Secondary 1 Math Perimeter and Area Free · no sign-up

Secondary 1 Perimeter and Area Worksheet

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

Perimeter is a length round the outside, area is a covering of the inside, and nearly every lost mark on this topic comes from mixing the two up or from mixing up the units they are measured in. The set works through the standard figures — triangle, parallelogram, trapezoid, rhombus, regular polygon, circle and its sectors — then does the two harder things: cutting an awkward figure into pieces you can compute, and running a formula backwards to recover a measurement you were not given. Metres, grams and minutes get the same treatment at the end. Read it on this page, or print the free PDF when you want to write on it.

Practice worksheet — free PDF

9 pages 15 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 14 harder problems come with the Secondary 1 Math bundle.

14 of the 15 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 14 of the 15 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

Q1Constructing a Circle

Zoe opens her compass to $4.5$ cm and draws a circle on her poster.
1. Give the radius and the diameter of her circle.
2. She then draws a chord that measures exactly $9$ cm. What is this chord called, and explain why no chord of her circle can be longer than $9$ cm.
3. Find the circumference of the circle, first as an exact multiple of $π$ and then as a decimal (use $π \approx 3.14$ ).
Q2Decomposable Geometric Figures

This question is built around a diagram or a table of values. Open it in the PDF.
Q3Finding Missing Measurements in Plane Figures From the Perimeter
1. An isosceles triangular banner has a perimeter of $74$ cm. Its base measures $24$ cm. Find the length of each of the two equal sides.
2. A rectangular window has a perimeter of $74$ cm and a width of $16$ cm. Find its length.
3. In one sentence, say what is the same about the first step of the two solutions.
Q4Methods to Decompose Figures

A workshop tabletop is a rectangle $120$ cm long and $80$ cm wide, with a semicircular bite of diameter $80$ cm cut out of one of the short ends.
1. Would you find this area by addition (cutting into pieces) or by subtraction (removing a piece from a bigger figure)? Say why.
2. Find the area of the tabletop, to the nearest square centimetre. Use $π \approx 3.14$ .
Q5Missing Measurements in Plane Figures

For each situation below, first say whether the measurement you are given is a perimeter or an area, then name the relation you will undo, and only then find the missing measurement.
1. A parallelogram has an area of $96 {cm}^{2}$ and a base of $12$ cm. Find its height.
2. A rectangle has a perimeter of $30$ cm and a length of $9$ cm. Find its width.
3. A triangle has an area of $45 {cm}^{2}$ and a height of $9$ cm. Find its base.
Q6The Arcs and Sectors of a Circle

A round pizza of radius $18$ cm is cut into slices. One slice is a sector with a central angle of $40^{\circ}$ . Give each answer as an exact multiple of $π$ and then as a decimal, using $π \approx 3.14$ .
1. Find the length of the crust on that slice (the arc).
2. Find the area of the slice.
3. Find the perimeter of the slice. Be careful — it is not the same as your answer to (a).
Q7The Area and the Perimeter of Triangles

A triangular flower bed sits in the corner of a park. Its three sides measure $9$ m, $12$ m and $15$ m, and the corner where the $9$ m and $12$ m sides meet is a right angle.
1. Find the area of the flower bed.
2. Find its perimeter.
3. Metal edging costs $$ 8.50$ per metre. Find the cost of edging the whole bed.
Q8The Perimeter and The Area of Decomposable Figures

A community-centre window is shaped like a rectangle $2.4$ m wide and $1.8$ m tall with a semicircle sitting on top of it, the flat side of the semicircle being the top of the rectangle. Use $π \approx 3.14$ and round each answer to the nearest hundredth.
1. Find the area of the glass.
2. Find the perimeter of the window frame that goes round the outside of the glass.
Q9The Perimeter and the Area of Plane Figures

Read each set of measurements carefully — one of them mixes two units.
1. A toy kite is shaped like a rhombus. Its diagonals measure $48$ cm and $0.2$ m, and each of its four sides measures $26$ cm. Find its area and its perimeter, in centimetres.
2. A wooden deck is a trapezoid. Its two parallel sides measure $10$ m and $4$ m, the distance between them is $4$ m, and its two slanted sides each measure $5$ m. Find its area and its perimeter.
Q10The Perimeter and the Area of Quadrilaterals

