6. Volumes of Pyramids
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A regular hexagon consists of six equilateral triangles.
About 77.99 cubic inches
The deck prism shown here is composed of the following three solids (from the bottom):
We are asked to find the volume of the deck prism. The base of each solid is a regular hexagon. A regular hexagon consists of six equilateral triangles.
Therefore, the area of a regular hexagon of an edge length of a is 6 times the area of an equilateral triangle with the same edge. By the formula for the area of an equilateral triangle, we get the following. c Area of Regular Hexagon &=6 * c Area of Equil. Triangle &=6 * a^2sqrt(3)/4 &= 3a^2sqrt(3)/2 Now, we will use the above formula and the formulas for the volume of a prism and for the volume of a pyramid, and we will find the volumes of Top, Middle, and Bottom parts of the given composite solid.
| Solid | Top | Middle | Bottom |
|---|---|---|---|
| Edge Length (a) | 3 | 3.25 | 3.5 |
| 3a^2sqrt(3)/2 | 3( 3)^2sqrt(3)/2≈ 23.38 | 3( 3.25)^2sqrt(3)/2≈ 27.44 | 3( 3.5)^2sqrt(3)/2≈ 31.83 |
| Area of Base | B≈ 23.38 | B≈ 27.44 | B≈ 31.83 |
| Height | h= 3 | h= 0.25 | h= 1.5 |
| Volume | V_1=1/3 B h | V_2= B h | V_3= B h |
| V_1=1/3( 23.38)( 3)=23.38 | V_2= 27.44( 0.25)=6.86 | V_3= 31.83( 1.5)≈ 47.75 |
Now, let's add the volumes of Top, Middle, and Bottom parts, to find the volume of the deck prism. c Volume of Deck Prism &= 23.38+6.86+47.75 &= 77.99 Finally, the volume of the nautical deck prism is about 77.99 cubic inches.