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Grade 10 · unit review

Review: Geometry: Circle Theorems, Arcs, and Sectors

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.

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Unit review questions
1. An inscribed angle intercepts an arc of 60°. Find the angle.
Need a hint?
  • An inscribed angle has its vertex on the circle.
  • Locate the angle vertex before choosing a theorem.
2. A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 27°. Find its third angle.
Need a hint?
  • A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.
  • The radius must end at the point of tangency.
3. A circle has radius 7 cm. Find the exact length of its 90° arc.
Need a hint?
  • A degree-measured arc covers its angle divided by 360 of the full circumference.
  • The angle fraction uses a full turn of 360 degrees.
4. A circle has radius 14 cm. Find the exact area of its 90° sector.
Need a hint?
  • A sector bounded by two radii occupies a fraction of the full disk.
  • The radius is squared before applying the sector fraction.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. An inscribed angle intercepts an arc of 68°. Find the angle. Candidate answers: (A) 33 ° (B) 34 ° (C) 35 ° (D) 68 °. 2. A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 29°. Find its third angle. Candidate answers: (A) 122 ° (B) 60 ° (C) 61 ° (D) 62 °. 3. 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. 4. 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.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 34 °. An inscribed angle is half the measure of its intercepted arc. Divide 68 by 2 to get 34.0°. Its vertex lies on the circle; a central angle has a different relationship. 2. 61 °. Radius and tangent are perpendicular at the contact point, giving 90°. The triangle angle total gives the remaining angle 180−90−29=61°. 3. 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. 4. 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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