dec 20. 2008 concept by kalid azad slide show by audrey mcgoldrick kalid’s “gentle introduction...
TRANSCRIPT
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
Kalid’s “Gentle introduction to calculus”
Using the circle
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
Take any circle:
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
“Unroll” the rings and line them up like this:
Draw as many rings inside the circle as you can:
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
If you split the circle into more rings...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
If you split the circle into more rings...they’d be skinnier than these...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
...and this would become...
If you split the circle into more rings...they’d be skinnier than these...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
...and this would become...
...this!...
If you split the circle into more rings...they’d be skinnier than these...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
...and this would become...
...this!...
...which is even prettier, and closer to a right triangle!
If you split the circle into more rings...they’d be skinnier than these...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
If you do this long enough, eventually you will have...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
If you do this long enough, eventually you will have...
...this triangle...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
...whose area must equal the area of the original circle!
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
This side must be the
radius of the circle
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
r
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
r
..and this side must be...
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
r
..and this side must be...
..its circumference!
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
r 2r
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
r
Use that good old formula for area of a triangle....
2r
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
Use that good old formula for area of a triangle...
b hArea
2
r·2r 2r 2r
r
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick
r
r
b hArea
2
r·2r
r2
2r
Use that good old formula for area of a triangle...and with a bit of algebra:
Dec 20. 2008 Concept by Kalid AzadSlide Show by Audrey McGoldrick