Cube and Cuboid

Cube and Cuboid: Complete Beginner Guide with Easy Examples

A cube and a cuboid are solid shapes. A solid shape has length, breadth and height, so it takes up space.

A dice is shaped like a cube. A brick, room, carton or book is usually shaped like a cuboid.

The two shapes are closely related. A cube is a special cuboid in which length, breadth and height are all equal.

This article begins with faces, edges and vertices. It then builds through surface area, volume, diagonals, open boxes, painting, cutting, painted small cubes, recasting, cavities and packing.

Basics: Face, Edge and Vertex

A face is a flat surface of a solid.

An edge is a line where two faces meet.

A vertex is a corner where three edges meet. The plural of vertex is vertices.

Both a cube and a cuboid have:

• $6$ faces
• $12$ edges
• $8$ vertices

The difference is in their dimensions.

Cube

Every face is an equal square. Every edge has the same length, usually called $a$.

$$l=b=h=a$$

Cuboid

Every face is a rectangle. Opposite faces are equal. Its length $l$, breadth $b$ and height $h$ may be different.

Length, Area and Volume Units

The unit tells us what kind of measurement we have.

• An edge uses a length unit, such as cm or m.
• A surface uses a square unit, such as ${\text{cm}}^2$ or ${\text{m}}^2$.
• Space inside a solid uses a cubic unit, such as ${\text{cm}}^3$ or ${\text{m}}^3$.

Why does area use a square unit? A rectangle uses two directions:

$$\text{length}\times \text{breadth}$$

Why does volume use a cubic unit? A solid uses three directions:

$$\text{length}\times \text{breadth}\times \text{height}$$

Writing $216{\text{cm}}^2$ for a volume is wrong even if the number $216$ is correct.

Volume of a Cube

A cube has equal length, breadth and height. Therefore:

$$V=a\times a\times a=a^3$$

Example: Find the volume of a cube with edge $6$ cm.

Solution:

1. Base area:

$$6\times 6=36{\text{cm}}^2$$

1. Multiply by height:

$$36\times 6=216{\text{cm}}^3$$

The same calculation is:

$$6^3=216$$

Answer: $216{\text{cm}}^3$

Meaning: The cube can contain $216$ unit cubes of size $1\text{cm}\times 1\text{cm}\times 1\text{cm}$.

Finding a Cube Edge from Volume

If $V=a^3$, then the edge is the cube root of the volume:

$$a=\sqrt[3]{V}$$

Example: A cube has volume $125{\text{cm}}^3$. Find its edge.

Solution:

We need a number that gives $125$ when multiplied by itself three times:

$$5\times 5\times 5=125$$

Therefore:

$$a=\sqrt[3]{125}=5\text{cm}$$

Answer: $5$ cm

Meaning: A cube of edge $5$ cm has dimensions $5\times 5\times 5$ cm.

Total Surface Area of a Cube

Imagine cutting a cube along some edges and opening it flat. We see six equal square faces. Each face has area $a^2$.

$$\text{TSA of cube}=6a^2$$

TSA means total surface area, which includes every outside face.

Example: Find the total surface area of a cube with edge $7$ cm.

Solution:

1. Area of one face:

$$7^2=49{\text{cm}}^2$$

1. Area of six faces:

$$6\times 49=294{\text{cm}}^2$$

Answer: $294{\text{cm}}^2$

Meaning: Covering the entire cube without overlap needs $294$ square centimetres of material.

Lateral Surface Area of a Cube

The word lateral means side surfaces only. For a cube, this means four side faces. The top and bottom are left out.

$$\text{LSA of cube}=4a^2$$

Example: Only the four side faces of a cube with edge $9$ cm are painted. Find the painted area.

Solution:

One face area is:

$$9^2=81{\text{cm}}^2$$

Four faces have area:

$$4\times 81=324{\text{cm}}^2$$

Answer: $324{\text{cm}}^2$

Meaning: The result measures the belt-like area around the cube.

Volume of a Cuboid

The base of a cuboid is a rectangle with area $lb$. Multiplying by height gives volume:

$$V=lbh$$

Example: Find the volume of a cuboid with length $8$ cm, breadth $5$ cm and height $4$ cm.

