NeokaalStudy

Areas Related to Circles: Competency Practice

Sector and segment areas, clock sweeps, wiper sweeping regions, and tethered field problems.

6 problems·20–25 min·★★★★☆
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  1. Problem 1
    sector-area·★★★★☆
    An arc in a circle of radius 6 cm subtends $60^\circ$ at the centre. Find its length, sector area, and minor-segment area.
    ▶Answer
    arc $=6\pi/3$ cm; sector $=36\pi/6$ cm$^2$; segment $=36\pi/6-36\sqrt3/4$ 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.
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  2. Problem 2
    segment-area·★★★★☆
    A chord subtends $90^\circ$ at the centre of a circle of radius 6 cm. Find the minor segment area.
    ▶Answer
    $\frac{36}4\pi-\frac{36}2$ 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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  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.
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  4. Problem 4
    sector-area·★★★★☆
    Two non-overlapping wipers, each 28 cm long, sweep through $130^\circ$. Find their total cleaned area.
    ▶Answer
    $\frac{5096\pi}{9}$ 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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  5. Problem 5
    sector-area·★★★☆☆
    A grazing rope of length 13 m is tied at a corner of a square field of side 16 m. Find the accessible area.
    ▶Answer
    $\frac{169}4\pi$ 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.
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  6. Problem 6
    segment-area·★★★★★
    A chord subtends $120^\circ$ at the centre of a circle of radius 15 cm. Find the corresponding minor segment area.
    ▶Answer
    $\frac{225}3\pi-\frac{225}4\sqrt3$ 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 →