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Secondary 3 Area of Solids and Plane Figures Worksheet

By Marius — private math tutor in Montreal. Updated 30 August 2026

Every question on this sheet answers the same question twice: which surfaces are actually there, and what is each one worth? The first half is where the marks are lost — a hidden disc counted, a diameter used as a radius, a height used where the slant height belongs — and the arithmetic afterwards is short. The sheet covers the lateral and total area of cones, cylinders and spheres, decomposable solids with a join, truncated solids handled as whole minus removed, regular polygons through the apothem, a plane figure decomposed two different ways, and a surface area written as an algebraic expression in one variable. Read it here, print it free when you want to write on it.

Practice worksheet — free PDF

6 pages 9 questions Letter size, print-ready

Download the worksheet Read the questions here

No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 9 harder problems come with the Secondary 3 Math bundle.

All 9 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

Q1Finding the Area of a Solid Using Algebra

A shop packs board games in closed rectangular boxes. Every box is $8$ cm wide and $5$ cm high, but the length changes from game to game. Let $x$ be the length of a box, in centimetres.
1. Write a simplified expression for the total surface area of a closed box.
2. Use your expression to find the surface area of a box $12$ cm long, and check the result by adding the six faces one at a time.
3. The travel edition ships with no lid. Write a simplified expression for the area of cardboard that version needs.
Q2Methods to Decompose Figures

The splash pad at a municipal park is a rectangle $12$ m long and $8$ m wide, with a semicircle attached to one of its $8$ m ends. The straight $8$ m end is the one opposite the semicircle.
1. Decompose the figure and find its area, first exactly in terms of $π$ , then to the nearest tenth of a square metre.
2. A rubber safety strip runs all the way around the outside. Find the length of strip needed, to the nearest tenth of a metre.
Q3The Area of Cones

A party hat is a cone made from a single piece of cardstock, open at the bottom. Its base has a diameter of $18$ cm and its slant height is $25$ cm.
1. Find the area of cardstock the hat needs, exactly in terms of $π$ and then to the nearest tenth of a square centimetre.
2. A second version has a cardstock disc closing the bottom. Find the total area of that version.
Q4The Area of Cylinders

A can of hot chocolate mix is a closed cylinder with a diameter of $14$ cm and a height of $20$ cm.
1. The paper label wraps around the curved side only, with no overlap. Find its area, exactly and to the nearest tenth of a square centimetre.
2. Find the total surface area of the can.
Q5The Area of Decomposable Solids

A grain bin is a cylinder with a cone on top. The cylinder has a radius of $3$ m and a height of $8$ m; the cone has the same radius and a slant height of $5$ m. The bin rests on the ground, and only the surfaces you can see from outside are painted.
1. Find the painted area, exactly in terms of $π$ and then to the nearest tenth of a square metre.
2. One pail of paint covers $30 m^{2}$ . How many pails must be bought?
Q6The Area of Truncated Solids

A lampshade is a truncated cone, open at the top and at the bottom. It began as a cone with a base radius of $12$ cm and a slant height of $20$ cm; a cut parallel to the base removed a small cone whose radius is $6$ cm.
1. Explain why the small cone's slant height must be $10$ cm.
2. Find the area of fabric the lampshade needs, exactly and to the nearest tenth of a square centimetre.
Q7The Area of a Sphere

A paper lantern is a sphere with a diameter of $30$ cm, covered on the outside with decorative paper.
1. Find the area of paper needed, exactly in terms of $π$ and then to the nearest tenth of a square centimetre.
2. The paper comes on a roll holding $3000 {cm}^{2}$ . Is one roll enough? Say how much is left over or missing.
Q8The Perimeter and the Area of Regular Polygons

A patio is laid with regular octagonal stones. Each stone has a side of $12$ cm and an apothem of $14.5$ cm.
1. Find the perimeter of one stone.
2. Find the area of one stone.
3. The patio uses $40$ stones. Find the total area they cover, in square metres.
Q9Synthesis — drawing on several sheets in this topic

A bubble-tea shop displays its syrup in a container built from a closed cylinder with a hemispherical dome on top. The cylinder has a radius of $5$ cm and a height of $14$ cm, and the dome has the same radius. The outside is frosted: the bottom disc, the curved side, and the dome.
1. Find the frosted area, exactly in terms of $π$ and then to the nearest tenth of a square centimetre.
2. Explain why the top disc of the cylinder is not part of your answer.
3. A taller model keeps the radius at $5$ cm but has cylinder height $h$ cm. Write a simplified expression for its frosted area, and check that $h = 14$ returns your answer from a).