Solution:

1. Base area:

$$8\times 5=40{\text{cm}}^2$$

1. Stack $4$ such unit-height layers:

$$40\times 4=160{\text{cm}}^3$$

Answer: $160{\text{cm}}^3$

Meaning: The cuboid can hold $160$ one-cubic-centimetre blocks.

Finding a Missing Cuboid Dimension

From $V=lbh$, any missing dimension equals volume divided by the product of the other two.

$$h=\frac{V}{lb}$$

Example: A cuboid has base $15$ cm by $8$ cm and volume $720{\text{cm}}^3$. Find its height.

Solution:

$$\text{Base area}=15\times 8=120{\text{cm}}^2$$ $$h=\frac{720}{120}=6\text{cm}$$

Answer: $6$ cm

Meaning: Six layers of volume $120{\text{cm}}^3$ make the full $720{\text{cm}}^3$.

Total Surface Area of a Cuboid

A cuboid has three pairs of equal opposite faces:

• Two faces of area $lb$
• Two faces of area $bh$
• Two faces of area $hl$

Therefore:

$$\text{TSA}=2(lb+bh+hl)$$

Example: Find the total surface area of a cuboid measuring $10$ cm by $6$ cm by $4$ cm.

Solution:

$$lb=10\times 6=60$$ $$bh=6\times 4=24$$ $$hl=4\times 10=40$$

Add one of each type, then double:

$$2(60+24+40)=2\left(124\right)=248{\text{cm}}^2$$

Answer: $248{\text{cm}}^2$

Meaning: This is the cardboard area needed for a completely closed box, before allowing extra material for joining.

Lateral Surface Area of a Cuboid

The four side walls contain two $l\times h$ faces and two $b\times h$ faces.

$$\text{LSA}=2lh+2bh=2h(l+b)$$

Another way to understand it is:

$$\text{LSA}=\text{perimeter of base}\times \text{height}$$

Example: Four walls of a room measure $12$ m by $8$ m by $5$ m. Find the wall area, excluding floor and ceiling.

Solution:

$$\text{Wall area}=2h(l+b)$$ $$=2\times 5(12+8)$$ $$=10\times 20=200{\text{m}}^2$$

Answer: $200{\text{m}}^2$

Meaning: This is the paintable area of the four walls before subtracting doors and windows.

Open-Top Cuboid

An open-top box has no top face. It includes the bottom and four side faces.

$$\text{Open-top area}=lb+2h(l+b)$$

Example: Find the material area of an open tray with $l=14$ cm, $b=10$ cm and $h=6$ cm.

Solution:

Bottom area:

$$14\times 10=140{\text{cm}}^2$$

Four wall area:

$$2\times 6(14+10)=12\times 24=288{\text{cm}}^2$$

Total material:

$$140+288=428{\text{cm}}^2$$

Answer: $428{\text{cm}}^2$

Meaning: The missing top saves exactly $14\times 10=140{\text{cm}}^2$ compared with a closed cuboid.

Wire Frame of a Cube or Cuboid

A frame uses edges, not faces. A cube has $12$ equal edges.

$$\text{Cube frame length}=12a$$

A cuboid has four edges of each dimension:

$$\text{Cuboid frame length}=4(l+b+h)$$

Example: A wire is bent into a cube frame of edge $11$ cm. Find the wire length.

Solution:

$$12\times 11=132\text{cm}$$

Answer: $132$ cm

Meaning: Four top edges, four bottom edges and four vertical edges together use $132$ cm of wire.

Face Diagonal and Body Diagonal

A face diagonal lies on one rectangular face. A body diagonal joins opposite vertices through the inside of the solid.

For a rectangle of sides $x$ and $y$, Pythagoras gives:

$$d=\sqrt{x^2+y^2}$$

For a cube:

$$\text{Face diagonal}=a\sqrt{2}$$ $$\text{Body diagonal}=a\sqrt{3}$$

For a cuboid:

$$\text{Body diagonal}=\sqrt{l^2+b^2+h^2}$$

Body Diagonal of a Cube

Example: Find the body diagonal of a cube with edge $10$ cm.

Solution:

Base diagonal:

$$\sqrt{{10}^2+{10}^2}=10\sqrt{2}$$

Body diagonal:

$$\sqrt{{\left(10\sqrt{2}\right)}^2+{10}^2}=\sqrt{200+100}=\sqrt{300}=10\sqrt{3}\text{cm}$$

Answer: $10\sqrt{3}$ cm

Meaning: The body diagonal is longer than an edge and a face diagonal because it crosses all three dimensions.

