How to derive the area of a circle using geometry?

Question

I think it can be done with triangles of height = to the radius. I dont know any further what to do.

Answer 1

Area of circle = π r² where r is the radius and π = 22/7 = 3.14

Example
if r = 10 cm
A = π x 10² cm² = 100π = 314 cm²

Answer 2

I believe you are right. Think of a square inscribed in the circle and draw lines from the center to the corners of the square. You will have 4 triangles of which the area for each is 1/2 base x height.

If you consider a many sided polygon inscribed in the circle you will have a number of these triangles. The sum of the bases will approach the circumference of the circle, and the heights will approach the radius of the circle.

What you will get when the polygon is infinite sided is a circle.

the sum of the areas of all the triangles will be:
1/2 x base1 x height + 1/2 base2 x height + 1/2 base3 x height....

which becomes 1/2 (base1 + base2 + base3+.....) x height

Now the sum of the bases is the circumference = 2 pi radius,
and the height is equal to radius. The sum becomes:

1/2 x circumference x radius = 1/2 ( 2 pi radius) x radius

= pi radius^2

Answer 3

Again, the theorem of Pappus-Guldinus, derived from geometry long before calculus existed, can be used. In one form, it says that the area swept out by a line is the product of the length of the line and the length of the path of the center of the line.

Here your line extends from 0 to r with a center at r/2. The path of the center is the circumference of a circle of radius r/2 or 2 π (r/2) The length of the line is r so the area of the circle is:

A = 2 π (r/2) r = π r^2

Answer 4

Imagine that you cut a circle into pie shaped pieces and then layed them out into a rectangle shape. As you cut more and more pieces (eventually reaching an infinite amount) you will form a rectangle that has a height is equal to the radius of the circle and a width that is equal to pi*r.

The pi*r is found by knowing that the top and bottom of the rectangle is equal to the circumference which equals 2*pi*r, half of that is pi*r.

Find the area of the rectangle (which equals the area of the circle at this point) to be (pi*r) * r or pi*r^2.

Answer 5

You can wave your hands about symmetry and similarity to say that the area is a constant times the square of the radius. That's what the ancient Greeks did, IIRC.

All else is estimating the value of pi.

What you're probably looking for is this:

You could come up with estimates by partitioning the circle into lots of narrow parallel rectangles and looking at the ratio between the size of one and its neighbor. I'm pretty sure that's historical too.

Obviously, trigonometry+calculus is a much easier way to go, if you already have the mechanisms of those subjects anyway.

Answer 6

Inscribe a regular polygon of n sides inthe circle and circumscribe another a similar polygon abut the circle. The area of the circle must fall between the areas of the two polygons.

Now increase the number of sides of the two polygons as large as you like and you will get the area of the circle to whatever accuracy you want.

If you use a square as the polygon, you get Area of circle lies between 4r^2 and 2r^2 which says that pi lies somwhere between
2 and 4. Not veryaccurate, but accuracy increases dramatically as you increase the number of sides of the polygons to 1000.

If there are 1000 sides there will be 1000 triangles making up each of the two polygons. The areas are easily computed using trig formulas for area.

Answer 7

No there is no finite linear combination of triangles that add up to a circle. This is a consequence of PI being an irrational number.

the area is PI r^^2

Answer 8

(pie) r (squared)--area