What is the ratio of the area of the circle to the area of the rectangle ?

Question

Prepp Team · Accepted Answer

Understanding the Problem: Ratio of Circle Area to Inscribed Rectangle Area

The problem asks for the ratio of the area of a circle to the area of a rectangle inscribed within it. We are given the radius of the circle, and two angles related to the figure: $\angle DAE = 30^\circ$ and $\angle ACD = 30^\circ$. The rectangle is ABCD.

Analyzing the Geometry

When a rectangle is inscribed in a circle, its diagonals are diameters of the circle. In rectangle ABCD, the diagonal AC is a diameter of the circle. We are given that the radius of the circle is $r$. Therefore, the diameter AC is $2r$.

In a rectangle, all angles are $90^\circ$. So, $\angle ADC = 90^\circ$. Triangle ADC is a right-angled triangle with the hypotenuse AC.

We are given that $\angle ACD = 30^\circ$. In the right-angled triangle ADC, we can use trigonometry to find the lengths of the sides AD and CD (which are the width and length of the rectangle).

Calculating the Dimensions of the Rectangle

In right-angled triangle ADC:

The side AD is opposite to angle $\angle ACD$.
The side CD is adjacent to angle $\angle ACD$.
The hypotenuse is AC.

Using trigonometric ratios:

Sine of $\angle ACD$: $ \sin(\angle ACD) = \frac{AD}{AC} $
Cosine of $\angle ACD$: $ \cos(\angle ACD) = \frac{CD}{AC} $

We have $\angle ACD = 30^\circ$ and $AC = 2r$.

Let's calculate the length of AD:

\[ AD = AC \times \sin(30^\circ) \]

We know that $\sin(30^\circ) = \frac{1}{2}$.

\[ AD = 2r \times \frac{1}{2} = r \]

So, the width of the rectangle AD is $r$.

Now, let's calculate the length of CD:

\[ CD = AC \times \cos(30^\circ) \]

We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.

\[ CD = 2r \times \frac{\sqrt{3}}{2} = r\sqrt{3} \]

So, the length of the rectangle CD is $r\sqrt{3}$.

The sides of the rectangle are $r$ and $r\sqrt{3}$.

Note: The angle $\angle DAE = 30^\circ$ is not needed to determine the dimensions of the rectangle from the given information about the inscribed rectangle and $\angle ACD$.

Calculating the Area of the Rectangle and the Circle

The area of the rectangle ABCD is given by:

\[ \text{Area of Rectangle} = \text{Length} \times \text{Width} = CD \times AD \] \[ \text{Area of Rectangle} = (r\sqrt{3}) \times r = r^2\sqrt{3} \]

The area of the circle with radius $r$ is given by:

\[ \text{Area of Circle} = \pi r^2 \]

Calculating the Ratio of Areas

We need to find the ratio of the area of the circle to the area of the rectangle:

\[ \text{Ratio} = \frac{\text{Area of Circle}}{\text{Area of Rectangle}} \] \[ \text{Ratio} = \frac{\pi r^2}{r^2\sqrt{3}} \]

We can cancel out the $r^2$ terms:

\[ \text{Ratio} = \frac{\pi}{\sqrt{3}} \]

Summary of Calculations

Item	Formula/Value	Calculated Value
Circle Radius	$r$	$r$
Circle Diameter (AC)	$2 \times \text{Radius}$	$2r$
Angle $\angle ACD$	Given	$30^\circ$
Rectangle Width (AD)	$AC \sin(30^\circ)$	$2r \times \frac{1}{2} = r$
Rectangle Length (CD)	$AC \cos(30^\circ)$	$2r \times \frac{\sqrt{3}}{2} = r\sqrt{3}$
Area of Rectangle	$AD \times CD$	$r \times r\sqrt{3} = r^2\sqrt{3}$
Area of Circle	$\pi r^2$	$\pi r^2$
Ratio (Circle Area / Rectangle Area)	$\frac{\pi r^2}{r^2\sqrt{3}}$	$\frac{\pi}{\sqrt{3}}$

The ratio of the area of the circle to the area of the rectangle is $\frac{\pi}{\sqrt{3}}$.

Revision Table: Key Concepts

Concept	Description	Relevance to Problem
Inscribed Rectangle	A rectangle drawn inside a circle such that all its vertices lie on the circle's circumference.	Its diagonal is the circle's diameter.
Circle Area	The space occupied by a circle, calculated as $\pi r^2$.	Required for the numerator of the ratio.
Rectangle Area	The space occupied by a rectangle, calculated as Length $\times$ Width.	Required for the denominator of the ratio.
Trigonometry (Sine and Cosine)	Used to relate angles and side lengths in right-angled triangles.	Used to find rectangle dimensions from diagonal and angle.

Additional Information: Inscribed Shapes

Inscribed shapes are common in geometry problems. Knowing the properties of how certain shapes relate to the circle when inscribed is crucial.

For an inscribed rectangle, the diagonal is always the diameter.
For an inscribed right-angled triangle, the hypotenuse is always the diameter.
For an inscribed equilateral triangle, there are specific relationships between its side length and the circle's radius.
For an inscribed square, its diagonal is the circle's diameter, and its side length is $r\sqrt{2}$ (where $r$ is the radius).

These properties help in finding the dimensions of the inscribed shape based on the circle's properties, or vice versa, which is often the first step in solving such problems.

Prepp Team · Answer

$\frac{\pi}{\sqrt 2}$

Prepp Team · Answer

$\frac{2\pi}{\sqrt 3}$

Prepp Team · Answer

$\frac{3\pi}{\sqrt 2}$

Item	Formula/Value	Calculated Value
Circle Radius	\(r\)	\(r\)
Circle Diameter (AC)	\(2 \times \text{Radius}\)	\(2r\)
Angle \(\angle ACD\)	Given	\(30^\circ\)
Rectangle Width (AD)	\(AC \sin(30^\circ)\)	\(2r \times \frac{1}{2} = r\)
Rectangle Length (CD)	\(AC \cos(30^\circ)\)	\(2r \times \frac{\sqrt{3}}{2} = r\sqrt{3}\)
Area of Rectangle	\(AD \times CD\)	\(r \times r\sqrt{3} = r^2\sqrt{3}\)
Area of Circle	\(\pi r^2\)	\(\pi r^2\)
Ratio (Circle Area / Rectangle Area)	\(\frac{\pi r^2}{r^2\sqrt{3}}\)	\(\frac{\pi}{\sqrt{3}}\)

Understanding the Problem: Ratio of Circle Area to Inscribed Rectangle Area

Analyzing the Geometry

Calculating the Dimensions of the Rectangle

Calculating the Area of the Rectangle and the Circle

Calculating the Ratio of Areas

Summary of Calculations

Revision Table: Key Concepts

Additional Information: Inscribed Shapes

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