直观感受：圆面积

Suppose that you have a circle of radius $R$, and you would like to find its area.

We will use an intuitive formula that can be proved later using calculus.

Consider a circle with radius $r$ $(0\le r\le R)$ that has the same center as the given circle.

By the definition of $\pi$, the circumference of this circle is $2\pi r$.

Now, we may "unwrap" the circle by taking the circumferences of all such circles and straightening them out.

Since the circumference is proportional to the radius, we actually get a triangle!

The height of this triangle is simply the radius $R$ of the original circle. The base is just the largest circumference, $2\pi R$!

Thus, we may find the area of the triangle, which is also the area of the circle: $$A=\frac{1}{2}bh$$ $$A=\frac{1}{2}(2\pi R)(R)$$ $$A=\pi R^2$$ We have just demonstrated an amazing result: that the area of a circle is equal to the circle constant $\pi$ times the square of the radius!

Of course, you still should not mistake this for a proof. You would use calculus to formalize the method above, but the calculus proof uses essentially the same idea.

This should immediately go into your bank of formulas which you should never, ever forget.

Suppose that you have a circle of radius $r.$ Inside this circle, inscribe a regular polygon with $n$ sides; we will use this polygon to approximate the area of the circle.

The polygon has apothem $a$ and side length $s.$ Draw $n$ radii connecting the center of the circle (also the center of the polygon) to each of the vertices of the polygon. This splits the polygon into $n$ triangles with base $s$ and height $a.$ Thus, the area $A_p$ of the polygon is: $$A_p = n\times \frac{1}{2}as$$ $$A_p = \frac{1}{2}aP,$$ where $P = ns$ is the perimeter of the polygon.

Now, we let $n$ grow without bound. The apothem $a$ approaches the radius $r$ of the circle, and the perimeter $P$ approaches the circle's circumference $C = 2\pi r.$ As this occurs, the regular polygon approaches the circle it is inscribed in, so it makes intuitive sense that $A_p$ approaches the area of the circle $A_c.$

We can thus intuitively derive the area of a circle (which turns out to be correct): $$A_c = \frac{1}{2}r\times 2\pi r$$ $$A_c = \pi r^2.$$

(Videos) Area of a Circle

by Art of Problem Solving

Here's a set of amazing videos on circular areas from Art of Problem Solving. The first illustrates a more intuitive basis for the circle area formula, while the second one elaborates on some interesting circle area problems and properties.

Here's a great video that provides some intuition behind the circle area formula!

Here's an awesome video that goes through some unusual problems involving circular areas.