- Thread starter X049
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It's one definition of the diameter. You know it's not a diameter if the area of the two parts aren't equal.so i know that the diameter can divide a circle in two equal pieces (they have the same area) but what is the the mathematical proof to that?

-Dan

What's the formula for the area in question? It should point you to a proof.so i know that the diameter can divide a circle in two equal pieces (they have the same area) but what is the the mathematical proof to that?

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The diameter of a circle is a line segment having its endpoints on the circle and which contains the center of the circle. A diameter divides the circle into two arcs of equal length. Because the center divides a diameter is two line segments of equal length, that being the radius, \(r\) of the circle. It is well known that the area of the circle is \(\pi r^2\) square units. Also well is the the length of the circumference of the circle is \(\pi D=2\pi r\).so i know that the diameter can divide a circle in two equal pieces (they have the same area) but what is the the mathematical proof to that?

So the fact is that a diameter divides a circle into two areas of equal measures results from a diameter passing through the center.

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There are various ways you might prove this formally, depending on what theorems you have about congruent sectors, or something like that.

But it's quite mathematical just to observe the

The diameter (length) of a closed bounded shape like a circle, is the

If we take that as the definition of the diameter of a circle, then we can try to prove that the area is bisected by a diameter line.

Consider a

Suppose it does not. Then we can draw a triangle ABC:

with BC≠0.

It is easily proved that the angle ABC is 90º

Now \(\displaystyle AC^2=AB^2+BC^2\)

\(\displaystyle \implies\) \(\displaystyle AC>AB\) (since\(\displaystyle BC≠0\))

Since A, C are points on the circle and AB is a diameter line of length

Therefore, any diameter line goes through the centre.

Now you can use e.g. Dr.Peterson's symmetry argument, to show that the area of the circle is bisected by the diameter line.