Secondary 3 Solids and Their Nets Worksheet

11 of the 14 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 14 questions are printed below. The other 3 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. Q1Cones

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

2. Q2Congruent and Similar Solids

   A dépanneur sells iced tea in two similar cylindrical cans. The small can has a base radius of $4\text{ cm}$ and a height of $10\text{ cm}$; the large can has a base radius of $6\text{ cm}$.

   1. Find the ratio of similarity $k$ from the small can to the large one.
   2. Find the height of the large can.
   3. The small can holds $160\pi {\text{ cm}}^3$ of iced tea, which is about $503\text{ mL}$. How much does the large one hold? Leave your answer in terms of $\pi$.
3. Q3Constructing the Nets of Cones

   A cone-shaped party hat has a base radius of $5\text{ cm}$ and a slant height of $12\text{ cm}$. Its net is a disc together with a sector of a circle.

   1. Give the radius of the disc and the radius of the sector.
   2. Find the length of the sector's arc, to the nearest tenth of a centimetre.
   3. Find the central angle of the sector.
4. Q4Constructing the Nets of Cylinders

   A tin of maple butter is a right circular cylinder with a base radius of $5\text{ cm}$ and a height of $18\text{ cm}$. Its net is made of one rectangle and two circles.

   1. Give the radius of each circle.
   2. Give the two dimensions of the rectangle, rounding any length to the nearest tenth of a centimetre.
   3. Which dimension of the rectangle would change if the tin were made taller but no wider?
5. Q5Cylinders

   The salt tank at a snow-clearing depot is a right circular cylinder standing on one of its bases.

   1. How many flat faces, curved surfaces, edges and vertices does it have?
   2. The tank is sliced by a horizontal plane, parallel to its base. What shape is the cut surface, and does its size depend on how high up the cut is made?
   3. Which two measurements are enough to describe the tank completely?
6. Q6Decomposable and Truncated Solids

   A water tank at an arena is made of a cylinder of radius $40\text{ cm}$ and height $90\text{ cm}$, with a hemisphere of the same radius closing it at the top. It rests flat on the floor.

   1. Name the two simple solids the tank decomposes into.
   2. List the surfaces that make up the outside of the tank, and name one surface of the two simple solids that is not part of the outside.
   3. Find the total height of the tank.
7. Q7Geometry

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

8. Q8Similarity, Congruence, and Equivalence

   For each pair of solids, say whether the two solids are congruent, similar but not congruent, equivalent but not similar, or none of these. Justify each answer.

   1. Two cubes, each with edges of $5\text{ cm}$.
   2. A cylinder with $r=2\text{ cm}$, $h=9\text{ cm}$, and a cylinder with $r=6\text{ cm}$, $h=27\text{ cm}$.
   3. A cube with edges of $6\text{ cm}$, and a rectangular prism measuring $4\text{ cm}\times 9\text{ cm}\times 6\text{ cm}$.
9. Q9Solids

   Consider these seven solids: a cube, a cone, a hexagonal prism, a sphere, a square-based pyramid, a cylinder, a regular tetrahedron.

   1. Sort them into polyhedra and non-polyhedra, and say what decides which group a solid goes into.
   2. Check Euler's relation, $F-E+V=2$, on the hexagonal prism.
10. Q10Solids with Curved Surfaces

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

11. Q11Tetrahedrons

    A wooden puzzle piece is a regular tetrahedron with all its edges $9\text{ cm}$ long.

    1. Give its number of faces, edges and vertices, and name the shape of each face.
    2. Check Euler's relation on it.
    3. A ribbon is glued along every edge. What total length of ribbon is needed?
12. Q12The Net and Drawing of a Sphere

    A globe-maker and a mathematics student are discussing spheres.

    1. Explain why a sphere has no exact net, unlike a cylinder or a cone.
    2. Globes are in fact covered with printed paper. Describe what shape those pieces of paper have and why the result is only an approximation.
    3. When a sphere is drawn on paper, the great circle around it is drawn as an oval rather than as a circle. Why?
13. Q13The Sphere

    A basketball is a sphere with a diameter of $24\text{ cm}$.

    1. Give its radius.
    2. What is a great circle of a sphere?
    3. Find the circumference of a great circle of this basketball, to the nearest tenth of a centimetre.
14. Q14Synthesis — drawing on several sheets in this topic

    A grain silo on a farm in the Beauce is a decomposable solid: a cylinder of radius $3\text{ m}$ and height $8\text{ m}$, with a cone of slant height $5\text{ m}$ sitting on top of it. The silo stands on the ground and its whole outside surface, including the disc it stands on, is to be covered in sheet metal.

    1. Name the three pieces that make up the net of the outside surface, and give the radius of each circular piece.
    2. Give the two dimensions of the rectangular piece, to the nearest tenth of a metre.
    3. Find the central angle of the sector.

The 10 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.