Body Diagonal of a Cuboid

Example: Find the body diagonal of a cuboid measuring $6$ cm by $8$ cm by $24$ cm.

Solution:

First find the base diagonal:

$$\sqrt{6^2+8^2}=\sqrt{36+64}=10$$

Then combine it with height:

$$\sqrt{{10}^2+{24}^2}=\sqrt{100+576}=\sqrt{676}=26\text{cm}$$

Answer: $26$ cm

Meaning: The dimensions form two familiar right triangles, $6$-$8$-$10$ and $10$-$24$-$26$.

Dimensions Given in a Ratio

If $l:b:h=2:3:4$, write dimensions as $2x,3x,4x$. The value $x$ is the common scale.

Example: A cuboid has dimension ratio $2:3:4$ and volume $192{\text{cm}}^3$. Find its total surface area.

Solution:

$$\left(2x\right)\left(3x\right)\left(4x\right)=192$$ $$24x^3=192\Rightarrow x^3=8\Rightarrow x=2$$

Dimensions are:

$$\mathrm{4,6,8}\text{cm}$$

Now:

$$\text{TSA}=2(4\times 6+6\times 8+8\times 4)$$ $$=2(24+48+32)=208{\text{cm}}^2$$

Answer: $208{\text{cm}}^2$

Meaning: The ratio gives relative dimensions, while the volume tells how large the cuboid really is.

Painting Cost

Painting cost is area multiplied by rate per square unit.

$$\text{Cost}=\text{painted area}\times \text{rate}$$

Example: A closed tank measures $12$ m by $9$ m by $5$ m. All outside faces are painted at ₹18 per ${\text{m}}^2$. Find the cost.

Solution:

$$\text{TSA}=2(12\times 9+9\times 5+5\times 12)$$ $$=2(108+45+60)=426{\text{m}}^2$$ $$\text{Cost}=426\times 18=\text{Rs.}7,668$$

Answer: ₹7,668

Meaning: Every one of the $426$ square metres costs ₹18 to paint.

Cutting a Large Cube into Equal Small Cubes

If large edge is $A$ and small edge is $a$, the number along one direction is $\frac{A}{a}$. For a cube:

$$N={\left(\frac{A}{a}\right)}^3$$

Example: A cube of edge $18$ cm is cut into cubes of edge $3$ cm. Find their number.

Solution:

$$\frac{18}{3}=6$$ $$N=6\times 6\times 6=216$$

Answer: $216$ small cubes

Meaning: There are six cubes in each row, six rows in each layer and six layers.

Painted Cube: Understanding Locations

Suppose a large cube is painted on all six faces and cut into $n$ equal parts along each edge.

Every small cube belongs to one of four location types:

• Corner cube: exactly $3$ painted faces
• Non-corner edge cube: exactly $2$ painted faces
• Face-middle cube: exactly $1$ painted face
• Inner cube: no painted face

For $n\ge 2$, there are always $8$ corner cubes.

Exactly Two Faces Painted in a Cube

Each of the $12$ edges has $n$ small cubes. The two corner cubes on an edge have three painted faces, so remove them.

$$N_2=12(n-2)$$

Example: A painted cube is cut into $5$ parts along each edge. Find cubes with exactly two painted faces.

Solution:

$$\text{Per edge}=5-2=3$$ $$N_2=12\times 3=36$$

Answer: $36$ cubes

Meaning: The $8$ corners are excluded because each corner has three painted faces.

Exactly One Face Painted in a Cube

On each large face, remove one row along every edge. The remaining middle square has ${(n-2)}^2$ cubes. There are six faces.

$$N_1=6{(n-2)}^2$$

Example: A painted cube is cut into $8$ parts along each edge. Find cubes with exactly one painted face.

Solution:

One face contributes:

$$6\times 6=36$$

Six faces contribute:

$$6\times 36=216$$

Answer: $216$ cubes

Meaning: Each counted cube touches one outside face only, so it has exactly one painted face.

No Face Painted in a Cube

Remove one outer layer from every side. The inner cube has $n-2$ cubes along each direction.

$$N_0={(n-2)}^3$$

Example: A painted cube is cut into $6$ parts along each edge. Find cubes with no painted face.

