Explaining the Area of a Circle: Understanding A(r) = πr^2

[mathdad]

The area of a circle as a function of its radius.

The textbook answer is A(r) = πr^2.

I cannot make the connection between the words and the equation.

What does A(r) mean?

I know that πr^2 means "pi times (radius) squared" but what does it really mean?

Why is the radius squared in the equation?

What words in the statement hint that we must multiply
π by the radius squared?

 

[skeeter]

$A(r)$ is read as "$A$ of $r$" or area as a function of radius ...

simple derivation w/o calculus ...

 

[mathdad]

Cool. I do not know calculus.

 

[HallsofIvy]

The problem said "find area as a function of radius". Surely before you got this question your textbook defined the word "function" and explained function notation?

"f(x)" is the standard way to write "f is a function of x". It would have been good to has first said "A is area and r is radius" but those are so commonly (surely, you recognized that "A" is the first letter of "area" and "r" is the first letter of "radius) used (complete to the use of the capital for "area" but the small letter for "radius") that the author may have felt it was not necessary to say that.

\(\displaystyle A(r)= \pi r^2\) says "the area of a circle with radius r is pi times the square of the radius".

 

[mathdad]

Other examples would be

g is a function of x = g(x)

y is a function of theta = y(theta)

h is a function of x = h(x)

Correct?