A parallelogram-shaped banner has a base of $1.2$ m. Its two slanted sides each measure $60$ cm, and the perpendicular distance between the base and the side opposite it is $45$ cm.
1. Find the area of the banner in ${cm}^{2}$ .
2. Find its perimeter, in centimetres and then in metres.
3. The $60$ cm slanted side is not the number used in the area formula. Explain why not.
Q11The Perimeter and the Area of Regular Polygons –
1. A paving stone is a regular hexagon with sides of $8$ cm and an apothem of $6.9$ cm. Find its perimeter and its area.
2. A blank metal plate is a regular octagon with sides of $15$ cm and an apothem of $18.1$ cm. Find its perimeter and its area.
3. In your own words, say what the apothem of a regular polygon is.
Q12Units for Measuring Mass and Their Conversion

A factory packs granola bars. Each bar has a mass of $45$ g, a carton holds $24$ bars, and the empty carton itself has a mass of $180$ g.
1. Find the mass of a full carton, in grams and then in kilograms.
2. A pallet holds $500$ full cartons. Find its mass in kilograms and then in tonnes.
3. A vitamin tablet has a mass of $250$ mg. How many tablets would it take to match the mass of one granola bar?
Q13Units of Length
1. Put these five lengths in order from smallest to largest, converting them all to metres first: $4500$ mm, $0.004$ km, $45$ cm, $4.5$ dm, $0.9$ m. Two of them are equal — say which.
2. A hallway is $12.5$ m long. Floor tiles are $25$ cm wide. How many tiles fit in one row along the hallway?
Q14Units of Time and Their Conversion
1. Convert $2$ h $45$ min to minutes, and convert $500$ s to minutes and seconds.
2. A hockey tournament starts its first game at $8$ : $15$ . Each game lasts $1$ h $25$ min, and the ice is resurfaced for $10$ min between games. At what time does the third game end?
Q15Synthesis — drawing on several sheets in this topic

A new concrete skatepark pad is a rectangle $12$ m by $7$ m, with one corner rounded off: a quarter-circle of radius $4$ m has been cut away from that corner. Use $π \approx 3.14$ .
1. Find the area of the pad, to the nearest hundredth of a square metre.
2. Find the perimeter of the pad, to the nearest hundredth of a metre.
3. Rubber edging is sold only in straight $2.5$ m lengths that can be bent round the curve. How many lengths must the town buy?

The 14 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.

Perimeter or area? Decide before you reach for a formula

Q5 makes that decision the first step of the answer, and it is worth doing on every question on the sheet. A perimeter is a distance, measured in cm or m, and it is what you buy fencing, baseboard or edging by. An area is a covering, measured in cm² or m², and it is what you buy paint, tiles or matting by. The question almost always tells you which by naming what is being bought.

Running a formula backwards. When the area is given and a dimension is missing, undo the formula instead of guessing at it. A parallelogram's area is base × height, so the height is the area divided by the base. A triangle's area is (base × height) ÷ 2, so the missing base is twice the area divided by the height — the halving has to be undone as well, and forgetting it is the standard error.

Same idea for a perimeter: a rectangle's perimeter is two lengths plus two widths, so subtract the two known sides from the total and halve what is left. In an isosceles triangle with a known base, subtract the base and halve. Say the sentence out loud before you write it — “the two equal sides share what is left over” — and the arithmetic follows.

The height in the formula is not always a side of the figure (Q10)

Q10 gives a parallelogram-shaped banner with a 60 cm slanted side and a 45 cm perpendicular distance between the base and the side opposite it, and then asks why the 60 does not appear in the area. It is the question that separates the students who know the formula from the students who know what it means.

The height is a perpendicular distance, always. The slanted side is longer than the height, so using it inflates the area — you would be claiming to cover more than the figure covers. A parallelogram can be sheared until it is almost flat while its sides stay exactly the same length; only the perpendicular distance changes, which is why only the perpendicular distance can be in the area formula.