A net is a promise about lengths, not a picture

Most net questions on this sheet come down to one sentence: the edge that wraps must equal the circle it wraps around. Everything else follows from it. In Q4 the tin is unrolled into a rectangle and two circles, and one side of that rectangle is the base circumference. In Q3 the cone is unrolled into a disc and a sector, and the sector's arc is the base circumference. Nothing stretches when a net folds up, so any length on the flat pattern is a real length on the solid.

Height and slant height are different lengths (Q1)

Q1 asks you to name three marked measurements of a traffic cone, and the naming is the question. The height runs from the apex straight down to the centre of the base and passes through the inside of the solid. The slant height runs from the apex to a point on the rim, along the surface. The radius joins the centre of the base to that same rim point.

Cut the cone vertically through the apex and those three lengths form a right triangle, with the slant height opposite the right angle. That picture is worth drawing once and remembering, because it settles a lot: the slant height is always the longest of the three, so a set of measurements where it is not describes a cone that cannot be built. The Pythagorean relation arrives this year and applies here directly — r² + h² = s² — and either the ordering argument or the Pythagorean one is a complete answer.

Congruent, similar, equivalent — and the three ratios (Q2, Q8)

Q8 asks you to sort three pairs of solids using these words, so they have to be kept apart. Congruent means every corresponding measurement is equal, so the ratio of similarity is k = 1. Similar means every corresponding measurement is in the same ratio k. Equivalent means the same volume and says nothing about shape at all — which is why a cube and a rectangular prism with entirely different edges can be equivalent.

Polyhedra, curved surfaces, and where Euler's relation actually applies (Q9, Q10)

Q9 sorts seven solids into polyhedra and non-polyhedra, and the deciding test is a single property: is every surface a flat polygon? One curved surface anywhere — the cone, the cylinder, the sphere — and the solid is out. Q10 then counts the flat faces, curved surfaces, edges and vertices of exactly those three, which is the useful exercise it looks like: a sphere has one surface, no edge and no vertex; a cone has one flat face, one curved surface, one curved edge and one vertex; a cylinder has two flat faces, one curved surface, two edges and no vertices at all.

Euler's relation, F − E + V = 2, is checked on the hexagonal prism in Q9 and on the tetrahedron in Q11. It is a statement about convex polyhedra, and that restriction is not decoration. Test it on a cylinder and you get 1, not 2; test it on a sphere and you get 1 again. Neither is a failure of the relation — they are simply not polyhedra, so nothing ever promised it would work on them.

The surface that is not there (Q6, Q14)

A decomposable solid is two simple solids joined, and the join costs you a surface. Q6 stacks a hemisphere on a cylinder; Q14 puts a cone on one. In both cases the top disc of the cylinder is sealed under the solid above it — you cannot see it, cannot paint it, and it is not in the net.

So the habit is: list the outside surfaces before computing anything, and say out loud which surface of each simple solid is not on that list. Q14 makes the point cleanly, because its net has exactly three pieces — one disc for the ground, one rectangle for the cylinder wall, one sector for the roof — and a student who lists four has counted the hidden disc.

Why a sphere has no net (Q12)

A net folds flat to curved without stretching, tearing or overlapping. A cylinder's side and a cone's side were rolled from something flat to begin with, so they can be laid out again exactly. A sphere's surface curves in two directions at once, and no piece of it can be flattened without being stretched or split — which is what an orange peel demonstrates every time one is squashed.

Globes are still covered in paper, using narrow tapering strips pointed at both ends, and Q12 asks you to say why that is an approximation rather than a net: each strip is stretched, just too little to see. The same fact is why every flat map of the world distorts something. Part (c) is a different kind of question — a great circle drawn on paper looks like an oval because a circle seen at an angle appears as an ellipse, and drawing it as a true circle would make the sphere read as a flat disc.

Q14: the synthesis question

A grain silo — a cylinder with a cone on top — asked for as a net. It chains three of the sheet's ideas in order: decide which surfaces are on the outside, unroll each one, then find the sector's angle from the arc it has to meet. Work it in that order and the arithmetic is short. Reach for a formula first and you will almost certainly include a disc that is not there.

Getting the most out of it

Draw the vertical cut through a cone

One sketch settles most cone questions: the axis, a radius and a slant line make a right triangle with the slant height as the longest side. Draw it and you can never again confuse the height with the slant height, or accept a set of measurements that cannot exist.

Name the solid's parts before you count anything

Faces, edges, vertices, apex, net, slant height: the matching question and the counting table look like the easy items on the sheet and are marked strictly. Write the definitions out once — an edge is where two surfaces meet, so a sphere has none — and the counting questions stop being guesswork.

Ask "length, area or volume?" before scaling

Every similarity question here is decided by that one question. Lengths take k, surfaces take k², capacities take k³, and angles take nothing. Say which one you are scaling out loud before you multiply.

Common questions