3D solids · word problems

Solid Heuristics

10 question types · PSLE Paper 1 and 2 · one figure that follows the deduction

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?

123456
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.

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

answer3; 6

Common pitfalls

  • Taking neighbours 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 neighbours.
02

Neighbours 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?

ABCDEF
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.

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

answerF; 4

Common pitfalls

  • Answering 1 neighbour because E touches only B in the flat net.
  • Counting F among the neighbours.
03

A Painted Cube Cut into Small Cubes

propertyCorners have 3 painted faces, edges 2, face centres 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.

  1. 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.
  2. Exactly one face painted: 6 faces × 4 = 24 cubes.
  3. No paint: the inner block, one layer in from every face, is 2 × 2 × 2 = 8 cubes.
  4. Check the rest: 8 corners with 3 faces, 12 edges × 2 = 24 with 2 faces; 8 + 24 + 24 + 8 = 64.

answer24; 8

Common pitfalls

  • Taking 16 cubes per face as painted on one face only: the 12 round the border are also painted on a neighbouring 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: 2, 3, 2front row: 1, 1, 1
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.

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

answer10 cubes; 80 cm3

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.

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

answer28 faces; 112 cm2

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?

front sees 3, 1, 2 · side sees 3, 2most: 11 cubes
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.

  1. 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).
  2. Back row: 3, 1, 2; front row: 2, 1, 2. Most = 3 + 1 + 2 + 2 + 1 + 2 = 11.
  3. 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.
  4. Fewest = 3 + 2 + 1 + 1 + 1 + 1 = 9.

answer11; 9

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.

435
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.

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

answer12 cm2; 20 cm2

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.

710
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.

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

answer44 cm by 10 cm; 748 cm2

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?

208 cm
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.

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

answer13 cm; 10 cm

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.

6459
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.

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

answer168 cm3; average height 7 cm

Common pitfalls

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