The same trap sits in the trapezoid, where the “distance between the parallel sides” is not either of the slanted sides. Q9 hands you a deck with two 5 m slanted sides sitting right beside the 4 m perpendicular distance, and only one of those numbers belongs in the area — while both of them belong in the perimeter.

Circles, arcs and sectors (Q1, Q6)

A circle's two numbers come from its radius, and the answers on this sheet are asked for twice — once as an exact multiple of π and once as a decimal, using π ≈ 3.14. Keep the exact form all the way through the working and put the decimal in only on the last line; rounding early and then multiplying is how an answer drifts.

A sector is a slice, and everything about it is the matching fraction of the whole circle. A 40° slice is 40÷360 of the circumference and 40÷360 of the area, with the same fraction used both times:

arc = (angle ÷ 360) × circumference sector area = (angle ÷ 360) × circle area

The arc is not the perimeter of the slice. The last part of Q6 turns on exactly this: a pizza slice is bounded by the curved crust and by two straight radii, so its perimeter is the arc plus two radii. The arc alone is the crust. Read which of the two the question wants, because the numbers are quite different and both look plausible.

A chord cannot beat the diameter. Q1 asks why no chord of a 4.5 cm-radius circle can exceed 9 cm. Because the longest chord is the one through the centre — that is what the diameter is — and any other chord cuts a shorter path across.

Regular polygons and the apothem (Q11)

The apothem is the perpendicular distance from the centre of a regular polygon to the middle of one of its sides. Slice the polygon into congruent triangles, one per side, each with a side as base and the apothem as height, and the area comes out as

A = (P × a) ÷ 2

with P the perimeter and a the apothem. Q11 asks you to say in your own words what the apothem is, and the reason that part is there is that students who cannot say it reliably use a radius, or a side, in its place. It is a distance to the middle of a side, not to a vertex.

Decomposable figures: sum or difference, decided before you compute (Q2, Q4, Q8)

Three questions cut an awkward shape into manageable ones, and they are asked in two different directions on purpose. Q2 splits an L-shaped floor into rectangles two different ways and makes you check that the two agree — if they disagree, one of your missing side lengths is wrong, and on an L-shape the missing sides are always found by subtracting the two full sides the notch sits between. Q4 goes the other way, taking a semicircular bite out of a rectangle, where subtraction is the only sensible route.

Areas add up. Perimeters do not. This is the single most expensive misunderstanding on the topic. When you cut a figure into two pieces, the cutting line becomes an edge of both pieces — so the two perimeters include it twice, and the real figure includes it not at all, since it is inside the room rather than round the outside. That is why Q2 asks you to explain where the cutting segment went. Areas are safe: the two pieces cover the figure exactly once between them, whichever way you cut.

The practical rule: compute an area by decomposition; compute a perimeter by walking round the outline once and adding the sides you actually walk along.

Q8 combines both skills in one figure — a rectangle with a semicircle on top. The area is a plain sum, but the perimeter drops the top of the rectangle, because the glass covers it and the frame does not go round it. Trace the outline with a finger before adding anything.

Units: the conversion that is not what students expect (Q12, Q13, Q14)

Three questions do nothing but convert — lengths in Q13, masses in Q12, hours and minutes in Q14 — and they are there because a measurement given in the wrong unit quietly ruins a geometry answer that is otherwise perfect. Q13 opens by asking you to order five lengths written in five different units, which cannot be done at all until every one of them is in metres. That is the discipline the rest of the topic depends on.

First rule: convert before you multiply. Bring both measurements to one unit and only then compute. Almost every unit error in a geometry problem is committed in that one step.

Second rule: an area unit is squared, and so is its conversion factor. Draw a square of side 1 m and mark it in centimetres: it is 100 cm along each edge, so it holds 100 × 100 = 10 000 small squares. 1 m² is 10 000 cm², not 100 cm². You never have to remember that number — the square of side 1 m regenerates it in ten seconds, which is the only reason it is worth knowing how it is built.

