Year 8 Mathematics | Victorian Curriculum 2.0
Circles: circumference & area
Topic 10 | Measurement & Space | Answer key

Year 8 core - answers

Fluency

Circumference and area

    1. 25.1225.12 cm. Method: 2×3.14×42 \times 3.14 \times 4.
    2. 31.431.4 m. Method: πd\pi d.
    3. 254.34254.34 cm2^2. Method: 3.14×813.14 \times 81.
    4. 154154 cm2^2. Method: r=7r = 7; 227×49\tfrac{22}{7} \times 49.
    5. 15.715.7 m.
    6. 78.578.5 cm2^2.
Fluency

Reverse problems

    1. 1010 cm. Method: r=62.86.28r = \dfrac{62.8}{6.28}.
    2. 44 m. Method: r2=50.243.14=16r^2 = \dfrac{50.24}{3.14} = 16.
    3. 0.50.5 m. Method: d=1.573.14d = \dfrac{1.57}{3.14}.
    4. 2020 cm. Method: r2=3143.14=100r^2 = \dfrac{314}{3.14} = 100; r=10r = 10; d=20d = 20.
Fluency

Semicircles and sectors

    1. Area 56.5256.52 cm2^2; perimeter 30.8430.84 cm. Method: 12πr2\tfrac{1}{2}\pi r^2; πr+2r\pi r + 2r.
    2. 12.5612.56 m2^2. Method: 14πr2\tfrac{1}{4}\pi r^2.
    3. 0.39250.3925 m2^2. Method: r=0.5r = 0.5; 12πr2\tfrac{1}{2}\pi r^2.
    4. 56.5256.52 cm2^2. Method: 18π(12)2=18×452.16\tfrac{1}{8}\pi (12)^2 = \tfrac{1}{8} \times 452.16.
Reasoning

Explain and spot the mistake

    1. Wrong. C=2πrC = 2\pi r (or πd\pi d). Kira used half of the correct formula.
    2. Quadruples. A∝r2A \propto r^2, so doubling rr multiplies area by 22=42^2 = 4.
    3. The formula is A=πr2A = \pi r^2, not π×2r\pi \times 2r. For r=5r = 5, A=π×25=78.5A = \pi \times 25 = 78.5 cm2^2. The student squared wrong.
    4. Because π\pi is defined as that ratio. Regardless of the circle’s size, C/dC/d always produces the same irrational number.
Problem solving

Real contexts

    1. 25.1225.12 m. Method: r=3.5+0.5=4r = 3.5 + 0.5 = 4; C=2π×4C = 2\pi \times 4.
    2. Pizza area 803.84803.84 cm2^2; one slice 100.48100.48 cm2^2.
    3. 157157 seconds ≈2.62\approx 2.62 min. Method: lawn area =π×25=78.5= \pi \times 25 = 78.5; 78.5/0.578.5 / 0.5.
    4. About 31.8531.85 m. Method: two straights =200= 200; two semicircles add to one full circle: C=400−200=200C = 400 - 200 = 200; r=200/(2π)≈31.85r = 200 / (2\pi) \approx 31.85.

Challenge - answers

Reasoning

Harder circles

    1. Perimeter =2×6+2π×2≈24.56= 2 \times 6 + 2\pi \times 2 \approx 24.56 cm (two long sides ++ one full circle from the two semicircles, with r=2r = 2). Area =24+π×4≈36.56= 24 + \pi \times 4 \approx 36.56 cm2^2.
    2. 50.2450.24 cm2^2. Method: π(52−32)=16π\pi(5^2 - 3^2) = 16\pi.
    3. 25π≈78.525\pi \approx 78.5 m. Method: 14×2πr=πr2\tfrac{1}{4} \times 2\pi r = \tfrac{\pi r}{2}.
    4. 3030 cm pizza: 14÷(π×225)≈0.019814 \div (\pi \times 225) \approx 0.0198 dollars per cm2^2. 4040 cm pizza: 22÷(π×400)≈0.017522 \div (\pi \times 400) \approx 0.0175 dollars per cm2^2. The 4040 cm pizza is cheaper per cm2^2.
Year 8 Mathematics study companion | Answer key