Case 1
- Pentagon side
- 4
- Prism length
- 10
275.276384
Open with these valuesPentagonal Prism Volume Calculator
275.276384units³
Result: 275.276384 units³A regular pentagon of side a covers ¼√(5(5 + 2√5)) × a², about 1.720 × a². Multiply by the length for the volume: side 4, length 10 gives about 275.3 cubic units. Enter one edge of the pentagon, not the distance from the centre to a side.
The numbers at a glance
Held fixed: Pentagon side 4.000.
| Prism length | Result |
|---|---|
| 2.500 | 68.819096 |
| 5.000 | 137.638192 |
| 7.500 | 206.457288 |
| 10.000Your value | 275.276384 |
| 12.500 | 344.095480 |
| 15.000 | 412.914576 |
| 17.500 | 481.733672 |
| 20.000 | 550.552768 |
Worked examples
Case 2
- Pentagon side
- 1
- Prism length
- 1
1.720477
Open with these valuesCase 3
- Pentagon side
- 5
- Prism length
- 10
430.119350
Open with these valuesHow it's calculated
V = ¼√(5(5 + 2√5)) × a² × L
- StepMeasure one of the five equal edges of the pentagon.
- StepMeasure the length between the two pentagonal ends, same unit.
- ResultRead the volume in the cube of the unit you entered.
Reference table
| Side, length | Exact | Volume |
|---|---|---|
| 0.5, 3 | 3√(5(5 + 2√5)) ÷ 16 | 1.290358 |
| 1, 1 | √(5(5 + 2√5)) ÷ 4 | 1.720477 |
| 2, 6 | 6√(5(5 + 2√5)) | 41.291458 |
| 4, 10 | 40√(5(5 + 2√5)) | 275.276384 |
| 5, 10 | 125√(5(5 + 2√5)) ÷ 2 | 430.119350 |
Questions
How do I calculate the volume of a pentagonal prism?
Find the area of the pentagonal end, ¼√(5(5 + 2√5)) × a², then multiply by the prism length. The area factor is about 1.720477, so a side of 4 and a length of 10 give 27.527638 × 10 = 275.276384 cubic units.
What is the area of a regular pentagon?
A regular pentagon with side a has area ¼√(5(5 + 2√5)) × a², about 1.720477 × a². For a side of 4 that is 27.527638 square units, and that cross-section is the same all the way along the prism.
Do I enter the side or the distance from the centre?
The side, meaning one of the five equal edges. The distance from the centre to the middle of a side — the apothem — is about 0.688 times the side, and entering it here would leave the volume at only 47 % of the truth.
Why does the volume equal the cross-section times the length?
A prism is its cross-section swept along a straight line, so its volume is always cross-section area times length. That holds for any prism — pentagonal, hexagonal or rectangular. Doubling the length doubles the volume while the cross-section stays put.
Which units does the answer use?
Whatever you entered, cubed. Side and length in centimetres give cubic centimetres, in inches cubic inches. Both inputs must use the same unit.
Sources and last check
Information, not professional advice.