Total skin of the solid, in square units.

Math CalculatorsRegular pentagon extruded

Pentagonal Prism Calculator

2026-09-29 8 min

Free pentagonal prism calculator: regular 5-gon base times height with full surface.

Enter side and height for pentagonal prism calculator volume and surface with base-polygon math.

Grade level: 7–11; Curriculum band
Difficulty: Beginner; For this topic
Time to master: 1 week; With daily practice
Tools on page: 10 on page; Calculator + practice

Regular pentagon extruded

Try:

Base side (s)

Prism height (H)

Results update live as you type

Status:V = 165.16583 · S = 175.055277

Pentagonal Prism Calculator Summary

Side4

Height6

Base area27.527638

Volume165.16583

Surface175.055277

Breakdown

Measure	Value
Side (s)	4
Height (H)	6
Apothem	2.752764
Base area	27.527638
Volume	165.16583
Surface	175.055277

Substitution: n=5

Results

Optimal

Volume

165.16583cubic units

baseA x H

Total surface

175.055277sq units

lateral + 2 bases

Base area

27.527638sq units

n-gon

Apothem

2.752764units

base

Closed-form solid geometry on the dimensions given. Invalid solids report no volume rather than a guess.

Statistics

Live volume and surface from your current dimensions.

Volume

165.16583

cubic units

Surface

175.055277

sq units

Base area

27.527638

sq units

Apothem

2.752764

units

Visualizations

Volume vs surface

165.2Volume

Volume scale

Volume165.2

Formula

Regular 5-gon base extruded; apothem unlocks base area.

V = baseA x H (baseA = n s a/2)

Alternative Form

S = baseP H + 2 baseA

Variables

Symbol	Name	Unit	Description
s	Base side	units	Edge
H	Height	units	Extrusion

Explanation

Base area via P a /2; volume multiplies by height; band wraps perimeter.

Derivation

Lateral unwraps to baseP-by-H rectangle.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Record the given dimensions.

side s, height H, n=5

s:4

H:6

Result: Dims recorded

Step 2

Validate positivity.

all dims > 0

Result: Dimensions accepted

Step 3

Apply the volume formula.

V = baseA x H, baseA = n s a/2

Intermediate: baseA=27.527638; V=165.16583

Result: V = 165.16583

Step 4

Apply the surface-area formula.

S = baseP x H + 2 baseA

Intermediate: S=175.055277

Result: S = 175.055277

Step 5

Compute auxiliaries (diagonals, slant heights).

a = s/(2 tan(pi/n))

Result: Base area = 27.527638; Apothem = 2.752764

Step 6

State the finished sentence.

dims to V + S

Result: 5-prism s=4 H=6: V=165.16583, S=175.055277

Final Answer

165.16583

Worked Examples

See how the calculator works with different scenarios.

Scenario

Defaults.

Inputs

side:4

Calculation

see live volume

Result

computed live

Interpretation

One side plus height fixes all.

Understanding Your Result

Your Result: Volume: 165.16583 cubic units

Typical range: Keep units consistent: lengths in, cubes out for volume, squares out for surface. Tripling edge multiplies volume 27x.

What This Means

Volume measures capacity (cubic units); surface area measures skin (square units). Both come from the same dimensions.

Why It Matters

Concrete, water, packaging, paint and insulation all bill from these two numbers — fill vs cover.

Professional Interpretation

Report volume in cubic units and surface in square units; add waste to both material orders.

Recommendations

•Keep one unit across inputs
•Add 5-10 percent waste
•Distinguish lateral vs total surface
•Round only finals

Learn — The Complete Guide

Pentagonal Prism Calculator computes volume and surface area from direct dimensions using closed-form solid geometry.

The core concept

Base area via P a /2; volume multiplies by height; band wraps perimeter.

Volume is cubic; surface is quadratic
Prism V=Bh; pyramid/cone divide by 3
Lateral excludes bases

Derivations — why the formulas work

Where volume comes from

1Slice into thin slabs of base area
2Integrate along height (Cavalieri)
3Tapered solids average to one third

Where you meet this in real life

▸Concrete and fill volumes

▸Tank and box capacity

▸Paint and cladding coverage

▸Packaging and 3D printing

▸Exam solid geometry

▸Concrete and water volumes

▸Tank capacity

▸Box and packaging design

▸Paint and insulation coverage

Learn this first — prerequisites in order

1Base areas cold
2Root and power fluency
3Length unit conversion

Careers that use this daily

•Mason — concrete loads
•Plumber — tank sizes
•Packer — box dims

Why it helps

•Instant volume plus surface
•Auxiliaries included (diagonals, slants)
•Labeled solid diagram
•Copy-ready breakdown

Best practices

•Sketch with labels first
•Measure in one unit
•Separate lateral vs total surface
•Add waste to orders

Limitations — what this tool does not do

•Single solid per run
•Ideal geometry only
•Exact inputs assumed

Curriculum map — where this topic lives in school

Level	Grades	Standards	Skills
Middle school	7–8	7.G, 8.G	Boxes and cylinders · Surface nets
High school	9–11	G-GMD, G-MG	Cones, spheres, platonic solids · Modeling

Timeline — how this idea grew up

c. 300 BCE
Euclid volumes
Elements proves prism and pyramid volumes.
250 BCE
Archimedes sphere
Sphere is 2/3 of its cylinder — his proudest result.
1600s
Calculus confirms
Integration reproduces every classic volume.