Solution:

$$n-2=6-2=4$$ $$N_0=4^3=64$$

Answer: $64$ cubes

Meaning: The unpainted centre is a $4\times 4\times 4$ cube.

Painted Cuboid Cut into Small Cubes

Let the cuboid contain $x$, $y$ and $z$ small cubes along its three directions.

Exactly two faces painted:

$$N_2=4\left[\right(x-2)+(y-2)+(z-2\left)\right]$$

Exactly one face painted:

$$N_1=2\left[\right(x-2\left)\right(y-2)+(y-2\left)\right(z-2)+(z-2\left)\right(x-2\left)\right]$$

No face painted:

$$N_0=(x-2)(y-2)(z-2)$$

These formulas assume all six outside faces are painted and the divisions make equal small cubes.

Exactly Two Painted Faces in a Cuboid

Example: A painted cuboid is divided into $6$, $5$ and $4$ small-cube parts along its dimensions. Find cubes with exactly two painted faces.

Solution:

Edges along the $6$ direction contribute:

$$4(6-2)=16$$

Edges along the $5$ direction contribute:

$$4(5-2)=12$$

Edges along the $4$ direction contribute:

$$4(4-2)=8$$

Total:

$$16+12+8=36$$

Answer: $36$ cubes

Meaning: The formula counts all edge-middle cubes but excludes all eight corners.

Exactly One Painted Face in a Cuboid

Example: A painted cuboid is divided into $7$, $6$ and $5$ parts. Find cubes with exactly one painted face.

Solution:

Two $7\times 6$ faces:

$$2(7-2)(6-2)=2\left(5\right)\left(4\right)=40$$

Two $6\times 5$ faces:

$$2(6-2)(5-2)=2\left(4\right)\left(3\right)=24$$

Two $5\times 7$ faces:

$$2(5-2)(7-2)=2\left(3\right)\left(5\right)=30$$

Total:

$$40+24+30=94$$

Answer: $94$ cubes

Meaning: None of these cubes touches an edge, so every one has only its outward face painted.

At Least One Face Painted

“At least one” includes one, two or three painted faces. It is usually quickest to subtract the unpainted inner cubes from all cubes.

Example: A painted cuboid is divided into $9$, $7$ and $5$ parts. Find cubes with at least one painted face.

Solution:

Total cubes:

$$9\times 7\times 5=315$$

Inner cubes:

$$(9-2)(7-2)(5-2)=7\times 5\times 3=105$$

At least one face painted:

$$315-105=210$$

Answer: $210$ cubes

Meaning: The count includes every small cube touching the outside surface in any way.

Probability with Painted Cubes

When one small cube is selected at random:

$$P=\frac{\text{number with required paint}}{\text{total small cubes}}$$

Example: A cube is painted and cut into $6$ parts along each edge. Find the probability of exactly two painted faces.

Solution:

$$\text{Total}=6^3=216$$ $$\text{Favourable}=12(6-2)=48$$ $$P=\frac{48}{216}=\frac{2}{9}$$

Answer: $\frac{2}{9}$

Meaning: Out of every $9$ equally sized groups of small cubes, the probability represents $2$ such groups.

Melting and Recasting Without Loss

When solids are melted and reshaped without losing material, total volume remains the same. Surface area usually changes, but volume does not.

Example: Cubes of edges $3$ cm, $4$ cm and $5$ cm are recast into one cube. Find its edge.

Solution:

$$V=3^3+4^3+5^3$$ $$=27+64+125=216{\text{cm}}^3$$

Let new edge be $a$:

$$a^3=216=6^3$$ $$a=6\text{cm}$$

Answer: $6$ cm

Meaning: The new cube has a different outside area, but it contains exactly the same total volume.

Recasting Cubes into a Cuboid

Example: Thirty cubes of edge $4$ cm are recast into a cuboid with base $20$ cm by $12$ cm. Find its height.

Solution:

Old volume:

$$30\times 4^3=30\times 64=1920{\text{cm}}^3$$

New base area:

$$20\times 12=240{\text{cm}}^2$$

Height:

$$h=\frac{1920}{240}=8\text{cm}$$

Answer: $8$ cm

Meaning: Eight layers of base area $240{\text{cm}}^2$ use the entire old volume.

