Take a Fraction of the Circle Area

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A circle has radius 4 cm. Find the exact area of its 90° sector. Candidate answers: (A) 16π cm² (B) 2.0π cm² (C) 4π cm² (D) 8π cm².
• Independent check inputs: A circle has radius 6 cm. Find the exact area of its 90° sector. Candidate answers: (A) 18π cm² (B) 36π cm² (C) 4.5π cm² (D) 9π cm².

Learn the idea

Circle angle theorems distinguish central, inscribed and tangent configurations. A sector or arc occupies a stated fraction of the whole circle, but its area and length use different whole-circle measures and units. A sector bounded by two radii occupies a fraction of the full disk. Its angle divided by 360 multiplies πr² to give the area enclosed by that sector.

Here is the method on the supplied teaching case. Take one quarter of the circle’s area: (90/360)π(4)²=4π cm². An arc bounds part of the sector but does not itself have area.

The radius is squared before applying the sector fraction. Exact answers can retain π; using arc length as area confuses a boundary measurement with the region that boundary helps enclose.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A circle has radius 18 cm. Find the exact area of its 90° sector. Candidate answers: (A) 162π cm² (B) 324π cm² (C) 40.5π cm² (D) 81π cm². Show the mathematical reason, not only the selected answer. Expected result: 81π cm². Take one quarter of the circle’s area: (90/360)π(18)²=81π cm². An arc bounds part of the sector but does not itself have area.

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