Area and perimeter of a composite outdoor space, measured with a tape and trundle wheel

The idea

The area of an irregular site is found by dissecting it into rectangles, triangles and trapeziums whose dimensions are measured; two different dissections must give the same total, and the perimeter is measured and computed separately because area and perimeter do not change together.

Safety card

What you need

• 30 m fibreglass measuring tape graduated in centimetres
• Trundle wheel (1 m circumference, clicking each metre)
• 8 marker cones and chalk
• Clipboard and 1 cm grid paper for the sketch
• A site in the school grounds bounded by straight edges, such as an L-shaped courtyard, a garden bed or a hard-court area

How to do it

1. Walk the boundary and sketch the site on grid paper, placing a cone at every corner.
2. Measure every straight edge with the tape pulled taut along the ground, to the nearest centimetre, and write each length on the sketch.
3. Check each corner drawn as a right angle with a 3-4-5 test: 3.00 m along one edge, 4.00 m along the other, 5.00 m across.
4. Dissect the sketch into rectangles, triangles or trapeziums, compute the area of each piece and the total in square metres.
5. Choose a different dissection (or a bounding rectangle minus the missing parts), compute the total again and compare.
6. Measure the perimeter with the trundle wheel, reading the final part-metre from the wheel's scale, and compare with the sum of the taped edges.
7. Treat each taped length as uncertain by ±5 cm, compute the largest and smallest possible areas, and state the area to a sensible number of figures.

What you should see

As a worked check, an L-shaped courtyard made from a 12.00 m by 8.00 m rectangle with a 5.00 m by 3.00 m corner missing has area 96 − 15 = 81.0 m², and the alternative dissection 12.00 × 5.00 + 7.00 × 3.00 = 60 + 21 = 81.0 m² agrees. Its perimeter is 40.0 m, the same as the full 12 m by 8 m rectangle, although its area is 15 m² smaller. With ±5 cm on every length the area lies between 79.6 and 82.4 m² (about ±1.7 per cent), so 81 m² is the honest statement; those bounds come from the bounding-rectangle dissection, 12.05 × 8.05 − 4.95 × 2.95 = 82.40 and 11.95 × 7.95 − 5.05 × 3.05 = 79.60, and the two-rectangle dissection gives a slightly narrower 79.66 to 82.36, because the same readings enter different products in the two forms, so the reported bounds must name the dissection they came from. The trundle wheel total and the taped total are compared and their difference reported, since the wheel follows every bump in the ground. The learner knows it worked when the two dissections agree exactly on paper and the reported area carries its uncertainty.

What changes

This activity lists no variables to change, measure and keep the same.

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

• A shape with a larger perimeter always has a larger area.
• Cutting a corner out of a rectangle always shortens its perimeter.
• Two different dissections of the same shape can give different areas.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

• Mathematics K–10 Syllabus (2022), Stage 4 Area; page read 2026-09-22MA4-ARE-C-01
• Mathematics K–10 Syllabus (2022), Stage 4 Length; page read 2026-09-22MA4-LEN-C-01
• Australian Curriculum v9AC9M8M01

Sources

The pages the author read to write this activity.

1. curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-4/fa3b012530
2. curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-4/fa498c8819
3. amsi.org.au/teacher_modules/area_volume_surface_area.html