Question 1: Calculate the surface area of a spherical ball with a radius of 3 cm. Use 4πr² and leave your answer in terms of π.

Answer: 36π

Question 2: A globe has a radius of 12 cm. Find its surface area using 4πr². Provide your answer as a decimal approximation (use π ≈ 3.14).

Answer: 1806.72

Question 3: If the surface area of a sphere is 1256 cm², what is its radius? Use 4πr² and π ≈ 3.14.

Answer: r ≈ 10

Question 4: Construct a circle (on the grid) that could represent a sphere with a radius of 5 units. You do not need to measure; draw a circle with the correct radius on the grid.

Answer: Student's drawing

Question 5: Explain why the surface area increases more rapidly than the radius as the sphere gets larger.

Answer: Because surface area is proportional to r squared (4πr²), it grows faster than the radius itself, which is linear. Doubling the radius increases surface area by a factor of four.

Question 6: A spherical tank has a radius of 8 meters. Find the surface area. Then, if the radius increases to 10 meters, calculate the new surface area. What is the percentage increase?

Answer: Surface area at 8m: 256π ≈ 804.25 m²; at 10m: 400π ≈ 1256.64 m²; Percentage increase ≈ 56%

Question 7: A spherical mirror has a surface area of 502.4 cm². Determine its radius, using 4πr² and π ≈ 3.14.

Answer: r ≈ 4

Question 8: Identify and correct the error in this calculation: A sphere with radius 7 cm has a surface area calculated as 4π×7²= 4×3.14×49= 615.86 cm², but the student reports the answer as 433.86 cm².

Answer: The student used 4πr² correctly but made a calculation mistake; the correct value is approximately 615.86 cm², not 433.86 cm².

Question 9: A spherical fruit has a radius of 6 cm. Calculate its surface area. Then, determine the surface area if the radius increases by 50%.

Answer: Initial: 4π×6²= 144π ≈ 452.39 cm²; increased radius: 9 cm; new surface area: 4π×81= 324π ≈ 1017.88 cm²; percent increase ≈ 125%

Question 10: Challenge: Derive the formula for the surface area of a sphere starting from the volume formula V = (4/3)πr³, explaining the steps involved.

Answer: This is a derivation; students should discuss that surface area relates to the rate of change of volume with respect to radius, leading to SA = 4πr².