The 9 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

Which stream is this for? Secondary 3 has no streams. Every student in Québec follows the same programme this year — the common ground the CST, TS and SN options all build on from Secondary 4 — and this set is written to the Progression of Learning for that shared year: the areas of solids with curved surfaces, decomposable and truncated solids, regular polygons, and area handled with algebra.

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.

Make a list of surfaces before you write a single formula

This is the whole method, and it is the difference between the students who find this topic easy and the ones who do not. Before any arithmetic, write down each surface the question is actually asking about, one per line. Then price each line. Q5, Q9 and the truncated-solid questions are all decided at that step.

Which surfaces count

Read the wording, not the shape: a solid has surfaces the question may not want.

1
Closed or open?
Q3's party hat is a cone with no base, so it is lateral area only; the second version adds the disc. Q4's can is closed, so both bases count. "Open at the top", "no lid", "the label only" are all instructions about your list.
2
Is a surface hidden by a join?
In Q5 and Q9, the top disc of the cylinder is sealed under the cone or the dome. It is not removed the way a hole is — it is simply counted zero times.
3
Does the solid stand on something?
Q5 sits on the ground, so its base is not painted. Q9 is frosted underneath, so its base is included. Two nearly identical solids, two different lists.
4
Has a cut created a new surface?
A truncated solid gains an opening, and whether that opening is a surface depends on the object: an open lampshade has none, a closed planter has a base.
5
Price each line, then add once
Keep π as a symbol down the whole column and round only at the end. Several questions ask for the exact value and the rounded one precisely so that this habit gets built.

The two measurements that get swapped (Q3, Q4, Q7)

Diameter is not radius. Q3's hat is given as 18 cm across, Q4's can as 14 cm across, Q7's lantern as 30 cm across — three questions in a row that hand you a diameter while every formula wants r. Halving takes a second; forgetting to multiplies a lateral area by 2 and a base area by 4.

Height is not slant height. The area of a cone's curved surface is πrs, built on the slant height, because that is the radius of the sector the cone unrolls into. The height runs through the inside of the solid and is not part of any surface. Since the slant height is the longest side of the right triangle formed by r, h and s, using h always gives an answer that is too small.

A quick check for a curved surface: the lateral area of a cylinder, 2πrh, is just the rectangle you get by unrolling it — circumference times height. If your number does not match circumference × height, you have not computed a lateral area.

Truncated solids: whole minus removed, and similarity supplies the missing piece (Q6)

A truncated cone or pyramid is never handled by a formula of its own. It is the whole solid minus the piece cut off, and the only difficulty is that the question rarely gives you the piece's measurements outright. Similarity gives them to you.

A cut parallel to the base leaves a small solid similar to the whole one, so all its lengths are k times the original's. In Q6 the small cone's radius is 6 cm against 12 cm, so k = ½ and its slant height must be half of 20 cm. That single deduction is the whole question; the subtraction afterwards is one line. Note that both slant heights appear and neither height does — this is an area question throughout.

Halving every length does not halve the area. It quarters it. So a cut halfway up removes only a quarter of the lateral surface, leaving three quarters — and it takes nothing at all from the base, which the cut never touched. That is why "cut halfway up, so half the area" is wrong twice over.

Regular polygons: the apothem, and the unit trap after it (Q8)

A regular polygon's area is P × a ⁄ 2, where a is the apothem — the perpendicular distance from the centre to the middle of a side. The formula is really the polygon cut into congruent triangles: each has base one side and height the apothem, and adding them gives exactly that expression. Knowing where it comes from is what stops the apothem from being confused with the radius out to a vertex, which is longer.

Converting an area squares the factor. Q8 finishes in square metres, and 1 m² is 100 × 100 = 10 000 cm², not 100 cm². Dividing by 100 is treating an area like a length, and it is the most common single error on this sheet. The same reasoning is why a volume conversion cubes the factor.

Decomposing a plane figure (Q2)

Q2's splash pad is a rectangle with a semicircle on one end, and it separates two things students often merge. For the area, add the pieces: the rectangle plus half a disc. For the perimeter, walk around the outside — and the side where the semicircle is attached is not on the outside at all, because the arc replaced it.

The other detail is that the 8 m side the semicircle sits on is its diameter, so the radius is 4 m. Sketch the figure, mark the pieces on the drawing, and decide before computing whether the shape is a sum or a difference. A composite region that can be read both ways — add two rectangles, or subtract a notch from the enclosing one — is worth doing both ways, because the two answers check each other.

