Take a Fraction of the Circumference

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A circle has radius 2 cm. Find the exact length of its 90° arc. Candidate answers: (A) 1π cm (B) 2π cm (C) 4π cm (D) 5π cm.
• Independent check inputs: A circle has radius 3 cm. Find the exact length of its 90° arc. Candidate answers: (A) 10π cm (B) 3/2π cm (C) 3π cm (D) 6π cm.

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 degree-measured arc covers its angle divided by 360 of the full circumference. Multiplying that fraction by 2πr gives arc length, retaining an exact multiple of π if requested.

Here is the method on the supplied teaching case. The arc is 90/360=1/4 of circumference 2π2. Its length is (2/2)π cm. Arc length uses linear units, not square units.

The angle fraction uses a full turn of 360 degrees. An arc length is one-dimensional; do not multiply the sector fraction by πr² or report its answer in square units.

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

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: A circle has radius 9 cm. Find the exact length of its 90° arc. Candidate answers: (A) 18π cm (B) 82π cm (C) 9/2π cm (D) 9π cm. Show the mathematical reason, not only the selected answer. Expected result: 9/2π cm. The arc is 90/360=1/4 of circumference 2π9. Its length is (9/2)π cm. Arc length uses linear units, not square units.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.