Recasting with Material Loss

If loss is $p\%$, usable volume is:

$$\text{Usable volume}=(1-\frac{p}{100})\times \text{original volume}$$

Example: Forty-eight cubes of edge $5$ cm are recast into a cuboid with base $30$ cm by $20$ cm. Ten percent volume is lost. Find height.

Solution:

Original volume:

$$48\times 5^3=48\times 125=6000{\text{cm}}^3$$

Usable volume:

$$0.90\times 6000=5400{\text{cm}}^3$$

Base area:

$$30\times 20=600{\text{cm}}^2$$

Height:

$$h=\frac{5400}{600}=9\text{cm}$$

Answer: $9$ cm

Meaning: The lost $600{\text{cm}}^3$ cannot appear in the new cuboid.

Joining Cubes Face to Face

When two cubes join, the touching face of each cube becomes hidden. One join hides two faces.

For $n$ cubes in one straight line, there are $n-1$ joins.

Example: Five cubes of edge $3$ cm are joined in a straight line. Find the outside surface area.

Solution:

$$\text{Exposed faces}=6\left(5\right)-2(5-1)$$ $$=30-8=22$$

One face area:

$$3^2=9{\text{cm}}^2$$

Outside area:

$$22\times 9=198{\text{cm}}^2$$

Answer: $198{\text{cm}}^2$

Meaning: The four contacts hide exactly eight square faces from view.

Cutting a Cube from a Corner

Removing a smaller cube from a corner has a surprising surface-area result.

Example: A cube of side $3$ cm is removed from one corner of a cube of side $12$ cm. Find the new total surface area.

Solution:

Original area:

$$6\times {12}^2=6\times 144=864{\text{cm}}^2$$

Removed area:

$$3\times 3^2=27{\text{cm}}^2$$

New exposed area:

$$3\times 3^2=27{\text{cm}}^2$$

Net change is zero.

Answer: $864{\text{cm}}^2$

Meaning: Volume decreases, but surface area stays unchanged for this exact corner-cube cut.

Cutting a Cubical Cavity into One Face

A cavity opening removes one square from the old surface. A cube-shaped hollow creates five new faces inside: four walls and a bottom.

Example: A cubical cavity of side $4$ cm is cut into the middle of one face of a cube of side $15$ cm. Find the new surface area.

Solution:

Original area:

$$6\times {15}^2=6\times 225=1350{\text{cm}}^2$$

One small square area:

$$4^2=16{\text{cm}}^2$$

Net increase:

$$(5-1)16=64{\text{cm}}^2$$

New area:

$$1350+64=1414{\text{cm}}^2$$

Answer: $1414{\text{cm}}^2$

Meaning: A hollow can increase surface area because it exposes new inside walls.

Largest Equal Cubes from a Cuboid

For the largest cube that divides a cuboid exactly, its edge must divide all three cuboid dimensions. Use their HCF.

HCF means highest common factor, the largest whole number that divides every given number exactly.

Example: Cut a $30\times 18\times 12$ cm cuboid into the largest equal cubes. Find unused volume.

Solution:

$$\text{HCF}(30,18,12)=6$$

Counts along dimensions:

$$\frac{30}{6}=\mathrm{5,}\frac{18}{6}=\mathrm{3,}\frac{12}{6}=2$$

Total cubes:

$$5\times 3\times 2=30$$

Used volume:

$$30\times 6^3=30\times 216=6480{\text{cm}}^3$$

Original volume:

$$30\times 18\times 12=6480{\text{cm}}^3$$

Unused volume:

$$6480-6480=0$$

Answer: $0{\text{cm}}^3$

Meaning: Choosing an HCF-sized cube makes every dimension divide exactly, so no space remains.

Packing Fixed-Size Complete Cubes

If cube size is fixed and does not divide a dimension, take only the whole-number part of each division. This is the floor value.

Example: Pack complete cubes of edge $30$ cm inside a $64\times 45\times 30$ cm cuboid. Find unused volume.

