Math // Measurement

Perimeters of polygons

Solve problems involving perimeters of polygons: find perimeter from side lengths, find an unknown side length, and explore rectangles with same perimeter but different areas (or vice versa)

Ages
8 to 9
Type
Method
Builds on
2 topics
Leads to
3 topics
Standards
2 codes

Quick check

If a rectangle has a perimeter of 24 cm and one side is 8 cm, can the learner work out the length of the other side — without drawing it?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Calculate the perimeter of a rectangle with sides 8 cm and 5 cm
  • Find the missing side of a pentagon with perimeter 30 cm and four known sides
  • Draw two rectangles both with perimeter 24 cm but different areas

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Measuring PerimetersRequired

    Measurement // Ages 7 to 8

    Measuring perimeter of simple shapes is prerequisite to perimeter problem solving

  • Area of rectangles by multiplicationHelpful

    Measurement // Ages 8 to 9

    Area of rectangles supports exploring same-perimeter-different-area relationships

You are here

Perimeters of polygons

Ages 8 to 9 // Method

Leads to

3 topics

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Perimeter problem-solving for ages 8–9 extends 'perimeter = sum of all sides' in two directions: working backwards to find an unknown side (total minus the known sides), and separating perimeter from area (rectangles with the same perimeter can have very different areas). It rests on labelling every side before adding and keeping units explicit (cm for perimeter, cm² for area).

Common misconceptions

  • Adds only the two given adjacent sides of a rectangle, forgetting the opposite pair
  • Confuses perimeter with area — multiplies the sides or counts interior squares when asked for perimeter
  • Believes a fixed perimeter forces a fixed area (can't produce two same-perimeter rectangles with different areas)
  • Drops the unit or mixes cm and m within a problem

Useful representations

  • A polygon with EVERY side labelled (not just the given ones)
  • Grid paper for drawing and comparing same-perimeter rectangles
  • A total-minus-parts bar / number line for the unknown-side step
  • A table of rectangles with columns for length, width, perimeter and area

Teaching practices

  • Insist on labelling ALL sides before adding — the documented remedy for the 'add just the two given sides' error
  • For an unknown side, reason 'perimeter total − sum of known sides = missing side' (a total-minus-parts / bar-model move)
  • Explore same-perimeter-different-area on grid paper (e.g. the 'robot on grid paper' activity): sketch several rectangles of fixed perimeter and tabulate their areas so the two attributes visibly come apart
  • Keep the unit with every number and answer in cm for perimeter, cm² for area, so the length/area distinction stays explicit

Activities to try

  1. 1

    Watch the area change while the perimeter stays the same

    Why it works: A fixed perimeter does not fix the area: rectangles with the same perimeter can have very different areas (although the perimeter still limits how big the area can get, e.g. no 24 cm rectangle has an area above 36 cm²).

    1. On a 1 cm square grid, draw or take a rectangle (e.g. one with perimeter 24 cm, starting as 5 cm × 7 cm) and record its perimeter and its area in a table with columns for length, width, perimeter and area.
    2. Reshape it to become long and thin while keeping the same perimeter of 24 cm — e.g. go from 5×7 to 1×11 — and record the new area in a fresh row each time.
    3. Compare the rows: notice the perimeter stays at 24 cm while the area changes (e.g. from 35 cm² down to 11 cm²).
    4. Answer: if I only tell you a rectangle's perimeter is 24 cm, can you tell me its area for certain? Explain why not.
  2. 2

    Label every side, then add

    Why it works: Perimeter is the sum of ALL sides; marking every side first prevents the common error of adding only the two given numbers.

    1. Take a rectangle marked only 8 cm along the top and 5 cm down one side.
    2. Before adding anything, label the two unmarked sides — each equals its opposite side (so 8 cm and 5 cm).
    3. Add all four sides: 8 + 5 + 8 + 5 = 26 cm, and write the answer with its unit (cm).
    4. Repeat for a polygon where only some sides are printed: fill in every side length you can determine, then add them all.
  3. 3

    Find the missing side by reversing

    Why it works: An unknown side is the perimeter total minus the sum of the known sides — a total-minus-parts (bar-model) reversal of finding the perimeter.

    1. Take a shape with a known total perimeter but one side blank — e.g. a pentagon with perimeter 30 cm and four sides of 5 cm, 6 cm, 7 cm and 4 cm.
    2. Add the known sides: 5 + 6 + 7 + 4 = 22 cm.
    3. Subtract from the total to find the missing side: 30 − 22 = 8 cm.
    4. Use a total-minus-parts bar or number line to narrate: 'All sides must total 30 cm; I already have 22 cm; so the last piece is 8 cm.'
    5. Explain in your own words why you subtract rather than add here.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
3.MD.8
The national curriculum in England: Key stages 1 and 2 framework document
Ma/KS2/Y4/M/2

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