Mathematics 7–10 · Year 8
Area of a circle: cutting sectors into a near-rectangle
Measurement and space
The idea
A circle cut into many equal sectors and laid alternately point up and point down forms a shape close to a rectangle with height r and base half the circumference, which is where A = πr² comes from.
What you need
- 2 paper circles of radius 10.0 cm per learner (printed with 8 and 16 equal sectors marked)
- Scissors and glue stick
- 1 cm grid paper, A3 or two A4 sheets taped end to end (the rearranged shapes are about 34 cm wide)
- Ruler
How to do it
- Trace one circle onto the grid paper with its centre on a grid corner and estimate its area first by counting whole and part squares.
- Cut the first circle into 8 sectors and glue them onto grid paper alternating point up and point down. Measure the base and height of the shape formed.
- Cut the second circle into 16 sectors and repeat. Compare how much straighter the top and bottom edges have become.
- Record the base as half the circumference (πr = 31.4 cm) and the height as the radius (10 cm); multiply to get the area and compare with the grid count.
- Extension: cut one 16-sector piece in half lengthwise and place each half at an end to square the sides.
What you should see
For r = 10.0 cm the exact area is 314.2 cm². On a circle centred on a grid corner there are 276 whole squares inside and 344 squares touched, so the area lies between 276 and 344 cm², and counting each part square as a half gives 310 cm², within 1.3 per cent of 314.2. The 8-sector shape has scalloped top and bottom edges, each made of four arcs that bulge 0.76 cm beyond their chords; the four chords along the bottom span 4 × 7.65 = 30.6 cm, and the height is 9.2 cm from the bottom chords to the top chords and 10 cm from the bottom chords to the tops of the upper arcs. The 16-sector shape spans eight chords of 3.90 cm, 31.2 cm, with heights of 9.8 to 10 cm, and its arcs bulge only 0.19 cm beyond their chords, so both edges are much flatter and the shape is closer to the limiting rectangle of πr = 31.4 cm by r = 10 cm, and 31.4 × 10 = 314 cm². The learner knows it worked when the rectangle estimate and the grid count agree within a few per cent and the 16-sector shape is visibly closer to a rectangle than the 8-sector one.
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.
- Area of a circle is π times the diameter.
- Cutting and rearranging changes the area.
- The formula is a rule to memorise rather than something that can be seen.
Safety card
Hazards
- scissors
Controls
- cut on the desk, blades away from the body
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
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
- Australian Curriculum v9AC9M8M03
Sources
The pages the author read to write this activity.