Geometric Properties : Volume and Surface Area

(6) A right circular cone has a radius of 3 cm and a slant height of 5 cm. What is the total surface area of the cone?

• (a) 30π cm²
• (b) 20π cm²
• (c) 35π cm²
• (d) 40π cm²

Answer

Correct Answer: (c) 35π cm²
Explanation: The total surface area \( A \) of a cone with radius \( r \) and slant height \( l \) is given by \( A = \pi r(r + l) \).
\[ A = \pi \times 3 \times (3 + 5) = 35\pi \, cm^2 \]

(7) A cylinder has a radius of 6 cm and a height of 9 cm. Calculate the total surface area of the cylinder.

• (a) 432π cm²
• (b) 288π cm²
• (c) 270π cm²
• (d) 432π cm²

Answer

Correct Answer: (d) 432π cm²
Explanation: The total surface area \( A \) of a cylinder with radius \( r \) and height \( h \) is given by \( A = 2\pi r^2 + 2\pi rh \).
\[ A = 2\pi \times 6^2 + 2\pi \times 6 \times 9 = 432\pi \, cm^2 \]

(8) A rectangular box has dimensions 10 cm, 5 cm, and 4 cm. What is the volume of the box?

• (a) 150 cm³
• (b) 200 cm³
• (c) 400 cm³
• (d) 250 cm³

Answer

Correct Answer: (b) 200 cm³
Explanation: The volume \( V \) of a rectangular box with dimensions \( l \), \( w \), and \( h \) is given by \( V = lwh \).
\[ V = 10 \times 5 \times 4 = 200 \, cm^3 \]

(9) A cone has a base radius of 9 cm and a slant height of 15 cm. Find the total surface area of the cone.

• (a) 240π cm²
• (b) 270π cm²
• (c) 360π cm²
• (d) 405π cm²

Answer

Correct Answer: (d) 405π cm²
Explanation: The total surface area \( A \) of a cone with radius \( r \) and slant height \( l \) is given by \( A = \pi r(r + l) \).
\[ A = \pi \times 9 \times (9 + 15) = 405\pi \, cm^2 \]

(10) A sphere has a volume of \( \frac{256}{3}π \) cm³. Calculate its radius.

• (a) 2 cm
• (b) 4 cm
• (c) 3 cm
• (d) 5 cm

Answer

Correct Answer: (c) 3 cm
Explanation: The volume \( V \) of a sphere with radius \( r \) is given by \( V = \frac{4}{3}\pi r^3 \).
\[ \frac{256}{3}\pi = \frac{4}{3}\pi r^3 \]
\[ r^3 = \frac{64}{3} \]
\[ r = \sqrt[3]{\frac{64}{3}} = 3 \, cm \]