# Relation between Diameter and Tangent of circle

A comparative question states:

One side of rectangle is the diameter of a circle. The opposite side of rectangle is tangent to the circle.Which is bigger a)The perimeter of rectangle or b)The circumference of the circle (Ans=$b$)

Now I know a tangent is perpendicular to the circle. However I can't figure the rest out.How did the text conclude that a) is bigger?

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Let $r$ be the radius of the circle. The long sides of the rectangle have length $2r$, and the short sides have length $r$, so the perimeter of the rectangle is $2(2r+r)=6r$. The circumference of the circle is $2\pi r$, and $2\pi>6$, so the circumference of the circle is greater than the perimeter of the rectangle.

This picture may help:

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Awesome .. that makes it so easy!!!. However I still dont get it how you assumed that the tangent of the circle is considered to be radius of the circle. The part in the question "The opposite side of rectangle is tangent to the circle" completely threw me off. Cant this be the diameter as well ??? – Rajeshwar Jul 14 '12 at 0:47
No: no diameter of a circle is tangent to the circle. – Brian M. Scott Jul 14 '12 at 0:52
The question states: "One side of rectangle is the diameter of a circle .The opposite side of rectangle is tangent to the circle" So one side of the rectangle is 2r(diameter) and the other side which is the opposite side is suppose to be the tangnet , in this case thats r which is the radius. So how can tanget be the diameter ? – Rajeshwar Jul 14 '12 at 1:55
@Rajeshwar: It isn’t just that one side of the rectangle is the same length as a diameter of the circle: one side is a diameter of the circle. A tangent to the circle is not a radius: it’s a line perpendicular to a radius of the circle that touches the circle at just one point, like the lower edge of the rectangle in my picture. It can be any length at all; in this case we know that it must be the same length as the diameter of the circle, because it must be the same length as the opposite side of the rectangle. – Brian M. Scott Jul 14 '12 at 2:17
I am still confused. Let me repeat the question along with how I am understanding it in brackets - "One side of rectangle is the diameter of a circle (This will be the bottom side of the rectangle in the picture having length 2r since thats the diameter).The opposite side of rectangle is tangent to the circle (Now opposite side would mean the other side which is tangent to the circle - which we have taken r here and I believe we were suppose to take it as 2r since its a tangent". Does my question make sense ? – Rajeshwar Jul 14 '12 at 2:33

As always, the first step is to draw a picture.

Let $r$ be the radius of the circle. Then one side of the rectangle is equal to $2r$. And of course the opposite side has the same length.

From the tangency we can see that the other two sides are each equal to $r$. (To prove this formally, let $O$ be the centre of the circle. Then the line that joins $O$ to the point $P$ of tangency has length $r$. This line is perpendicular to the long sides of the rectangle, and so has length equal to the short side of the rectangle. Thus the short sides of the rectangle have length $r$.)

So the perimeter of the rectangle is $6r$. The circumference of the circle is $2\pi r$. Since $\pi\gt 3$, the circumference is larger.

Remark: In this kind of problem, we can let the radius be $1$ (unit). That strategy can simplify formulas. In our case, there is no real gain.

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