Grade 8 Problems and Questions on Circles with Detailed Solutions

Detailed solutions and full explanations to grade 8 problems and questions on circles are presented.

  1. The three circles \( C_1, C_2, C_3 \) have centers \( O_1, O_2, O_3 \) on line \( L \) and are tangent at the same point. If the diameter of the largest circle is \( 20 \) units, find the ratio of the area of the largest circle to the smallest.

    Diameter of \( C_1 \) is \( 20 \) ⇒ \( r_1 = 10 \). Area of \( C_1 \): \( A = \pi (10)^2 \) Diameter of \( C_2 = 10 \) ⇒ \( r_2 = 5 \) Diameter of \( C_3 = 5 \) ⇒ \( r_3 = 2.5 \) Area of \( C_3 \): \( B = \pi (2.5)^2 \)

    \[ \frac{A}{B} = \frac{\pi (10)^2}{\pi (2.5)^2} = \left( \frac{10}{2.5} \right)^2 = 4^2 = 16 \]

    Ratio: \( 16 : 1 \)

  2. Mrs. Parkinson's garden has 4 squares and 2 semicircles. Each small square has area \( 4\ \text{m}^2 \). Find the total area.

    Total square area: \( 4 \times 4 = 16\ \text{m}^2 \) Side length: \( 2 \ \text{m} \) ⇒ semicircle radius \( r = 2 \ \text{m} \) Two semicircles form one full circle: area \( = \pi (2)^2 = 4\pi \)

    \[ \text{Total area} = 16 + 4\pi \approx 28.56\ \text{m}^2 \]
  3. A sprinkler sprays water at max distance \( 12 \ \text{m} \). Find the irrigated area. \[ A = \pi (12)^2 = 144\pi \approx 452\ \text{m}^2 \]
  4. A circular garden (diameter \( 10\ \text{m} \)) has a walkway width \( 1\ \text{m} \). Find walkway area.

    Inner radius: \( 10 \ \text{m} \), Outer radius: \( 11 \ \text{m} \)

    \[ A = \pi (11)^2 - \pi (10)^2 = 121\pi - 100\pi = 21\pi \ \text{m}^2 \]
  5. A circular pizza costs \$19.99, diameter \( 36 \ \text{cm} \). Find cost per \(\text{cm}^2\).

    Radius: \( 18\ \text{cm} \) Area: \( \pi (18)^2 = 324\pi \approx 1017 \ \text{cm}^2 \)

    \[ \text{Cost/cm}^2 \approx \frac{19.99}{1017} \approx 0.02\ \text{USD} \ (\text{2 cents}) \]
  6. Robinsons' garden area is \( 5\ \text{m}^2 \). Find fencing length.

    \(\pi r^2 = 5 \ \Rightarrow \ r \approx 1.26 \ \text{m}\)

    \[ C = 2\pi r \approx 8\ \text{m} \]
  7. A disk's radius increases by \( 20\% \). Find percent increase in area.

    Old area: \( \pi r^2 \) New radius: \( 1.2r \), New area: \( 1.44\pi r^2 \)

    \[ \% \text{ change} = \frac{1.44\pi r^2 - \pi r^2}{\pi r^2} \times 100\% = 44\% \]
  8. A circular table diameter \( 100 \ \text{in} \), tablecloth hangs \( 15\ \text{in} \) over edges. Find tablecloth area.

    Diameter: \( 130\ \text{in} \) ⇒ radius \( 65\ \text{in} \)

    \[ A = \pi (65)^2 = 4225\pi \approx 13267\ \text{in}^2 \]
  9. Square \( ABCD \) has one vertex at the center of a circle and two vertices on the circle. Circle area is \( 100\ \text{cm}^2 \). Find \( AC \). \[ \pi r^2 = 100 \ \Rightarrow \ r^2 = \frac{100}{\pi} \] \[ AC^2 = r^2 + r^2 = 2r^2 = \frac{200}{\pi} \] \[ AC = 10\sqrt{\frac{2}{\pi}}\ \text{cm} \]
  10. Perimeter ratio of circle \( A \) to \( B \) is \( 3:1 \). Find area ratio. \[ \frac{R_a}{R_b} = 3 \quad \Rightarrow \quad A_a = \pi (3R_b)^2 = 9\pi R_b^2 \] \[ \frac{A_a}{A_b} = 9:1 \]

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