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Volume of Pentagonal Prism: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Volume of Pentagonal PrismVolume of Pentagonal Prism: Definition and ExamplesTable of ContentsVolume of a Pentagonal Prism Definition of Volume of a Pentagonal Prism

The volume of a pentagonal prism is the amount of space occupied by this three-dimensional solid in cubic units. A pentagonal prism features two pentagonal bases (top and bottom) connected by rectangular lateral faces. To calculate the volume, we multiply the area of the pentagonal base by the height of the prism, which gives us the formula V=Base Area×HeightV = \text{Base Area} \times \text{Height}V=Base Area×Height.

A pentagonal prism is a type of heptahedron with 15 edges, 10 vertices, and 7 faces. It has a pentagonal cross-section and can be classified as a right pentagonal prism when the bases are aligned directly on top of each other. For regular pentagonal prisms, the base area can be calculated as half the product of the perimeter and the apothem (the perpendicular distance from the center of the pentagon to any of its sides).

Examples of Volume of a Pentagonal Prism Example 1: Finding the Volume with Known Dimensions Problem:

Find the volume of a regular pentagonal prism whose apothem length of 3 inches, base length of 12 inches, and height of 15 inches.

Step-by-step solution:

Step 1, Identify the given values.

Apothem length (a) =3= 3=3 inches Base length (b) =12= 12=12 inches Height (h) =15= 15=15 inches

Step 2, Apply the formula for the volume of a regular pentagonal prism.

Volume = Base Area×Height=52×a×b×h \text{Base Area} \times \text{Height} = \frac{5}{2} \times a \times b \times hBase Area×Height=25​×a×b×h

Step 3, Substitute the values into the formula.

Volume = 52×3×12×15 \frac{5}{2} \times 3 \times 12 \times 1525​×3×12×15

Step 4, Calculate the final volume. =1,350  inch3= 1,350\; \text{inch}^{3}=1,350inch3

Therefore, the volume of the regular pentagonal prism is 1,350  inch31,350\; \text{inch}^{3}1,350inch3.

Example 2: Finding the Apothem Length Using Volume Problem:

Find the apothem length of the regular pentagonal prism if the height is 20 feet, the base length is 7 feet, and its volume is 1,680  ft31,680\; \text{ft}^{3}1,680ft3.

Step-by-step solution:

Step 1, Write down the known values.

Volume of pentagonal prism =1,680  ft3= 1,680\; \text{ft}^{3}=1,680ft3 Height (h) =20= 20=20 feet Base length (b) =7= 7=7 feet

Step 2, Use the formula for the volume of a regular pentagonal prism and rearrange it to find the apothem. Volume =52×a×b×h= \frac{5}{2} \times a \times b \times h=25​×a×b×h

Step 3, Substitute the known values into the formula. 1,680=52×a×7×201,680 = \frac{5}{2} \times a \times 7 \times 201,680=25​×a×7×20

Step 4, Solve for the apothem length (a). a=2×1,6805×7×20=4.8a = \frac{2 \times 1,680}{5 \times 7 \times 20} = 4.8a=5×7×202×1,680​=4.8 feet

Therefore, the regular apothem length of the pentagonal prism is 4.84.84.8 feet.

Example 3: Finding the Height Using Volume and Base Area Problem:

If the volume of a pentagonal prism is 528  ft3528\; \text{ft}^{3}528ft3 and the base area is 24  ft224\; \text{ft}^{2}24ft2, then find the height of the prism.

Step-by-step solution:

Step 1, Identify the given information.

Volume =528= 528=528 cubic feet Base area =24= 24=24 square feet

Step 2, Recall the formula relating volume, base area, and height. Volume of prism =base area×height= \text{base area} \times \text{height}=base area×height

Step 3, Substitute the known values and solve for height. 528=24×height528 = 24 \times \text{height}528=24×height

Step 4, Calculate the height by dividing. height=52824=22  ft\text{height} = \frac{528}{24} = 22\; \text{ft}height=24528​=22ft

Therefore, the height of the pentagonal prism is 22  ft22\; \text{ft}22ft.

Comments(1)CCricketFollowerVioletNovember 6, 2025I've been struggling to explain volume of pentagonal prisms to my students. This page made it so easy! The examples really helped them grasp it.

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