Solution:

Complete cube counts:

$$2,1,1$$

Total complete cubes:

$$2\times 1\times 1=2$$

Cuboid volume:

$$64\times 45\times 30=86400{\text{cm}}^3$$

Used volume:

$$2\times {30}^3=2\times 27000=54000{\text{cm}}^3$$

Unused volume:

$$86400-54000=32400{\text{cm}}^3$$

Answer: $32400{\text{cm}}^3$

Meaning: Empty strips remain because $30$ does not divide $64$ or $45$ exactly.

Open Box Made from a Rectangular Sheet

If squares of side $x$ are cut from all four corners of an $L\times B$ sheet, folding creates:

$$\text{new length}=L-2x$$ $$\text{new breadth}=B-2x$$ $$\text{height}=x$$

Therefore:

$$V=x(L-2x)(B-2x)$$

Example: A $54$ cm by $36$ cm sheet has $6$ cm squares removed from every corner. Find the open box volume.

Solution:

New length:

$$54-2\left(6\right)=42\text{cm}$$

New breadth:

$$36-2\left(6\right)=24\text{cm}$$

Height:

$$6\text{cm}$$

Volume:

$$42\times 24\times 6=6048{\text{cm}}^3$$

Answer: $6048{\text{cm}}^3$

Meaning: The removed square side becomes the box height, while both sheet dimensions shrink by twice that side.

Common Mistakes

Confusing cube and cuboid

A cube has all equal dimensions. A cuboid may have three different dimensions.

Mixing square and cubic units

Surface area uses square units. Volume uses cubic units.

Using total surface area for four walls

Lateral area excludes top and bottom.

Counting a top face on an open box

List the faces physically present before using a formula.

Using face diagonal when body diagonal is required

Cube face diagonal is $a\sqrt{2}$, while body diagonal is $a\sqrt{3}$.

Counting painted corners as exactly two painted faces

Corner cubes have three painted faces when all six large faces are painted.

Counting edge cubes inside the exactly-one-face group

Exactly-one-face cubes must lie away from all edges.

Forgetting to remove two outer layers

The inner count uses $n-2$, not $n-1$, because there is an outer layer on both ends.

Using cube formulas for a divided cuboid

When direction counts differ, use $x,y,z$ formulas rather than replacing all with one $n$.

Adding surface areas during recasting

Recasting preserves volume, not surface area.

Applying loss to the new dimensions instead of volume

First reduce the available volume by the loss percentage, then use the new-shape formula.

Counting joined faces as exposed

Every face-to-face join hides two faces.

Assuming every cut lowers surface area

A cut can expose new faces. Compare removed outside area with newly exposed inside area.

Using HCF when cube size is already fixed

HCF finds the largest exact cube size. For a fixed size, count complete cubes with whole-number division.

Subtracting the corner square only once from a sheet dimension

One square is cut from each end, so base length and breadth each lose $2x$.

Cheat Sheet

Concept	Formula	Meaning or condition
Cube volume	$a^3$	Three equal dimensions
Cube total surface area	$6a^2$	All six faces
Cube lateral area	$4a^2$	Four side faces
Cube face diagonal	$a\sqrt{2}$	Across one square face
Cube body diagonal	$a\sqrt{3}$	Through the inside
Cube frame length	$12a$	All twelve edges
Cuboid volume	$lbh$	Base area times height
Cuboid total surface area	$2(lb+bh+hl)$	All six faces
Cuboid lateral area	$2h(l+b)$	Four walls
Open-top cuboid area	$lb+2h(l+b)$	Bottom and four walls
Cuboid body diagonal	$\sqrt{l^2+b^2+h^2}$	Opposite vertices
Cuboid frame length	$4(l+b+h)$	Four edges of each dimension
Small cubes from a cube	${\left(\frac{A}{a}\right)}^3$	Exact division required
Cube, three painted faces	$8$	All corners, for $n\ge 2$
Cube, exactly two painted	$12(n-2)$	Non-corner edge cubes
Cube, exactly one painted	$6{(n-2)}^2$	Face interiors
Cube, no paint	${(n-2)}^3$	Inner cube
Cuboid, exactly two painted	$4\left[\right(x-2)+(y-2)+(z-2\left)\right]$	Edge interiors
Cuboid, exactly one painted	$2\left[\right(x-2\left)\right(y-2)+(y-2\left)\right(z-2)+(z-2\left)\right(x-2\left)\right]$	Face interiors
Cuboid, no paint	$(x-2)(y-2)(z-2)$	Inner cuboid
At least one painted	Total cubes $-$ inner cubes	Easier than separate addition
Recasting without loss	Old total volume $=$ new volume	Shape changes only
Recasting with loss $p\%$	Usable volume $=(1-\frac{p}{100})V$	Apply loss to volume
Joined cubes exposed faces	$6n-2J$	$J$ is number of joins
Largest exact cube edge	$\text{HCF}(l,b,h)$	No unused volume
Open box from sheet	$x(L-2x)(B-2x)$	Corner square side is $x$

