Mathematics • Year 8 • Unit 3 • Lesson 4
Perimeter — Mixed Challenge
Mix regular polygons, rectangles, working backwards from a known perimeter, and composite L-shapes. Six mixed problems, one "find the mistake", and one open-ended design challenge.
1. Mixed problems — different shapes, same rule
Each question is a different perimeter scenario. Decide your approach BEFORE writing. Show working. 3 marks each
1.1 A regular nonagon (9-sided polygon) has side length 11 cm. Find P.
1.2 A rectangle has P = 44 m and width 8 m. Find the length.
1.3 A regular hexagon has P = 72 cm. Find the side length.
1.4 An isosceles triangle has two equal sides of 13 cm and a base of 10 cm. Find P.
1.5 An L-shape has total width 20 cm, total height 15 cm, with a 12 cm × 10 cm rectangular notch removed from one corner. Find the perimeter.
1.6 A square has the same perimeter as a regular hexagon with side length 6 cm. What is the side length of the square?
2. Find the mistake
A student is asked to find the width w of a rectangle with P = 56 cm and length l = 18 cm. Their working is below. Exactly one line contains a mistake — spot it, explain why, and re-do correctly. 3 marks
Student's working — find w when P = 56 and l = 18:
Line 1: P = 2l + 2w
Line 2: 56 = 18 + 2w
Line 3: 2w = 56 − 18 = 38
Line 4: w = 38 ÷ 2 = 19 cm
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § Card 9 — "forgetting to double in the rectangle formula" is exactly this trap. P = 2l + 2w means TWO lengths and TWO widths.3. Open-ended challenge — design rectangles with P = 36 m
This question has more than one valid answer. 4 marks
3.1 A council wants to fence rectangular dog runs using exactly 36 m of fencing for each run (no leftover or waste).
Find three different rectangular dog runs that each have perimeter exactly 36 m. For each design:
- State the length and width in metres (whole numbers only).
- Check that 2l + 2w = 36.
- State whether the rectangle is also a square.
Bonus: for the rectangle that IS a square, explain why a square uses fencing most efficiently if you also want to maximise the floor area (we haven't done area yet, but think about which shape feels "biggest").
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Regular nonagon, s = 11
P = 9 × 11 = 99 cm.
1.2 — Rectangle P = 44, w = 8
Half-perimeter = 22 = l + w, so l = 22 − 8 = 14 m.
1.3 — Regular hexagon, P = 72
s = P ÷ n = 72 ÷ 6 = 12 cm.
1.4 — Isosceles triangle, two sides 13, base 10
P = 13 + 13 + 10 = 36 cm.
1.5 — L-shape 20 × 15 with 12 × 10 notch
Missing horizontal = 20 − 12 = 8 cm; missing vertical = 15 − 10 = 5 cm. Six sides: 15, 20, 5, 12, 10, 8.
P = 15 + 20 + 5 + 12 + 10 + 8 = 70 cm.
1.6 — Square with same P as hexagon (s = 6)
Hexagon P = 6 × 6 = 36 cm. Square P = 4s = 36, so s = 36 ÷ 4 = 9 cm.
2 — Find the mistake
(a) The mistake is on Line 2 (carried into Lines 3–4).
(b) The student wrote 18 instead of 2 × 18 = 36. A rectangle has TWO lengths, not one — so on Line 2 the right-hand side should be 2(18) + 2w = 36 + 2w, not 18 + 2w. They forgot to double the length.
(c) Corrected working:
P = 2l + 2w
56 = 2(18) + 2w = 36 + 2w
2w = 56 − 36 = 20
w = 20 ÷ 2 = 10 cm. ✓
Sanity check: 2(18) + 2(10) = 36 + 20 = 56 cm ✓
3 — Three rectangular dog runs with P = 36 m (sample solution)
Half-perimeter = l + w = 18 m. Any whole-number pair (l, w) with l + w = 18 works:
Design 1: 12 m × 6 m. Check: 2(12) + 2(6) = 24 + 12 = 36 ✓. Not a square.
Design 2: 10 m × 8 m. Check: 2(10) + 2(8) = 20 + 16 = 36 ✓. Not a square.
Design 3: 9 m × 9 m. Check: 2(9) + 2(9) = 18 + 18 = 36 ✓. This IS a square.
Bonus: A square with side 9 m has area 81 m², while the 12 × 6 rectangle has area only 72 m² and the 10 × 8 rectangle has area 80 m². So with the same fence length, the square uses the fencing most efficiently — it encloses the most "ground". (This is true for any rectangle: the square gives the maximum area for a given perimeter.)
Marking: 1 mark per valid design with correct check (up to 3 marks). 1 bonus mark for explaining that the square encloses the most area for the same fencing.