Geometric Properties : Volume and Surface Area

(1) A cylindrical tank has a radius of 4 meters and a height of 10 meters. What is the volume of the tank?

• (a) 160π m³
• (b) 160π m³
• (c) 80π m³
• (d) 320π m³

Answer

Correct Answer: (b) 160π m³
Explanation: The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \).
\[ V = \pi \times 4^2 \times 10 = 160\pi \, m^3 \]

(2) A cube has a surface area of 54 cm². What is the length of one side of the cube?

• (a) 9 cm
• (b) 3 cm
• (c) 18 cm
• (d) 6 cm

Answer

Correct Answer: (b) 3 cm
Explanation: The surface area \( S \) of a cube with side length \( a \) is \( S = 6a^2 \).
\[ 54 = 6a^2 \]
\[ a^2 = 9 \]
\[ a = 3 \, cm \]

(3) A cone has a base diameter of 12 cm and a volume of 144π cm³. What is the height of the cone?

• (a) 12 cm
• (b) 6 cm
• (c) 18 cm
• (d) 24 cm

Answer

Correct Answer: (b) 6 cm
Explanation: The volume \( V \) of a cone with radius \( r \) and height \( h \) is \( V = \frac{1}{3}\pi r^2 h \).
\[ 144\pi = \frac{1}{3}\pi \times \left(\frac{12}{2}\right)^2 \times h \]
\[ h = \frac{144\pi \times 3}{\pi \times 36} \]
\[ h = 6 \, cm \]

(4) A rectangular prism has a length of 8 cm, a width of 6 cm, and a height of 4 cm. What is the total surface area of the prism?

• (a) 120 cm²
• (b) 160 cm²
• (c) 192 cm²
• (d) 208 cm²

Answer

Correct Answer: (d) 208 cm²
Explanation: The total surface area \( A \) of a rectangular prism with length \( l \), width \( w \), and height \( h \) is given by \( A = 2lw + 2wh + 2lh \).
\[ A = 2 \times 8 \times 6 + 2 \times 6 \times 4 + 2 \times 8 \times 4 = 208 \, cm^2 \]

(5) A sphere has a radius of 5 cm. What is the volume of the sphere?

• (a) \( \frac{125}{3}\pi \) cm³
• (b) \( \frac{500}{3}\pi \) cm³
• (c) \( \frac{500}{3}\pi \) cm³
• (d) \( \frac{625}{3}\pi \) cm³

Answer

Correct Answer: (c) \( \frac{500}{3}\pi \) cm³
Explanation: The volume \( V \) of a sphere with radius \( r \) is given by \( V = \frac{4}{3}\pi r^3 \).
\[ V = \frac{4}{3}\pi \times 5^3 = \frac{500}{3}\pi \, cm^3 \]