Safe Shortcuts

• Draw or list the faces included before selecting a surface-area formula.
• Use square units for area and cubic units for volume.
• Think “base area times height” for any cuboid volume.
• For a body diagonal, square all three dimensions, add and take the square root.
• In painted cubes, first identify location: corner, edge, face middle or inside.
• “Exactly two painted” means edge but not corner.
• “Exactly one painted” means face middle but not edge.
• “No paint” means remove one outer layer from both ends of every dimension.
• For “at least one painted,” subtract inner cubes from total cubes.
• In recasting, write a volume-balance equation before calculating dimensions.
• Apply material loss to total old volume before equating it to new volume.
• Every face-to-face join hides two faces.
• A corner cube cut removes and exposes three equal faces, so surface area stays unchanged.
• Use HCF only for the largest cube that divides all dimensions exactly.
• For a folded open box, subtract $2x$ from both sheet dimensions and use height $x$.

A shortcut is safe only after the included faces, inside volume or small-cube location is clear. It should shorten correct reasoning, not replace the picture of the solid.

Frequently Asked Questions

What is the difference between a cube and a cuboid?

A cube has six equal square faces and equal dimensions. A cuboid has rectangular faces and may have different length, breadth and height.

How many faces, edges and vertices does a cube have?

A cube has $6$ faces, $12$ edges and $8$ vertices. A cuboid has the same counts.

What is the volume formula for a cube?

For edge $a$, volume is $a^3$.

What is the volume formula for a cuboid?

For length $l$, breadth $b$ and height $h$, volume is $lbh$.

What is total surface area?

It is the combined area of every outside face of a closed solid.

What is lateral surface area?

It is the area of the side faces only. For a cuboid, it is the area of four walls without floor and ceiling.

What is the body diagonal of a cube?

It is the straight inside line between opposite vertices, with length $a\sqrt{3}$.

What is the body diagonal of a cuboid?

It is $\sqrt{l^2+b^2+h^2}$.

How many small cubes have three painted faces?

When all faces are painted and $n\ge 2$, the eight corner cubes have three painted faces.

How many small cubes have exactly two painted faces?

For a cube divided into $n$ parts per edge, the count is $12(n-2)$.

How many small cubes have exactly one painted face?

For a cube, the count is $6{(n-2)}^2$.

How many small cubes have no painted face?

For a cube, the inner count is ${(n-2)}^3$.

Why do painted-cube formulas use n minus 2?

One outside position is removed from each end of a direction, so two positions are removed in total.

What remains constant during melting and recasting?

Volume remains constant when there is no material loss. Surface area and dimensions can change.

How is recasting handled when material is lost?

Find original volume, keep only the percentage that remains, then equate usable volume to the new solid's volume.

Why does removing a cube from a corner not change surface area?

Three equal outside squares disappear and three equal inside cut squares appear, so the changes cancel.

How do I find the largest cube cut from a cuboid?

Find the HCF of length, breadth and height. That HCF is the largest exact cube edge.

How is an open box formed from a sheet?

Cut equal squares of side $x$ from four corners. The base becomes $(L-2x)\times (B-2x)$ and height becomes $x$.

Summary

A cube is a cuboid with all three dimensions equal. Both shapes have six faces, twelve edges and eight vertices. Surface area measures covering in square units, while volume measures inside space in cubic units.

Cube formulas follow from equal square faces and equal dimensions. Cuboid formulas come from three pairs of opposite rectangles. Body diagonals use Pythagoras across three perpendicular dimensions.

Painted-cube counting becomes simple when small cubes are grouped by location: corners, edge middles, face middles and the inner block. Recasting keeps volume when there is no loss. Joins hide faces, cavities expose new faces, HCF gives the largest exact cube size, and folded-sheet boxes lose twice the corner-cut size from each base dimension.