Geometric Properties : Volume and Surface Area

(11) A triangular pyramid has a base with sides of length 6 cm, 8 cm, and 10 cm, and a height of 12 cm. Calculate the volume of the pyramid.

• (a) 144 cm³
• (b) 192 cm³
• (c) 216 cm³
• (d) 240 cm³

Answer

Correct Answer: (b) 192 cm³
Explanation: The volume \( V \) of a triangular pyramid with base area \( B \) and height \( h \) is given by \( V = \frac{1}{3}Bh \).
\[ B = \frac{1}{2} \times 6 \times 8 = 24 \, cm^2 \]
\[ V = \frac{1}{3} \times 24 \times 12 = 192 \, cm^3 \]

(12) A cylinder has a volume of 1256 cm³ and a height of 10 cm. What is the radius of the cylinder?

• (a) 4 cm
• (b) 5 cm
• (c) 6 cm
• (d) 7 cm

Answer

Correct Answer: (d) 7 cm
Explanation: The volume \( V \) of a cylinder with radius \( r \) and height \( h \) is given by \( V = \pi r^2 h \).
\[ \pi r^2 \times 10 = 1256 \]
\[ r^2 = \frac{1256}{10\pi} \]
\[ r = \sqrt{\frac{1256}{10\pi}} \approx 7 \, cm \]

(13) A cube has a diagonal length of 10√2 cm. What is the volume of the cube?

• (a) 500 cm³
• (b) 100 cm³
• (c) 250 cm³
• (d) 625 cm³

Answer

Correct Answer: (c) 250 cm³
Explanation: The diagonal \( d \) of a cube with side length \( a \) is given by \( d = \sqrt{3}a \).
\[ \sqrt{3}a = 10\sqrt{2} \]
\[ a = \frac{10\sqrt{2}}{\sqrt{3}} \]
The volume \( V \) of the cube is \( a^3 \).
\[ V = \left(\frac{10\sqrt{2}}{\sqrt{3}}\right)^3 = 250 \, cm^3 \]

(14) A sphere has a surface area of 154 cm². Calculate its radius.

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

Answer

Correct Answer: (b) 3 cm
Explanation: The surface area \( A \) of a sphere with radius \( r \) is given by \( A = 4\pi r^2 \).
\[ 4\pi r^2 = 154 \]
\[ r^2 = \frac{154}{4\pi} \]
\[ r = \sqrt{\frac{154}{4\pi}} \approx 3 \, cm \]

(15) A cone has a volume of 150π cm³ and a base radius of 5 cm. Calculate its height.

• (a) 6 cm
• (b) 9 cm
• (c) 12 cm
• (d) 15 cm

Answer

Correct Answer: (b) 9 cm
Explanation: The volume \( V \) of a cone with radius \( r \) and height \( h \) is given by \( V = \frac{1}{3}\pi r^2 h \).
\[ \frac{1}{3}\pi \times 5^2 \times h = 150\pi \]
\[ h = \frac{150\pi \times 3}{\pi \times 25} \]
\[ h = 9 \, cm \]