Secondary 2 Perimeter and Area of Plane Figures

8 of the 10 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 8 of the 10 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. Q1Decomposable Geometric Figures

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

2. Q2Methods to Decompose Figures

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

3. Q3The Arcs and Sectors of a Circle

   A lawn sprinkler stands at the corner of a park. It turns back and forth through a central angle of ${120}^{\circ }$ and sprays water up to $9$ m away. Use $\pi \approx 3.1416$ and round each answer to one decimal place.

   1. What area of grass does the sprinkler water?
   2. How long is the arc traced by the far edge of the spray?
   3. What is the perimeter of the watered region?
4. Q4The Area and the Perimeter of Triangles

   A sailmaker cuts a right-angled sail whose two shorter sides measure $5$ m and $12$ m and whose longest side measures $13$ m.

   1. How much edging tape is needed to go all the way around the sail?
   2. How much cloth does the sail use?
   3. A second sail is cut as a triangle with a base of $12$ m and a height of $5$ m, but with no right angle anywhere. Does it use the same amount of cloth? Does it need the same amount of edging tape? Explain.
5. Q5The Perimeter and The Area of Decomposable Figures

   A neighbourhood skating rink is a rectangle $20$ m long and $12$ m wide, with a half-disc added at each of the two short ends. Use $\pi \approx 3.1416$.

   1. Find the perimeter of the rink, to one decimal place.
   2. Find its area, to one decimal place.
   3. Resurfacing the ice costs $6 per square metre. What does one resurfacing cost, to the nearest dollar?
6. Q6The Perimeter and the Area of Plane Figures
   1. For each task, state whether you need a perimeter or an area, and give the unit you would use: (i) buying baseboard for a bedroom; (ii) painting a wall; (iii) fencing a dog run; (iv) laying sod on a lawn.
   2. A rectangular dog run measures $7.5$ m by $4$ m. Find its perimeter and its area, with the correct units on each.
   3. Explain in one sentence why the two answers in b) cannot be compared to each other.
7. Q7The Perimeter and the Area of Plane Figures Using Algebra

   A rectangular vegetable bed is $(2x+3)$ metres long and $5$ metres wide.

   1. Write a simplified expression for its perimeter.
   2. Write a simplified expression for its area, in expanded form.
   3. Find the perimeter and the area when $x=4$. Check your area a second way, by first working out the length and the width.
8. Q8The Perimeter and the Area of Quadrilaterals

   Find the area of each quadrilateral below.

   1. A parallelogram with a base of $14$ cm and a height of $9$ cm.
   2. A trapezoid whose parallel sides measure $11$ cm and $19$ cm, with a height of $6$ cm.
   3. A rhombus whose diagonals measure $16$ cm and $10$ cm. Then find its perimeter, knowing that each of its sides measures $9.4$ cm.
9. Q9The Perimeter and the Area of Regular Polygons
   1. A regular hexagonal paving stone has sides of $8$ cm and an apothem of $6.9$ cm. Find its perimeter and its area.
   2. A regular octagonal road sign has sides of $25$ cm and an apothem of $30.2$ cm. Find its perimeter and its area.
   3. Explain in one or two sentences why the formula $A={\displaystyle \frac{P\times a}{2}}$ works for every regular polygon.
10. Q10Synthesis — drawing on several sheets in this topic

    A window is made of a rectangle $1.2$ m wide and $1.8$ m tall, with a half-disc of glass sitting on top of it. The flat side of the half-disc is the top of the rectangle. Use $\pi \approx 3.1416$ and round each answer to the nearest hundredth.

    1. Find the area of glass in the window.
    2. Wooden trim runs all the way around the outside of the window. What length of trim is needed?
    3. The glazier quotes the trim in whole metres. How many metres must be bought?

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.

Perimeter and area answer different questions, in different units

Q6 asks which of the two you need before it asks you to compute anything: baseboard for a bedroom, paint for a wall, fencing for a dog run, sod for a lawn. Anything that runs around a figure is a perimeter and is measured in metres or centimetres; anything that covers it is an area and is measured in square units. The two answers to the same question about the same dog run cannot be compared to each other at all, and saying why — they measure different kinds of thing — is part of the answer.

Decomposing a figure: add or subtract, and decide first (Q1, Q2)

A decomposable figure is one that can be cut into ordinary shapes whose areas you already know. Two routes are always available, and the questions ask for both on purpose: cut the figure into pieces and add, or enclose it in a larger rectangle and subtract what is not part of it. Q2 asks for the same area both ways, and the reason is that the two methods check each other — if they disagree, one of your missing side lengths is wrong.