Length, mass and time do not all behave alike. Metres and grams step by factors of ten, so those conversions are a decimal point moving a known number of places. Time is the exception and needs care: 1 h 15 min is 1.25 h, not 1.15 h, because the minutes are sixtieths, and an hours-and-minutes total has to be carried at 60 rather than at 100.

What the sheet does not need

Every side length you need is given to you. The relation that recovers a third side of a right triangle from the other two arrives later in the Québec Education Program, and so do the trigonometric ratios and the general methods for solving equations, so nothing here depends on any of them. The reverse-direction questions are solved by undoing a formula and by reasoning about equal shares, which is exactly how they are meant to be solved.

Q15: the synthesis question

The last free question rounds one corner off a rectangular pad with a quarter-circle, and then asks for area, perimeter and a number of edging lengths to order. It is the sum-versus- difference decision and the areas-add-but-perimeters-do-not warning in one figure: the corner is subtracted from the area, while in the perimeter two straight pieces are shortened and a curved piece is added in their place. Then the last part asks how many fixed-length strips must be bought, and buying is always rounded up — a shop does not sell two thirds of a strip, and a remainder still has to be covered.

Getting the most out of it

Write the unit on every line, not just the last one

Carry cm, m, cm² and m² through the working. A unit that changes halfway down a page is visible instantly if it is written down and invisible if it is not, and mismatched units are the most common cause of a wrong answer on this whole topic.

Trace the outline before computing a perimeter

Put a finger on the drawing and walk all the way round it, adding only the edges you actually travel along. Any segment you drew yourself to cut the figure up is not on that walk. This one habit removes most decomposition errors.

Keep π exact until the final line

Several questions ask for the exact multiple of π and then the decimal. Do the whole calculation with π in it, substitute 3.14 once at the end, and round only there. Rounding early and then multiplying moves the answer in a way that no later checking will catch.

Round the way the situation requires

When a question asks how many rolls, cans or lengths must be bought, round up, and say why: a remainder still has to be covered and shops do not sell part of a unit. That sentence is part of the answer, not decoration.

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 1 Math Solutions Bundle, which is what keeps the rest of the series free.

What else exists for Perimeter, Area and Units of Measurement

Three PDFs · 22 pages · all three are in the bundle below.

Answer key — 4 pages. All 15 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 13 pages, 14 problems. A separate sheet at exam-plus difficulty covering the same 14 concepts. Harder than anything on the free sheet.
Challenge answer key — 5 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

The one thing that's for sale

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Every Secondary 1 Math topic — the complete Solutions Bundle

One download, one payment, the whole program. Every answer key and every challenge set for all 15 Secondary 1 Math worksheet sets — including this one.

15 sets · 45 PDFs · 182 pages$19.99

Worked solutions, not answer lists — every step written out
Covers the whole year's program at this level
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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 1 Solutions Bundle, which covers every set at this level.

Why can't I use the slanted side of a parallelogram in the area formula?

Because the height in the formula is a perpendicular distance, and the slanted side is longer than that distance. A parallelogram can be pushed over until it is nearly flat while every side keeps its length — only the perpendicular distance changes, which is why only the perpendicular distance can decide the area.

How many square centimetres are in a square metre?

Ten thousand. Draw a square one metre on each side and mark the edges in centimetres: it is 100 by 100, so it holds 100 × 100 = 10 000 small squares. An area conversion factor is the square of the length one, which is why 1 m² is not 100 cm².

Why can't I add the perimeters of the pieces I cut a figure into?

Because the line you cut along becomes an edge of both pieces, so it gets counted twice — and it is not part of the real outline at all, since it runs through the inside of the figure. Areas can be added safely; perimeters have to be found by walking round the outside once.

What is the difference between the arc and the perimeter of a sector?

The arc is only the curved part. The perimeter of the sector is that arc plus the two straight radii that bound the slice. Both are reasonable-looking answers and one question on this sheet asks for each, so read carefully which one is wanted.

Do I need the Pythagorean relation for any of this?

No. It arrives later in the Québec Education Program, and every side length these questions need is given to you. Nothing on this sheet requires it, or trigonometry, or a general equation-solving method.

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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