Mathematics 11–12 · Years 11–12

Area of the school oval from an aerial photograph by the trapezoidal rule

Applications of measurement: Perimeter, area and volume (Mathematics Standard, Year 11: the trapezoidal rule for up to four applications, so the Standard pathway stops at four strips); Ratios and rates: Ratios (Mathematics Standard 2, Year 12); Integral calculus: Areas and the definite integral (Mathematics Advanced, Year 12: any number of strips)

Practical, model not builtLow risk

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The idea

Slicing an irregular region into strips and treating each as a trapezium gives an area estimate that improves as the strips narrow, which is the idea behind the definite integral.

What you need

A printed aerial photograph of the school oval (or a local pond) from NSW Spatial Services SIX Maps, with its scale bar
1 cm grid paper, a ruler, a compass for the semicircle check
A trundle wheel for a ground check of one offset

How to do it

Draw a straight baseline along the length of the oval on the print, divide it into four equal strips and measure the five offsets (widths perpendicular to the baseline); convert each to metres with the scale bar.
Apply A = h/2 (d_f + d_l) to each strip and add; four applications is the limit in the Standard syllabus. Mathematics Advanced learners repeat with offsets every 2 cm and every 1 cm.
Count whole and part grid squares for an independent estimate.
Check the method on a semicircle of radius 10 cm against the exact area pi r^2 / 2: with 4 strips (Standard and Mathematics Advanced), then with 8 and 20 strips (Mathematics Advanced).
Walk one offset on the ground with the trundle wheel and compare it with the value from the photograph.

What you should see

For the semicircle of radius 10 cm the trapezoidal estimates are 136.6 cm^2 with 4 strips, 149.8 cm^2 with 8 and 155.2 cm^2 with 20, against the exact 157.1 cm^2; the rule underestimates a boundary that bulges outward and the error falls as the strips narrow. For the oval, narrower strips move the estimate towards the grid count. The learner knows it worked when the four-strip estimate (and, for Mathematics Advanced learners, the 2 cm and 1 cm estimates) and the grid count agree more closely than the width of one grid square times the perimeter, and the trundle-wheel offset matches the scaled one.

What changes

What you change: strip width h
What you measure: estimated area
What you keep the same: the same outline
offsets measured perpendicular to the baseline
one scale conversion

Common misconceptions

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

The trapezoidal rule gives the exact area; it is exact only for straight boundaries.
More strips always overestimate; the rule underestimates a boundary that bulges outward and overestimates one that caves in.

Safety card

Low riskLearners carry it out

Hazards

walking the oval during sport

Controls

take the ground measurement when the oval is not in use

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).

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 Standard 11–12 Syllabus (2024), Year 11 focus area Applications of measurement; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-11-05
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Ratios and rates; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-05
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Integral calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-05
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities