3D Solids: 12 Applications on Cube Nets, Painted Cubes, Stacks, Views and Water Levels

01

Opposite Faces on a Cube Net

propertyIn a net, two squares in one line with exactly one square between them fold to opposite faces

A cube net is a column of four squares numbered 1, 2, 3, 4 from top to bottom, with square 5 attached to the left of square 2 and square 6 to its right. When the net is folded into a cube, which face is opposite face 1? Which face is opposite face 5?

In the column, 1 and 3 have one square between them: opposite faces. So do 2 and 4.

In the column, 1 and 3 have exactly one square between them: they fold to opposite faces. So do 2 and 4.

step 1 of 4

Folding a strip of squares wraps it round the cube, so squares one apart in a straight line land on opposite faces. The rule works for the column and for the row through square 2.

All steps

In the column, 1 and 3 have exactly one square between them: they fold to opposite faces. So do 2 and 4.
In the row, 5, 2, 6: squares 5 and 6 have 2 between them, so they are opposite.
Opposite face 1: face 3. Opposite face 5: face 6.
Check: the three pairs 1-3, 2-4, 5-6 use all six faces once.

answer3; 6

techniqueNets

examsPSLE · SAT · GCSE Higher

Common pitfalls

Taking neighbors in the net, such as 1 and 2, as opposite: touching squares fold to touching faces.
Pairing 4 with 1 because they are the two ends of the column: the ends fold round to become neighbors.

02

Neighbors of a Face on the Folded Cube

propertyEvery face of a cube touches four others and is opposite the one that is left

The same cross-shaped net has its squares lettered A, B, C, D down the column and E, F on the left and right of B. When it is folded, which face is opposite E, and how many faces share an edge with E?

E, B, F in a row: F is opposite E.

In the row E, B, F, the squares E and F have one square between them: F is opposite E.

step 1 of 4

A cube has six faces. Any face touches four of the others along its four edges, and the one it does not touch is opposite. The net's one-apart rule names that opposite face.

All steps

In the row E, B, F, the squares E and F have one square between them: F is opposite E.
A face has four edges and a different face along each: E touches 4 faces.
Those four are all the faces except E itself and F: A, B, C and D.
Check: E touches B in the net already; A, C and D fold round to meet its other three edges.

answerF; 4

techniqueNets · The Cuboid

examsPSLE · SAT · GCSE Higher

Common pitfalls

Answering 1 neighbor because E touches only B in the flat net.
Counting F among the neighbors.

03

A Painted Cube Cut into Small Cubes

propertyCorners have 3 painted faces, edges 2, face centers 1, and the hidden core 0

A wooden cube with edges of 4 cm is painted on all six faces and then cut into 64 cubes of edge 1 cm. How many small cubes have paint on exactly one face? How many have no paint at all?

Each face is 4 by 4 small squares.

Each face of the big cube is 4 × 4 small squares; the ones not on the face's border form a 2 × 2 block, painted on that face only.

step 1 of 4

Paint reaches only the outside. A small cube's painted faces depend on where it sat: at a corner, along an edge, in the middle of a face, or inside.

All steps

Each face of the big cube is 4 × 4 small squares; the ones not on the face's border form a 2 × 2 block, painted on that face only.
Exactly one face painted: 6 faces × 4 = 24 cubes.
No paint: the inner block, one layer in from every face, is 2 × 2 × 2 = 8 cubes.
Check the rest: 8 corners with 3 faces, 12 edges × 2 = 24 with 2 faces; 8 + 24 + 24 + 8 = 64.

answer24; 8

techniqueThe Cuboid · Volume of a Cuboid

examsPSLE · SAT · GCSE Higher

Common pitfalls

Taking 16 cubes per face as painted on one face only: the 12 round the border are also painted on a neighboring face.
Forgetting the unpainted core, or making it 3 × 3 × 3 by taking one layer off only once.

04

Counting the Cubes in a Stack

propertyCount column by column from a top view with heights; the hidden cubes hold up the visible ones

Identical cubes of edge 2 cm are stacked on a table. Seen from above, the stack covers 3 by 2 positions; the back row has columns 2, 3 and 2 cubes high, and the front row has columns 1, 1 and 1 high. How many cubes are there, and what is the volume of the stack?

Back row columns: 2 + 3 + 2 = 7 cubes.

Back row: 2 + 3 + 2 = 7 cubes.

step 1 of 4

A cube cannot float, so every column stands on cubes all the way down. Adding the column heights counts the hidden ones too.

All steps

Back row: 2 + 3 + 2 = 7 cubes.
Front row: 1 + 1 + 1 = 3 cubes.
Total = 7 + 3 = 10 cubes; the cubes under the tall columns are counted even though the view hides them.
Each cube is 2 × 2 × 2 = 8 cm³: volume = 10 × 8 = 80 cm³.

answer10 cubes; 80 cm³

techniqueDrawing Solids on a Dot Grid · Plans and Elevations

examsPSLE

Common pitfalls

Counting only the cubes that can be seen in the drawing.
Using 2 cm³ for a cube of edge 2 cm.

05

Exposed Faces of a Stack

propertyCount what each of the six directions sees; on a table, leave the bottom out

The same stack, with columns 2, 3, 2 at the back and 1, 1, 1 at the front, stands on a table. How many faces of the small cubes are exposed to the air? What is the exposed area, the cubes being 2 cm?

From above: one top per column, 6.

From above: one top face per column, 6.

step 1 of 5

Every exposed face points in one of six directions, and looking from that direction shows each such face once. From above and from the sides the view is a silhouette; the bottom sits on the table.

All steps

From above: one top face per column, 6.
From the front and from the back: the tallest column at each of the three positions, 2 + 3 + 2 = 7 each way.
From the left and from the right: the tallest column in each row, 1 + 3 = 4 each way.
The bottom faces rest on the table and are not exposed.
Exposed faces = 6 + 7 + 7 + 4 + 4 = 28; each face is 2 × 2 = 4 cm², so 112 cm².

answer28 faces; 112 cm²

techniqueSurface Area of a Cuboid · Plans and Elevations

examsO-Level · SAT · GCSE Higher

Common pitfalls

Multiplying 10 cubes by 6 faces: most faces are pressed against another cube or the table.
Counting the front view once and forgetting the back sees the same silhouette from behind.

06

Fewest and Most Cubes for Three Views

propertyEach column can be as tall as both its views allow, or just tall enough for one of them to be true

A stack of cubes covers a 3 by 2 grid of positions seen from above. Seen from the front, the three columns of the silhouette are 3, 1 and 2 cubes high. Seen from the right side, the two columns are 3 and 2 high. What is the greatest number of cubes the stack can have? What is the least?

Most: each column as tall as both its views allow.

Most: at each position the height is the smaller of its front limit and its side limit. Front limits 3, 1, 2 across; side limits 3 (back row) and 2 (front row).

step 1 of 4

A position's column is limited by both the front view of its file and the side view of its row. For the most cubes, take the smaller of the two limits everywhere. For the fewest, keep every position at one cube except the ones needed to reach each view's height.

All steps

Most: at each position the height is the smaller of its front limit and its side limit. Front limits 3, 1, 2 across; side limits 3 (back row) and 2 (front row).
Back row: 3, 1, 2; front row: 2, 1, 2. Most = 3 + 1 + 2 + 2 + 1 + 2 = 11.
Fewest: every position has at least 1 cube (the top view). The front's 3 and the back row's 3 can be one column of 3 at the back-left; the front's 2 and the front row's 2 can be one column of 2 at the front-right.
Fewest = 3 + 2 + 1 + 1 + 1 + 1 = 9.

answer11; 9

techniquePlans and Elevations

examsPSLE · SAT · GCSE Higher

Common pitfalls

Adding the two views' heights, 6 + 5, as if every view showed different cubes.
Letting a column exceed one of its views: a 3 in the front row would show as 3 from the side.

07

The Shadow of a Block

propertyLight from straight ahead casts the face it meets, at full size

A rectangular block is 4 cm long, 3 cm wide and 5 cm tall. Light shining straight down casts its shadow on the table; light shining horizontally at its 4 cm by 5 cm face casts a shadow on the wall behind. Find the area of each shadow.

A block 4 by 3 by 5.

From above the light sees the top face, 4 cm by 3 cm: shadow = 12 cm².

step 1 of 3

Parallel light straight at a block casts a shadow the size of the face it sees, because every point of that face travels straight to the screen.

All steps

From above the light sees the top face, 4 cm by 3 cm: shadow = 12 cm².
From the front it sees the 4 cm by 5 cm face: shadow = 20 cm².
The third face, 3 by 5, would be the shadow from the side: 15 cm², not asked.

answer12 cm²; 20 cm²

techniquePlans and Elevations · Area and Perimeter

examsPSLE · SAT · GCSE Higher

Common pitfalls

Using the volume or a mix of all three dimensions for a shadow.
Casting the front-lit shadow with the top face.

08

Unrolling a Cylinder

propertyThe curved surface unrolls to a rectangle whose length is the circle's circumference

A closed cylinder has radius 7 cm and height 10 cm. Take π = 227. When its curved surface is unrolled flat, what rectangle does it make? Find the total surface area of the cylinder.

Once round the circle: 2 × 227 × 7 = 44 cm.

Circumference = 2 × 227 × 7 = 44 cm.

step 1 of 4

Cut the curved surface along its height and unroll it: one side is the height, the other is the distance once round the circle. The two circular ends are added for the total.

All steps

Circumference = 2 × 227 × 7 = 44 cm.
The curved surface unrolls to a rectangle 44 cm by 10 cm: 440 cm².
Each end is a circle of area 227 × 7 × 7 = 154 cm².
Total surface = 440 + 154 + 154 = 748 cm².

answer44 cm by 10 cm; 748 cm²

techniqueSurface Area of a Cylinder · Circumference and Diameter

examsO-Level · SAT · GCSE Higher

Common pitfalls

Making the rectangle's length the diameter, 14 cm.
Adding one end only, as if the cylinder were open.

09

Water Rising Round a Sunken Solid

propertyThe water level rises by the solid's volume spread over the tank's base

A rectangular tank has a base 20 cm by 10 cm and holds water 8 cm deep. A solid metal cube of edge 10 cm is lowered in and ends up fully under water. Find the new depth of the water. A stone of volume 2000 cm³ is then added and sinks; by how much does the level rise again?

Base 20 × 10 = 200 cm²; water 8 cm deep.

Base area = 20 × 10 = 200 cm².

step 1 of 4

A sunken solid pushes aside exactly its own volume of water, and that water has nowhere to go but up, across the whole base of the tank.

All steps

Base area = 20 × 10 = 200 cm².
Cube volume = 10 × 10 × 10 = 1000 cm³; rise = 1000 ÷ 200 = 5 cm.
New depth = 8 + 5 = 13 cm, which does cover the 10 cm cube, so the cube is fully under water as stated.
Stone: rise = 2000 ÷ 200 = 10 cm.

answer13 cm; 10 cm

techniqueVolume of a Cuboid

examsPSLE · SAT · GCSE Higher

Common pitfalls

Dividing the volume by the tank's height or by a side length instead of the base area.
Forgetting to check that the water really covers the cube; if it did not, only the wet part would count.

10

A Block Cut on a Slant

propertyA slanted top averages out: base area × the mean of the two heights

A rectangular block with a base 6 cm by 4 cm is cut by a slanting plane so that along the 6 cm length its height runs from 5 cm at one end to 9 cm at the other, the same at both 4 cm edges. Find the volume of the piece below the cut.

Base 6 × 4 = 24 cm²; the top slants from 5 up to 9.

Base area = 6 × 4 = 24 cm².

step 1 of 4

Two such pieces put together, one turned over on the other, make a full block of height 5 + 9 = 14 cm. So one piece is half of that, which is the base times the average height.

All steps

Base area = 6 × 4 = 24 cm².
Two copies of the piece, one inverted on the other, make a block 24 cm² by 5 + 9 = 14 cm.
One piece = 12 × 24 × 14 = 168 cm³.
The same as base × average height: 24 × 7 = 168 cm³.

answer168 cm³; average height 7 cm

techniqueVolume of a Cuboid · The Mean

examsPSLE · SAT · GCSE Higher

Common pitfalls

Using the taller height for the whole block, 24 × 9.
Averaging the heights and then halving again.

11

The Net of an Open Box on Grid Paper

propertyEvery face of the net is a rectangle of whole grid squares: turn each edge into squares, then add the five faces

An open box has no lid. Its external length is 14 cm, its breadth 10 cm and its height 8 cm, and its walls are so thin that their thickness can be ignored. A student draws its net on a large sheet of grid paper whose squares are 2 cm by 2 cm, with every edge of the net along the grid lines. (a) How many grid squares does the net cover? (b) The box is then filled with cubes of edge 2 cm. How many cubes fit inside it?

An open box: a base and four walls, no lid.

The net has five faces: the base, two long walls and two short walls. There is no lid.

step 1 of 5

An open box unfolds into five rectangles: the base and four walls, with no lid. Every edge is a whole number of grid squares long, so each face can be counted as rows of squares, and the same squares line up the cubes inside the box.

