Circumference and area of a circle – roll a wheel, rearrange sectors
MathematicsPlane GeometryAges 14–15
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Sign in to playRoll a wheel along a ruler to measure its circumference, record results for several radii and plot C against d to discover π. Cut the circle into n sectors and rearrange them into an almost-rectangle πr × r to derive A = πr². Work out arc length and sector area from the angle θ, and the area and perimeter of composite shapes with semicircles and rings.
Lesson: Circumference and area of a circle; arc length, area of a sector and of an annulus; composite shapes
What it shows
A wheel that rolls without slipping travels exactly one circumference in one turn, so the distance on the ruler measures C. Recording C for several diameters gives points on a straight line through the origin whose gradient is π, so C = πd = 2πr. Cutting the disc into n equal sectors and placing them alternately up and down gives a shape whose base tends to half the circumference, πr, and whose height tends to r, so A = πr². A sector with angle θ is θ/360 of the circle. Composite shapes combine these results with rectangles, triangles and squares.
How to use
Choose a tab. In Roll to measure C, press Roll one turn or drag the wheel, then Record, change Radius r and repeat to build the table and graph. In Cut and rearrange, press the button or drag Progress, and increase Number of sectors n. In Arcs and sectors, drag point B or set Angle at centre θ. In Composite shapes, pick a Shape and change r and Length a.
Parameters you can change
- Mode Roll a wheel to measure the circumference, Cut and rearrange to find the area, Arcs and sectors, Composite shapes
- Radius r 1–6 cm
- Number of sectors n 4–48
- Angle at centre θ 5–360 °
- Composite shape Window (rectangle + semicircle), Running track (rectangle + two semicircles), Ring (annulus), Ice-cream cone (triangle + semicircle), Square with a quarter circle cut out
- Length a 1–8 cm
Questions to explore
- Why does the ratio C : d stay the same for every wheel you roll?
- As n increases, which lengths do the base and height of the rearranged shape approach?
- What fraction of the circle is a sector with an angle of 45°, and how long is its arc?
