Secondary 1 Solids and Their Nets Worksheet

11 of the 13 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 11 of the 13 questions are printed below. The other 2 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

1. Q1Constructing the Net of Prisms

   A confectioner in Rimouski packs a chocolate log in a box shaped like a prism. Its two bases are equilateral triangles with sides of $6\text{ cm}$, and the box is $20\text{ cm}$ long.

   1. How many pieces would you cut out to make the net of this box, and what shape is each one?
   2. Give the dimensions of every rectangle in the net.
   3. Find the total area of the rectangular pieces only.
2. Q2Constructing the Nets of Pyramids

   A park in Sherbrooke has a monument shaped like a pyramid with a square base. The base measures $10\text{ m}$ on each side, and each triangular face has a height of $12\text{ m}$ measured from the middle of a base edge up to the top of the monument.

   1. Describe the net of this pyramid: how many pieces, and what shape and size is each one?
   2. The four triangular faces are to be covered in copper. What area of copper is needed?
3. Q3Constructing the Nets of Solids

   This question is built around a diagram or a table of values. Open it in the PDF.

4. Q4Decomposable and Truncated Solids

   A bird feeder is built by gluing a pyramid with a square base on top of a rectangular prism. The prism is $20\text{ cm}\times 20\text{ cm}$ at the base and $30\text{ cm}$ tall; the pyramid's square base is $20\text{ cm}\times 20\text{ cm}$ and matches the top of the prism exactly.

   1. Name the two solids the feeder decomposes into, and say which faces of them disappear inside the glued join.
   2. How many faces does the finished feeder show on the outside?
   3. The tip of the pyramid is then sliced off with a horizontal cut, making a truncated pyramid. What shape is the new face created by the cut?
5. Q5Euler's Theorem

   This question is built around a diagram or a table of values. Open it in the PDF.

6. Q6Prisms

   A shop displays its candles in three prism-shaped boxes: box $1$ has a rectangular base, box $2$ has a base that is a regular hexagon, and box $3$ has a triangular base.

   1. Name each prism.
   2. For each one, give the number of faces, vertices and edges.
   3. What do all three boxes have in common that makes them prisms rather than pyramids?
7. Q7Pyramids

   A garden lamp is shaped like a pyramid whose base is a regular octagon ($8$ sides).

   1. How many faces, vertices and edges does the lamp have?
   2. What shape is each lateral face?
   3. Of these three solids, which are pyramids: (i) a solid with one pentagonal face and five triangles meeting at a point; (ii) a solid with two identical triangular faces and three rectangles; (iii) a solid with four triangular faces?
8. Q8Solids

   Six objects are on a table: a Rubik's cube, a soup can, a beach ball, a party hat, a die shaped like a triangular pyramid, and a chocolate box shaped like a triangular prism.

   1. Name the solid each object represents.
   2. Sort the six into polyhedrons and solids that are not polyhedrons.
   3. Write the rule you used to sort them, in one sentence.
9. Q9Solids with Curved Surfaces
   1. For each of the cylinder, the cone and the sphere, say how many flat faces and how many curved surfaces it has.
   2. Which of the three has a net made of two circles and a rectangle?
   3. A can of maple syrup is a cylinder with a diameter of $10\text{ cm}$ and a height of $14\text{ cm}$. Its label is a rectangle that wraps exactly once around the can, with no overlap. Give the dimensions of the label, to the nearest tenth of a centimetre. (Use $\pi \approx 3.14$.)
10. Q10Tetrahedrons

    Jade folds a paper decoration in the shape of a tetrahedron whose six edges all measure $5\text{ cm}$.

    1. How many faces, vertices and edges does it have, and what shape is each face?
    2. What is the total length of all its edges?
    3. Her net is a single piece made of four triangles joined along three fold lines. When she folds it up, how many edges still have to be taped, and what total length of tape is that?
11. Q11The Area and the Volume of Solids

    A moving company uses a cardboard box shaped like a rectangular prism, measuring $30\text{ cm}$ long, $20\text{ cm}$ wide and $15\text{ cm}$ high.

    1. Find the total area of cardboard in the closed box.
    2. Find the volume of the box.
    3. Say in one sentence what each of the two answers tells the company.
12. Q12The Area of Prisms

    A ramp at a skatepark is a prism lying on its side. Its two bases are right triangles with sides of $6\text{ m}$, $8\text{ m}$ and $10\text{ m}$, and the ramp is $15\text{ m}$ long.

    1. Find the lateral area of the prism (the three rectangles).
    2. Find the area of the two triangular bases.
    3. Find the total area of the prism.
13. Q13Synthesis — drawing on several sheets in this topic

    A gift box is a prism whose two bases are identical triangles with sides of $5\text{ cm}$, $5\text{ cm}$ and $6\text{ cm}$. The height of each triangle, measured onto the $6\text{ cm}$ side, is $4\text{ cm}$. The box is $12\text{ cm}$ long.

