm has 8 equally spaced ribs. Find the area between consecutive ribs.
Two non-overlapping wipers, each 26 cm long, sweep through $113^\circ$. Find their total cleaned area.
A warning light covers a sector of radius 14 km and angle $90^\circ$. Find the warned area.
Write the formula for the area of a sector with radius $R$ and central angle $p^\circ$.
A chord subtends $90^\circ$ at the centre of a circle of radius 14 cm. Find the minor 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.
A chord subtends $120^\circ$ at the centre of a circle of radius 12 cm. Find the corresponding minor 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.

Neokaal Study

Answer Key

$\frac{64\pi}{3}$ cm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$256\pi$ cm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$\frac{147\pi}{4}$ cm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
arc $=24\pi/3$ cm; sector $=576\pi/6$ cm$^2$; segment $=576\pi/6-576\sqrt3/4$ cm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$\frac{324}4\pi$ m$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
length $=50\pi+200$ mm; sector area $=625\pi/8$ mm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$\frac{2916}{8}\pi$ cm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$\frac{19097\pi}{45}$ cm$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$49\pi$ km$^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$\frac{p}{360}\pi R^2$
1. Identify the central angle as a fraction of $360^\circ$.
2. Apply the arc or sector formula and simplify.
$\frac{196}4\pi-\frac{196}2$ cm$^2$
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.
minor $=36\pi/6-36\sqrt3/4$; major $=180\pi/6+36\sqrt3/4$ cm$^2$
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.
$\frac{144}3\pi-\frac{144}4\sqrt3$ cm$^2$
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.
$256\pi-\frac{3*radius*radius}2\sqrt3$ cm$^2$
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.