Spatial Geometry & Functions · Geometry & Area Equivalence
Area Cut & Paste Equidecomposition Visualizer
Geometry area formulas come alive through cut-and-paste transformations. Watch a circle dissect into 4 to 64 wedges reforming into a rectangle of πr × r, or observe a triangle's apex slide across parallel lines with invariant area.
- Client-side
- No Sign-up
- Live Derivation
- Projector Mode
- Print Worksheet
3D Simulation Stage● 60 FPS
Parameter Controls
type
sectors
cm
cm
cm
Live Derivations & Metrics
area_target
--
Exact analytical area
rect_base
--
Circle unrolls to half-circumference πr
rect_height
--
Corresponds to radius r or height h
perimeter_diff
--
Perimeter increases by 2r due to two exposed radial sides
Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Circle-to-Rectangle Perimeter Increase & Area Calculation
Problem: A circular paper is cut into equal sectors and rearranged into an approximate rectangle. The perimeter of the rectangle is 10 cm greater than the circle's original circumference. Find the area of the circle.
Analytical Strategy: The two long sides sum to 2 × (πr) = 2πr (the circle's perimeter). The additional perimeter comes entirely from the two short ends, which each equal the radius r. Thus ΔC = 2r = 10 cm, giving r = 5 cm.
Step-by-Step Derivation:
- Step 1: Let the circle radius be r. Circle circumference C = 2πr.
- Step 2: Equivalent rectangle length = πr, width = r. Rectangle perimeter = 2(πr + r) = 2πr + 2r.
- Step 3: Perimeter difference ΔC = 2r = 10 cm, so r = 5 cm.
- Step 4: Compute area S = πr² = π × 5² = 25π ≈ 78.54 cm².
⚠️ Common Misconception Pitfall: Mistaking perimeter increase for area change; area is strictly conserved during rearrangement, while perimeter increases by exactly 2r.
Student Inquiry Worksheet & Practice Problems
Q1. A circle with radius 6 cm is cut into 32 sectors to form a rectangle. By how many cm does the perimeter increase?
Answer Key: 12 cm (exactly 2r = 2 × 6)
Q2. A parallelogram (base 10 cm, height 6 cm) is cut along height and reshaped into a rectangle. How do area and perimeter change?
Answer Key: Area unchanged (60 cm²), perimeter decreases (slanted edge replaced by shorter vertical height)
Frequently Asked Questions & Misconceptions
Why are the top and bottom edges of the reassembled circle slightly wavy?
Each sector's outer edge is a circular arc. As slice count increases from 16 to 64 and infinity, the arc curvature flattens, and the boundary converges strictly to parallel line segments.