    1. Describe the net of the box: how many pieces, and the dimensions of each.
    2. Find the total area of the net.
    3. Find the volume of the box.

The 12 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Prism or pyramid? One question separates them (Q6, Q7, Q8)

A prism has two identical parallel bases joined by rectangles. A pyramid has one base and triangles rising from it to a single point. Q6 asks what three prisms have in common that makes them prisms and not pyramids, and the answer is that pair of bases: it is the presence of the second base, not the shape of the first, that decides the family.

Q8 sorts everyday objects into polyhedrons and solids that are not, and asks for the sorting rule in one sentence. A polyhedron is a solid whose faces are all flat polygons — so one curved surface anywhere, as on a can, a ball or a party hat, disqualifies it. Write the rule as a condition, not as a list of examples.

Nets: the right pieces are not enough (Q1, Q2, Q3)

Q1 and Q2 both ask the same question in advance of any drawing: how many pieces, and what shape and size is each one? Answering that before you pick up a pencil is the whole skill. A prism on a triangular base unfolds into two congruent triangles and one rectangle per side; a pyramid on a square base unfolds into one square and four triangles.

Q3 also asks how many edges are already joined as fold lines and how many still need taping, and the two must add up to the number of edges the cube has. Any solid built from a single-piece net obeys the same accounting, which makes it a free check on your own drawing.

Euler's relation, and what it is for (Q5)

Q5 has you count the vertices, edges and faces of three familiar solids and compute V − E + F each time. The last column comes out the same on every row:

That constant holds for every ordinary solid whose faces are flat and which has no hole through it, and its practical use is as a check and as a shortcut. As a check: count the three numbers independently, and if the combination is not 2, one of the counts is wrong — usually the edges, which are the hardest to count on a drawing because half of them are hidden. As a shortcut: any two of the three numbers give the third without counting anything, which is how you handle a solid too complicated to count reliably.

Decomposable and truncated solids (Q4)

Gluing two solids together does not give you a solid with the sum of their faces, and Q4 is built to make that visible. When a pyramid is set on top of a prism so that their squares match exactly, both of those squares vanish inside the join — they are no longer surfaces of anything. Count what is left on the outside, which is what the word “face” means for the finished object.

Truncating is the opposite operation: a cut removes a piece and, in exchange, creates a new face where the blade passed. The shape of that new face is the shape of the cut, so a horizontal cut through a pyramid on a square base leaves a square. Nothing about that requires calculation — it requires picturing the cut.

Curved solids: identified and unrolled, not measured (Q9)

The cylinder, the cone and the sphere appear here to be described and unfolded, and Q9 asks how many flat faces and how many curved surfaces each one has — a cylinder two flat and one curved, a cone one flat and one curved, a sphere none flat and one curved.

The part worth understanding rather than memorising is the cylinder's label. Unroll the curved surface and it becomes a rectangle. One side of that rectangle is the height of the can; the other is the distance once round the rim, which is the circumference of the circular base. That is the only way a curved measurement gets into a flat drawing at this level, and it is why the net of a cylinder is two circles and a rectangle.

Surface area is a net, and volume is a filling (Q11, Q12)

These are the two numbers a solid carries and they answer different questions, which is exactly what Q11 asks you to state in a sentence. Total surface area is how much material wraps round the solid — it is the area of the net, in cm², and it tells the company how much cardboard to buy. Volume is how much fits inside, in cm³, and it tells them how much the box will hold.

Every triangle whose area is needed on this sheet is handed to you with its own height measured, so nothing here depends on recovering a third side of a right triangle from the other two — that relation belongs to a later year of the Québec Education Program.

Q13: the synthesis question

The last free question is one gift box worked three ways: describe its net, total the areas of the pieces, and find the volume. The three answers are connected, and the connection is the lesson — the net's piece list is what makes the area a matter of addition rather than of memory, and the two bases you drew in the net are exactly the base whose area the volume needs. Do them in the order asked and the second and third parts are mostly done already.

Getting the most out of it

Describe the net before you draw it

How many pieces, what shape is each, what are its two dimensions. Written out as a list, that description answers the counting questions, sets up the surface area, and shows immediately if a face has been forgotten. Drawing without it is how a rectangle ends up with the wrong second dimension.

Count faces, then vertices, then edges — and check

Edges are the count that goes wrong, because half of them are hidden in any drawing of a solid. Get them from the structure rather than the picture, then test the three numbers against V − E + F = 2. A count that fails that test is wrong, and you have found it in one line instead of at the end.

Say which faces you are counting

Lateral area or total area, with a lid or without one, glued faces included or excluded: these are decisions, and the answer should state the one you made. Most lost marks on surface area are a face counted twice or a face left out, and both become visible the moment the faces are listed rather than added in your head.

Common questions