All steps

The net has five faces: the base, two long walls and two short walls. There is no lid.
On the grid, 14 cm is 14 ÷ 2 = 7 squares, 10 cm is 10 ÷ 2 = 5 squares and 8 cm is 8 ÷ 2 = 4 squares.
The base is 7 × 5 = 35 squares. Each long wall is 7 × 4 = 28 squares and each short wall 5 × 4 = 20 squares.
(a) The net covers 35 + 28 + 28 + 20 + 20 = 131 squares. Check by area: 140 + 2 × 112 + 2 × 80 = 524 cm², and one square is 2 × 2 = 4 cm², so 524 ÷ 4 = 131.
(b) The walls are too thin to take up room, so the cubes fill the box in rows: 7 along the length, 5 across the breadth and 4 layers up the height. That is 7 × 5 × 4 = 140 cubes.

answer(a) 131 (b) 140

techniqueNets · Surface Area of a Cuboid · Volume of a Cuboid

examsPSLE · SAT · GCSE Higher

Common pitfalls

Counting six faces, as for a closed box: the lid would add another 35 squares and give 166.
Dividing the area by 2, the side of a grid square, instead of by 4, its area: 524 ÷ 2 = 262 counts each square twice.

12

Most and Fewest Cubes on a 3 by 3 Grid, Then Painting the Largest

propertyEach position is capped by its front-view column and its side-view row; for the fewest, let one column meet a front peak and an equal side peak together

Identical cubes of edge 2 cm are glued together into a solid standing on a table over a 3 by 3 grid. Seen from above, all 9 squares of the grid are covered. Seen from the front, the three columns are 4, 2 and 3 cubes high, from left to right. Seen from the right side, the three columns are 1, 3 and 4 cubes high, from front to back. (a) What is the greatest number of cubes the solid can have? (b) What is the least number? (c) The solid with the greatest number of cubes is painted on all its outside faces, the bottom included, and then taken apart into single cubes. How many of the cubes have exactly three painted faces?

Caps: columns 4, 2, 3 from the front; rows 1, 3, 4 from the side; a cube on every square.

The front view caps the left column at 4, the middle at 2 and the right at 3. The side view caps the front row at 1, the middle row at 3 and the back row at 4. The top view puts at least 1 cube on every position.

step 1 of 6

Name the rows front, middle and back, and the columns left, middle and right. The front view caps the height at every position of a column and the side view caps every position of a row, and each view's number must be reached somewhere. Part (c) is a list: go through the cubes one by one and count the faces that nothing touches.

All steps

The front view caps the left column at 4, the middle at 2 and the right at 3. The side view caps the front row at 1, the middle row at 3 and the back row at 4. The top view puts at least 1 cube on every position.
(a) For the most cubes, each position is as tall as both its caps allow, the smaller of the two. Front row 1, 1, 1; middle row 3, 2, 3; back row 4, 2, 3. Every cap is reached, so both views are right, and the solid has 3 + 8 + 9 = 20 cubes.
For the fewest, start with one cube on each of the 9 positions and raise as few columns as the views need. The left 4 and the back 4 can be one column, at the back-left, and the right 3 and the middle 3 can be one column, at the middle-right. The middle column's 2 matches no row, so it needs a column of its own, in the middle or back row, since the front row is capped at 1.
(b) The fewest is 9 + 3 + 2 + 1 = 15 cubes: nine single cubes, then 3, 2 and 1 more to raise three columns to 4, 3 and 2.
(c) In the solid of 20 cubes, a face is painted when no cube touches it, and every bottom face on the table is painted. Going cube by cube, exactly three painted faces are found on the front-middle cube (front, top and bottom), the bottom cubes at the back-left and the back-right (back, outer side and bottom), and the third cube up at the back-left (back, left, and its right face, which rises above the back-middle column of 2).
So 4 cubes have exactly three painted faces. Check: the front-left and front-right cubes each stand alone with four painted faces (top, bottom, front and side), and so do the top cubes of the middle-left and middle-right columns (top, front and both sides).

answer(a) 20 (b) 15 (c) 4

techniquePlans and Elevations · Drawing Solids on a Dot Grid

examsPSLE · SAT · GCSE Higher

Common pitfalls

Raising a position past one of its caps to get more cubes, such as 4 at the back-middle: the front view would then show 4 in the middle column, not 2.
Counting three painted faces at the corners of the grid, as on a painted cuboid: this solid has steps, so a corner cube can have four painted faces and a cube in the middle of a side can have three. Each cube's six faces have to be checked.