Question

a gem is cut in the shape of two square pyramids connected at their base. the length of each side of the base is \\(6\text{ cm}\\) and the height of each pyramid is \\(4\text{ cm}\\).
determine the volume of the gem:

hint: use the formula sheet to determine the formula(s) needed to solve the problem.

a. \\(96\text{ cm}^3\\)
b. \\(144\text{ cm}^3\\)
c. \\(48\text{ cm}^3\\)
d. \\(88\text{ cm}^3\\)

Explanation:

⚡ Using what you learned: volume of prisms, pyramids, cylinders, cones, spheres

Step 1: Volume of a single square pyramid

The formula for the volume of a pyramid is:

$$ V_{\text{pyramid}} = \frac{1}{3} \cdot B \cdot h $$

where \( B \) is the area of the base and \( h \) is the height.

For a square base with side length \( s = 6\text{ cm} \):

$$ B = s^2 = 6^2 = 36\text{ cm}^2 $$

Given the height \( h = 4\text{ cm} \):

$$ V_{\text{pyramid}} = \frac{1}{3} \cdot 36 \cdot 4 = 12 \cdot 4 = 48\text{ cm}^3 $$

Step 2: Total volume of the gem

The gem consists of two identical square pyramids connected at their base:

$$ V_{\text{total}} = 2 \cdot V_{\text{pyramid}} $$

$$ V_{\text{total}} = 2 \cdot 48 = 96\text{ cm}^3 $$

Answer:

A \( 96\text{ cm}^3 \)