How the Area of a Disk Grows

radius R

2

increment ΔR

0.5

For R = 2, an increment ΔR = 0.500

yields actual annular area ΔA = 7.06858

approximate area 2π R ΔR = 6.28319

For R = 2, circumference = 2π R ≈ 12.60

Slope of secant line is the ratio

area ΔA

increment ΔR

≈ 14.10

An increase

ΔR

in the radius of a circular disk leads to a corresponding increase

ΔA

in the area of the disk. Geometrically,

ΔA

is the red annulus. Slicing and then unrolling this annulus would yield a shape close to a rectangle with base

2πR

and height

ΔR

. This gives the approximation

ΔA≈2πRΔR

. Using the slider, notice this approximation is quite accurate for

ΔR

close to zero. Graphing the area

A

as a function of the radius

R

, the ratio

ΔA/ΔR

is the slope of a secant line to the curve. Dividing both sides of

ΔA≈2πRΔR

by

ΔR

implies the slope

ΔA/ΔR

may be approximated by the circumference

2πR

.