Geometry

Circles

Circle tasks combine area formulas with angle rules. München names circles for its T-Kurs entrance test.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• Arc length b = (α/360°) · 2πr; sector area A = (α/360°) · πr².
• An inscribed angle is half the central angle over the same arc.
• Thales: an angle in a semicircle is 90°.
• A tangent is perpendicular to the radius at the point of contact.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. A central angle and an inscribed angle intercept the same arc of a circle. If the central angle measures 120°, what is the measure of the inscribed angle?

   • a70
   • b55
   • c120
   • d60
   Show solution

   Answer: 60

   Inscribed Angle Theorem: an inscribed angle is half the central angle subtending the same arc.

   120 / 2 = 60.

2. 2. A circle has radius 6. An arc on the circle subtends a central angle of 90°. What is the arc length, expressed as a multiple of π? (e.g. an arc length of 3π has answer 3)

   • a3
   • b2
   • c15
   • d4
   Show solution

   Answer: 3

   Arc length = (θ/360) × 2πr.

   (90/360) × 2π×6 = 3π.

3. 3. A ring lies between two circles with the same centre, of radius 7 cm and 4 cm. What is its area, as a multiple of π? (e.g. 12π cm² → 12)

   • a6
   • b9
   • c65
   • d33
   Show solution

   Answer: 33

   Ring = big disc minus small disc: π(R² − r²), not π(R − r)².

   π(7² − 4²) = π(49 − 16) = 33π.

4. 4. A circle has radius 3. A sector of the circle has a central angle of 120°. What is the area of the sector, expressed as a multiple of π? (e.g. an area of 5π has answer 5)

   • a2
   • b5
   • c12
   • d3
   Show solution

   Answer: 3

   Sector area = (θ/360) × πr².

   (120/360) × π×3² = (120/360) × 9π = 3π.

5. 5. A circle has radius 2. A sector of the circle has a central angle of 180°. What is the area of the sector, expressed as a multiple of π? (e.g. an area of 5π has answer 5)

   • a18
   • b1
   • c4
   • d2
   Show solution

   Answer: 2

   Sector area = (θ/360) × πr².

   (180/360) × π×2² = (180/360) × 4π = 2π.

6. 6. A circle has radius 4. An arc on the circle subtends a central angle of 90°. What is the arc length, expressed as a multiple of π? (e.g. an arc length of 3π has answer 3)

   • a22.5
   • b1.0
   • c3.0
   • d2.0
   Show solution

   Answer: 2.0

   Arc length = (θ/360) × 2πr.

   (90/360) × 2π×4 = 2π.

Common mistakes

• Forgetting the factor α/360°.
• Using the inscribed angle where the central angle is needed.
• Using the diameter as radius in πr².

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs