# Thread: Rectangle Inscribed in Semicircle...Part 1

1. ## Rectangle Inscribed in Semicircle...Part 1

A rectangle is inscribed in a semicircle of radius 1.

(a) Express the area A of the rectangle as a function of the angle theta.

(b) Show that A = sin(2theta)

2. Originally Posted by magentarita
A rectangle is inscribed in a semicircle of radius 1.

(a) Express the area A of the rectangle as a function of the angle theta.

(b) Show that A = sin(2theta)
um, what angle is theta?

in any case, i think it would be a good idea for you do draw this on a pair of axis. let the rectangle sit on the x-axis with its top vertices touching the upper-half circle of radius 1 centered at the origin. try to center the rectangle also. see if that gives you any ideas

3. Hello, magentarita!

A rectangle is inscribed in a semicircle of radius 1.

(a) Express the area $\displaystyle A$ of the rectangle as a function of $\displaystyle \theta.$

(b) Show that: .$\displaystyle A \:=\: \sin(2\theta)$
Code:
              * * *
*     |     *
*-------+-------*
*|       |     * |*
|       |  1*   |y
* |       | * θ   | *
*-+-------+-------+-*
x

The area of the rectangle is: .$\displaystyle A \:=\:2xy$

We have: .$\displaystyle x \:=\:\cos\theta,\;y\:=\:\sin\theta$

Therefore: .$\displaystyle A \;=\;2\sin\theta\cos\theta \;=\;\sin2\theta$

4. Originally Posted by Soroban
Hello, magentarita!

Code:
              * * *
*     |     *
*-------+-------*
*|       |     * |*
|       |  1*   |y
* |       | * θ   | *
*-+-------+-------+-*
x
The area of the rectangle is: .$\displaystyle A \:=\:2xy$

We have: .$\displaystyle x \:=\:\cos\theta,\;y\:=\:\sin\theta$

Therefore: .$\displaystyle A \;=\;2\sin\theta\cos\theta \;=\;\sin2\theta$
Good job Soroban. Nice try Jhevon.

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### a rectangle is to be inscribed in a semicircle of radius 2

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