Area written as an expression (Q1, Q9)

Q1 keeps two dimensions of a box fixed and lets the length be x. The six faces come in three pairs, so the total is 2(8x) + 2(5x) + 2(40), which simplifies to a first-degree expression in x. Two habits make these reliable: pair the faces rather than listing six separately, and check by substitution — evaluating the expression and adding the six faces one at a time must agree, and if they do not, the expression is where the error is.

Removing the lid is then subtracting one face, not rebuilding the expression from scratch. Q9 does the same thing for a curved solid: only the cylinder's height is free, so its frosted area is a constant plus a term in h, and setting h to the original value must return the original answer. That last check costs one line and catches nearly everything.

Materials round up (Q5)

Q5 ends by asking how many pails of paint must be bought. Six point six pails is not an answer: you cannot buy part of a pail, so the answer is seven. Whenever a question converts an area into a count of packets, tins, rolls or sheets, the rounding goes up, whatever the decimal. Q7 is the same idea in reverse — the roll is enough, but only just, and saying how much is left over is part of the answer.

Getting the most out of it

List the surfaces, then price them

Write each surface on its own line before touching a formula: bottom disc, curved side, dome. Almost every lost mark on this sheet is a surface counted that should not have been, or one missed that should. The list also makes your working readable, which is where partial credit comes from.

Halve the diameter the moment you read it

Write "r = …" as your first line whenever a question gives a width, a diameter or a distance across. Three questions on this sheet are built to catch the substitution of d for r, and the resulting answer is always wrong by a clean factor of 2 or 4 — plausible enough to go unnoticed.

Keep π to the last line

Carry π as a symbol through the whole computation and round only at the end. Several questions ask for both the exact and the rounded value, and rounding early then multiplying is how an answer drifts by a whole square centimetre before you have finished.

Want the solutions, or something more challenging?

The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the Secondary 3 Math Solutions Bundle, which is what keeps the rest of the series free.

What else exists for The Area of Solids and Plane Figures

Three PDFs · 11 pages · all three are in the bundle below.

Answer key — 2 pages. All 9 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 6 pages, 9 problems. A separate sheet at exam-plus difficulty covering the same 8 concepts. Harder than anything on the free sheet.
Challenge answer key — 3 pages. Every challenge problem worked to the same standard, with the checks shown.
PDF, letter size, print-ready.

In the bundle See what's in it Not sold separately

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Taking Secondary 1 Math as well? The Secondary 1 Math bundle covers all 15 of its sets — 45 PDFs, 182 pages — on the same terms.

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Common questions

Is this worksheet really free?

Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete Secondary 3 Solutions Bundle, which covers every set at this level.

Why does the area of a cone use the slant height and not the height?

Because the curved surface unrolls into a sector whose radius is the slant height — the distance from the apex to the rim measured along the surface. The height runs through the inside of the solid and is not part of any surface at all. Since the slant height is the longest side of the right triangle formed by the radius, the height and itself, using the height always gives an answer that is too small.

Do I count the surface where two solids are joined?

No. In a cylinder topped by a cone or a dome, the cylinder's top disc is sealed under the solid above it, so it is not part of the outside. It is not subtracted either, the way a hole would be — it is simply counted zero times. Whether the base counts depends on the wording: a solid standing on the ground usually is not painted underneath.

How do I find the area of a truncated cone?

As the whole cone's lateral area minus the removed cone's. The cut is parallel to the base, so the piece removed is similar to the whole solid, and its lengths are k times the original's — that is how you get its slant height when the question does not give it. Both slant heights appear in the calculation and neither height does.

Why is 1 m² equal to 10 000 cm² and not 100 cm²?

Because an area has two dimensions, so the length factor is squared: 1 m = 100 cm gives 1 m² = 100 × 100 = 10 000 cm². Dividing by 100 is treating an area like a length. The same reasoning cubes the factor for volumes, where 1 m³ = 1 000 000 cm³.

What exactly is the apothem?

The perpendicular distance from the centre of a regular polygon to the middle of one of its sides — not the distance out to a vertex, which is longer. The area formula P × a ⁄ 2 is the polygon cut into congruent triangles, each with a side as its base and the apothem as its height.

Is there a stream to choose in Secondary 3?

No. Secondary 3 is the last common year: every student follows the same mathematics programme, and the CST, TS and SN options begin in Secondary 4. The surface-area work here is assumed by all three of them.

Can teachers use this in class?

Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.

I'm stuck on one question. Can you help?

Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.

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