Volume and Surface Area of a Pyramid (Video & Practice) - Free Printable

To find the volume of this shape, we need to break it down into simpler parts. The image shows a triangular bipyramid (or a double pyramid), which is made of two pyramids joined at their bases.

Here is the step-by-step solution:

Step 1: Identify the Base Area

The two pyramids share a common base, which is the shaded triangle in the middle.
* The base of this triangle is 14 in.
* The height of this triangle (the dashed line perpendicular to the base) is 12 in.

The formula for the area of a triangle is:
$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Calculating the area of the shared base:
$$ \text{Base Area} = \frac{1}{2} \times 14 \times 12 $$
$$ \text{Base Area} = 7 \times 12 $$
$$ \text{Base Area} = 84 \text{ sq in} $$

Step 2: Calculate the Volume of the Top Pyramid

The volume of a pyramid is calculated using the formula:
$$ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$

For the top pyramid:
* Base Area = $84 \text{ sq in}$
* Height = 13 in (given by the vertical dashed line)

$$ V_{\text{top}} = \frac{1}{3} \times 84 \times 13 $$
$$ V_{\text{top}} = 28 \times 13 $$
$$ V_{\text{top}} = 364 \text{ cu in} $$

Step 3: Calculate the Volume of the Bottom Pyramid

Looking at the diagram, the bottom pyramid is a reflection of the top one. It shares the same base and has the same height dimensions implied by the symmetry of the figure.
* Base Area = $84 \text{ sq in}$
* Height = 13 in

$$ V_{\text{bottom}} = \frac{1}{3} \times 84 \times 13 $$
$$ V_{\text{bottom}} = 364 \text{ cu in} $$

*(Note: Even if the bottom height were different, we would calculate it separately. However, in standard geometry problems of this type involving a single height label on an axis of symmetry, the heights are equal.)*

Step 4: Add the Volumes Together

To get the total volume of the entire solid, add the volume of the top pyramid and the bottom pyramid.

$$ \text{Total Volume} = V_{\text{top}} + V_{\text{bottom}} $$
$$ \text{Total Volume} = 364 + 364 $$
$$ \text{Total Volume} = 728 \text{ cu in} $$

Alternative Method:
You can also think of this as one large shape with the formula:
$$ V = \frac{1}{3} \times \text{Base Area} \times \text{Total Height} $$
Total Height = $13 + 13 = 26$
$$ V = \frac{1}{3} \times 84 \times 26 $$
$$ V = 28 \times 26 $$
$$ V = 728 $$

Final Answer:
728 cubic inches