Area of a Circle Formula via Sector Partitioning
What you will learn
Derive and verify the formula for the area of a circle (Area = π·r²) by partitioning a circle into 16 and 32 congruent sectors and rearranging into an approximate rectangle.
Materials needed
- Cardboard sheet with pre-printed circular disc of radius r = 7 cmWhat is it?Cardboard sheet with pre-printed circular disc of radius r = 7 cm₹50
Laser-cut acrylic circle sector partitioning demonstration kit (16 and 32 sectors).
Specifications- Diameter
- 140 mm (radius r = 70 mm)
- Material
- 3 mm two-tone acrylic (red/yellow)
- Sectors
- 16 sectors + 32 sectors matching kits
- Tray
- Rectangular alignment boundary tray
- Protractor for measuring 11.25° and 22.5° sector anglesWhat is it?Protractor for measuring 11.25° and 22.5° sector angles₹95
Precision geometric drafting instrument box.
Specifications- Compass
- Self-centering geared mechanical compass
- Divider
- Hardened steel needle points divider
- Protractor
- 360° full circular protractor
- Two contrasting colored glaze papers (blue and white) eachWhat is it?Two contrasting colored glaze papers (blue and white) each₹140
Heavy 300 GSM duplex art card and color-fast adhesive.
Specifications- Weight
- 300 GSM smooth white art board
- Adhesive
- Acid-free solventless PVA glue stick
- Scissors and Craft Glue StickWhat is it?Scissors and Craft Glue Stick₹185
Standard laboratory apparatus used in Area of a Circle Formula via Sector Partitioning.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
- Geometry Compass and 30 cm rulerWhat is it?Geometry Compass and 30 cm ruler₹230
Standard laboratory apparatus used in Area of a Circle Formula via Sector Partitioning.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
Tick items as you collect them. Tap + to see the product, specifications and buy options.
Stay safe
- Cut strictly along circular arcs and radial radii lines without tearing paper edges.
- Ensure alternate up/down sector placement is tight without overlapping.
- Keep sector radius r constant throughout circle division.
- Handle sharp scissors with care.
Steps
- 1Draw a circle of radius r = 7 cm on colored cardboard.Observation
Draw a circle of radius r = 7.0 cm on heavy card paper using compass; area = π·r² = 153.94 cm².
SafetyTighten compass hinge so radius does not drift during drawing.
- 2Using a protractor, divide the circle into 16 congruent sectors, each with central angle θ = 360° / 16 = 22.5°.Observation
Divide circle into 16 equal sectors using protractor (each angle = 360°/16 = 22.5°); color half red and half yellow.
SafetyMark radial lines with fine 0.3 mm pen.
- 3Shade 8 alternate sectors in blue and the remaining 8 sectors in white.Observation
Cut out all 16 sectors using scissors; cut one sector in half (11.25° each) to form end pieces.
SafetyUse blunt-ended safety scissors when cutting sector curves.
- 4Cut out all 16 sectors carefully along the radial radii lines.Observation
Arrange the 16 sectors alternating up and down in an intermeshing strip; place half-sectors at both ends.
SafetyAlign sector tip vertices strictly along straight reference line.
- 5Cut one end sector into two equal half-sectors of 11.25° to form vertical borders.Observation
Observe that the intermeshing sectors form an approximate rectangle of length π·r (22.0 cm) and breadth r (7.0 cm).
SafetyMeasure rectangular dimensions with precision millimeter ruler.
- 6Arrange the sectors side by side alternating up and down in interlocking fashion on a display board.Observation
Repeat mentally or with 32 sectors; sectors become narrower and edges straighten, proving Area = πr².
SafetyMount rearranged sectors on chart paper with glue stick for laboratory record.
- 7Measure the dimensions of the formed parallelogram/rectangle: length = half circumference = (2πr)/2 = π·r, and breadth = radius r. Area = π·r × r = π·r².
Tap + on a step to see what you will observe and any safety tips.
Record your observations
| Sector Partitions (n) | Base Length of Rearranged Shape (cm) | Height of Shape (cm) | Shape Approximated | Computed Area (cm²) |
|---|---|---|---|---|
| 4 Sectors | 18.2 cm (Curved lobes) | 6.1 cm | Rough parallelogram (Wavy) | 111.0 cm² (Crude estimate) |
| 8 Sectors | 20.6 cm | 6.6 cm | Moderate parallelogram | 136.0 cm² |
| 16 Sectors | 21.6 cm | 6.9 cm | Near-flat parallelogram | 149.0 cm² |
| 32 Sectors | 21.9 cm (approaching π·r = 21.99 cm) | 7.0 cm (approaching r) | Almost perfect rectangle | 153.3 cm² |
| 64 Sectors (Limit) | 21.99 cm (= π · r) | 7.00 cm (= r) | Perfect Rectangle: Area = (πr) × r = πr² | 153.94 cm² (Exact π·r²) |
Tap a highlighted cell to write your own observation.
Result and conclusion
As the number of circular sectors increases, the rearranged shape approaches a rectangle of base length π·r (half circumference) and height r. Area = base × height = π·r × r = π·r².
| Observation | What it means |
|---|---|
| Calculus Method of Exhaustion | As sector count n approaches infinity, the curved outer circular arc sectors flatten into microscopic straight segments. |
| Dimensions of Limiting Rectangle | Length of rectangle equals half the circular circumference (L = 1/2 × 2πr = πr = 21.99 cm); Width equals radius (W = r = 7.0 cm). |
| Geometric Area Derivation | Area of rectangle = Length × Width = (π·r) × r = π·r² = 153.94 cm². |
| Conservation of Area | Because the rearranged sectors have identical area to the original circle, the formula A = πr² is incontrovertibly verified. |
Sample readings. Replace with your own results.