A little history

• Archimedes asked for sphere+cylinder on his tomb.
• Platonic solids named for Plato dialogues.

Exam-day checklist

• Sketch the base first.
• Circle lateral vs total explicitly.

Myths vs facts

Myth: Volume and surface grow together.

Truth: Volume is cubic, surface quadratic: big solids hold far more per skin (square-cube law).

Myth: Slant height equals vertical height.

Truth: Slant runs along the face; vertical drops straight — slant is always longer.

Myth: Any dims make a solid.

Truth: All lengths must be positive; prisms need consistent base and height.

Practical Tips

•Measure in one unit
•Name lateral vs total surface
•Check slant vs vertical height
•Add waste to concrete and paint
•Track cubic vs square units

Common Mistakes

Mixing radius and diameter

Halve diameters before pi r squared formulas.

Using vertical height as slant

Slant = sqrt(r^2+h^2) for cones, longer than h.

Reading volume units as area

Volume is cubic; surface is square.

Assumptions

This calculator makes the following assumptions to provide estimates:

Ideal solid: Perfect geometry; wall thickness ignored.

Positive inputs: All dimensions exceed zero.

Consistent units: One length unit in; cubes/squares out.

Limitations

Be aware of these limitations when using this calculator:

Single solid: Assemblies need multiple runs.

No tolerance: Manufacturing variation not modeled.

Static shape: No deformation or fill dynamics.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I get base area?

Regular 5-gon: baseA = n s a/2 with a = s/(2 tan(pi/n)).

Lateral vs total?

Lateral (band) is baseP x H; total adds both n-gon ends.

Where used?

Bars, columns, honeycomb cells and pencil bodies.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Concrete ordering by volume

From dims to truckloads.

Paint coverage by surface

Skin into liters and coats.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Volume: Three-dimensional capacity enclosed, in cubic units.
Surface area: Total skin of the solid, in square units.
Lateral surface: Side skin excluding bases.
Slant height: Distance along a cone/pyramid face from apex to base edge.
Space diagonal: Longest interior segment, corner to opposite corner.
Apothem (base): Base-polygon center-to-side distance for prisms/pyramids.

Authoritative References

Trusted sources and standards referenced by this calculator.

Math Is Fun — Solids Khan Academy — Volume and surface Wikipedia — Volume

Applicable Standards

Common Core G-GMD — volume and surfaceNCTM Geometry expectations

Diagrams & Illustrations

Visual intuition for every core idea. Select a card to enlarge it.

Did You Know?

Short intuition builders — great for class warm-ups.

Sphere wins capacity

Among closed shapes the sphere holds the most volume per surface — why bubbles, tanks and planets round off.

Platonic solids are five only

Tetrahedron, cube, octahedron, dodecahedron, icosahedron: no sixth regular solid exists.

Square-cube law rules biology

Elephants need thick legs because weight (volume) outruns bone area as size grows.

Compare Side by Side

Prism volume is always base times height.

Part	Value	Formula	Note
Volume	165.16583	baseA x H	extruded
Surface	175.055277	band + 2 bases	skin

Recent Calculations

Your last runs on this page — stored only in this browser, never uploaded.

5-prism s=4 H=6: V=165.16583, S=175.055277
Enter side and height for pentagonal prism calculator volume and surface with base-polygon math.

Study Plan

A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.

Prerequisites

▸Area of bases
▸Squares and roots
▸Pi and circles

Mastery progress — 0%

I pick V=Bh vs V=Bh/3 correctly
I separate lateral and total
I compute slants and diagonals
I track units

1
Week 1 · ~15 min/day
Prisms and cylinders
Goals
- Apply V=Bh fluently
- Separate lateral vs total
- Track cubic units
Practice
Solve 10 boxes and cans on this page.
2
Week 2 · ~20 min/day
Pyramids, cones, spheres
Goals
- Use third-factor 1/3 correctly
- Compute slant heights
- Size tanks from volume
Practice
Measure a box, can and ball at home.

Exam tips

•Draw the base first, then extrude.
•Write V and S formulas before numbers.

Practice Problems

0/4 solved

Try each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.

Basicseasy

Compute volume and surface with defaults.

Unitsmedium

Convert volume to liters.

Scalingmedium

Double every edge. What happens?

Choicehard

Which holds more per surface: cube or sphere?

Printable Worksheet

Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.

Problems Name

Arithmetic practice — set #1Name: __________ · Date: __________

1.18 × 47 =
2.8 ÷ 11 =
3.38 × 26 =
4.35 ÷ 4 =
5.21 − 16 =
6.2 + 26 =
7.14 ÷ 12 =
8.23 ÷ 3 =
9.49 ÷ 7 =
10.22 − 17 =
11.41 + 18 =
12.47 − 35 =

Formula Cheat Sheet

Every formula on this page in one printable table.

Formula	Meaning	Example
V = baseA x H (baseA = n s a/2)	Headline volume	Plug in defaults
S = sum of faces	Headline surface	Add every face
d = sqrt(l^2+w^2+h^2)	Box diagonal	Space Pythagoras

Tip: every interactive block on this page — checklist, practice, worksheet, history — works offline and saves progress in your browser.

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Page Information

Last Updated:

2026-09-29

Version:

3.1.0-math

References:

License:

Free to use for educational and professional purposes.