On an L-shaped figure the missing lengths are always recovered by subtraction: a short side of the notch is the difference between the two full sides it sits between. Mark those subtracted lengths on the drawing before computing anything, and treat a negative area as an immediate sign that you subtracted a piece larger than the whole.

Perimeters do not decompose the way areas do (Q1)

Cut a figure in two and the areas of the pieces add up to the area of the whole. The perimeters do not, and the reason is visible on the drawing: the cut creates a new edge that belongs to both pieces and to neither outline. Every internal cut adds its own length twice to the total, which is why adding the perimeters of the pieces always overshoots.

The same idea explains a result that surprises people: cutting a window out of the middle of a plaque removes area but leaves the outside perimeter completely unchanged. The edge that has to be filed smooth is the outside edge plus the new inside edge — two separate closed outlines, both of them real.

Arcs, sectors, and half-discs (Q3, Q5, Q10)

A sector is a fraction of a disc, and the fraction is the central angle over 360°. Take that fraction of the whole area for the sector's area, and the same fraction of the whole circumference for the arc length. One idea, applied twice.

Q10 adds a step that is easy to skip: the trim is quoted in whole metres, so the answer has to be rounded upwards, whatever the decimal says. A quantity you have to buy is always rounded up — buying 4.2 metres of trim means buying 5 — and rounding to the nearest metre would leave the window unfinished.

Equal area does not mean equal perimeter (Q4)

Q4 gives two triangles with the same base and the same height, one right-angled and one not, and asks whether they need the same cloth and the same edging. The areas match, because area depends only on the base and the perpendicular height. The perimeters do not, because the slanted sides differ.

This runs both ways and is worth holding on to: two figures with the same area can have very different perimeters, and two figures with the same perimeter can have very different areas. Neither quantity determines the other, and any claim that one does needs a counterexample rather than an opinion — one explicit pair of figures, with both numbers computed.

Writing a perimeter and an area as expressions (Q7)

This is the Secondary 2 step, and it is algebra doing geometry's bookkeeping. A rectangle (2x + 3) metres long and 5 metres wide has a perimeter of 2(2x + 3) + 2(5), which simplifies to 4x + 16, and an area of 5(2x + 3), which expands to 10x + 15. Nothing new is happening geometrically; the length simply has a letter in it.

Two things to watch. Distribute across the whole bracket — 5(2x + 3) is 10x + 15, never 10x + 3 — and simplify by collecting like terms only, since 4x and 16 are not like terms and 4x + 16 is a finished answer rather than an unfinished one. Then Q7 asks you to substitute a value and check the area a second way, by working out the length and the width first and multiplying. Agreement between the two routes is what tells you the expansion was right, and it is a check worth running every time you expand.

The apothem, and why one formula covers every regular polygon (Q9)

Join the centre of a regular polygon to each vertex and it falls into identical triangles, one per side. Each has the side as its base and the apothem as its height, so each has area (side × apothem) ÷ 2. Add up all of them and the sides collect into the perimeter, giving perimeter × apothem ÷ 2 — for a pentagon, an octagon or a dodecagon alike. That argument is what Q9 asks for, and it is more valuable than the formula it produces.

Run it backwards and it finds a missing length: given the area and the apothem, the perimeter is the area doubled and divided by the apothem, and dividing that by the number of sides gives one side. Rearranging a formula to solve for the quantity you were not given is the habit this sheet is quietly building.

Getting the most out of it

Draw it, mark every length, then choose a formula

Redraw the figure large enough to write on, mark every given length, and fill in the lengths you can recover by subtracting along the sides. Most wrong answers here are the right formula with a slanted side in place of a perpendicular height, and that is a mistake you can see in a marked drawing.

Say whether you are adding or subtracting before you compute

Write "area of the big rectangle minus the notch" or "left rectangle plus top rectangle" at the head of the working. It commits you to a route, it makes the arithmetic checkable by someone else, and where a question asks for both routes it gives you an exact cross-check.

Trace the outline with a finger before computing a perimeter

Especially with sectors, half-discs and figures with a piece cut out. Whatever your finger travels along belongs to the perimeter; an internal edge where two pieces meet does not. That thirty-second trace prevents nearly every perimeter error on this sheet.

Round at the end, and round the right way

Carry π through the calculation and round only in the final line. Then ask what the number is for: a quantity that has to be bought in whole units is rounded up, never to the nearest.

Common questions