Pentagonal Prism - Definition, Formula, and Examples
Pentagonal Prism - Definition, Formula, and Examples
TL;DR
A pentagonal prism is a prism whose two bases are pentagons - five-sided polygons - connected by five flat rectangular faces. Because the two ends are identical and parallel, and the sides are straight rectangles, the shape belongs to the family of prisms, the same family as the rectangular box and the triangular prism.
What Is A Pentagonal Prism?
A pentagonal prism is a prism whose two bases are pentagons - five-sided polygons - connected by five flat rectangular faces. Because the two ends are identical and parallel, and the sides are straight rectangles, the shape belongs to the family of prisms, the same family as the rectangular box and the triangular prism.
Count the parts and a pentagonal prism always gives the same three numbers: 7 faces (2 pentagons + 5 rectangles), 15 edges, and 10 vertices (5 corners on each pentagon). A solid with exactly seven faces is also called a heptahedron. If you would like the flat five-sided shape that forms each base on its own, see the pentagon shape.
A pentagonal prism can be a right prism (the rectangles meet the bases at 90 degrees, so it stands straight) or an oblique prism (the bases slide sideways, so it leans). It can also be regular (both pentagons are regular, all five sides equal) or irregular. Most school problems use the right regular pentagonal prism, and that is the shape we measure below.
How the volume and surface-area formulas come from the shape
Every prism follows one rule: fill the base, then push it up through the height. That is why the volume is always the base area multiplied by the height. You are stacking copies of the base until you reach the top.
Volume.
V=B×h
Here BBB is the area of one pentagon base and hhh is the height of the prism (the distance between the two bases). For a regular pentagon with side length bbb and apothem aaa (the distance from the centre to the middle of a side), the base area is
B=52×b×a
so the full volume becomes
V=52×b×a×h.
Surface area. The surface is the two pentagon bases plus the five rectangles that wrap around the side (the lateral surface).
Total Surface Area=2B+P×h
PPP is the perimeter of the base pentagon, so P×h is the combined area of the five rectangles unrolled into one strip. For a regular pentagonal prism this simplifies to
TSA=5ab+5bh
where 5ab is the two bases written out and 5bh is the five identical rectangles.
| Quantity | Formula | What each symbol means |
|---|---|---|
| Base area BBB | 52,b,a | b = base side, a = apothem |
| Volume VVV | B×h | h = prism height |
| Lateral surface | P×h | P = base perimeter =5b |
| Total surface area | 2B+P,h=5ab+5bh | sum of bases and side rectangles |
Examples Of Pentagonal Prism
Example 1
How many faces, edges, and vertices does a pentagonal prism have?
Count each family of parts.
Faces: 2 pentagon bases + 5 rectangular sides = 7 faces.
Vertices: each pentagon has 5 corners, and there are 2 pentagons, so 5×2 = 10 vertices.
Edges: 5 edges around the top pentagon, 5 around the bottom, and 5 vertical edges joining them, so 5 + 5 + 5 = 15 edges.
A quick check with Euler's formula for solids, V−E+F=2: here 10−15+7=2.
Example 2
A student is asked for the volume of a regular pentagonal prism with base side 4 cm, apothem 2.75 cm, and height 10 cm.
The tempting move is to treat the pentagon base like a square and compute base area as side times side: 4×4=16 cm², then 16×10=160 cm³. That answer is wrong, and you can see why: a pentagon is not a square, so multiplying one side by itself does not give its area. The correct method uses the base-area formula:
B=52×4×2.75=27.5 cm²
Then push the base up through the height:
V=B×h=27.5×10=275 cm³.
Example 3
Find the total surface area of a regular pentagonal prism with base side 6 cm, apothem 4.13 cm, and height 12 cm.
Base area:
B=52×6×4.13=61.95 cm².
Two bases:
2B=123.9 cm².
Lateral surface (perimeter times height, with P=5×6=30 cm):
P×h=30×12=360 cm².
Total surface area:
TSA=123.9+360=483.9 cm².
Example 4
A pentagonal prism has base area 30 cm² and volume 210 cm³. Find its height.
Start from the volume rule and solve for the unknown.
V=B×h
210=30×h
h=210/30=7 cm.
Example 5
A gift box is a regular pentagonal prism with base side 5 cm, apothem 3.44 cm, and height 8 cm.
Base area:
B=52×5×3.44=43 cm².
Volume:
V=43×8=344 cm³.
Surface area:
TSA=2(43)+25×8=286 cm².
Example 6
Find the lateral surface area only of a pentagonal prism whose base perimeter is 35 cm and height is 9 cm.
Lateral Surface=P×h=35×9=315 cm².
Why The Pentagonal Prism Matters - "Five Walls Give More Angles To Defend"
The pentagonal prism is not a classroom curiosity. Its five-sided base is chosen on purpose whenever a design needs more than four flat walls but wants every wall to stay straight and every corner to stay rigid.
Fortress design. Star forts and bastion forts built from the 1500s onward used pentagon footprints so defenders had five walls of sightlines and no blind corners - more firing angles than a square could give.
Packaging and crystals. Some gift boxes, pencils, and nuts are pentagonal prisms because five faces distribute pressure evenly and resist rolling. Certain mineral crystals also grow in prism form.
Where the maths is going. Once you can measure one prism by "fill the base, push it up," you can measure any prism - hexagonal, octagonal, or the general prisms family - and then move to curved solids like the cylinder and the cone.
Key Takeaways
A pentagonal prism has 7 faces, 15 edges, and 10 vertices - two pentagon bases and five rectangular sides.
Volume = B×h (base area times height); for a regular prism, V=52bah.
Total surface area = 2B+Ph; lateral surface is the five rectangles only.
The apothem is essential - the base is a pentagon, not a square.
Real pentagonal prisms include the Pentagon building, some packaging, and certain crystals.