Question

Find the ratio of the area inside the square but outside the circle to the area of the square

Answer 1

The ratio of the area inside the square but outside the circle to the area of the square is about 0.2146

Further explanation

The basic formula that need to be recalled is:

Circular Area = π x R²

Circle Circumference = 2 x π x R

where:

R = radius of circle

The area of sector:

$\text{Area of Sector} = \frac{\text{Central Angle}}{2 \pi} \times \text{Area of Circle}$

The length of arc:

$\text{Length of Arc} = \frac{\text{Central Angle}}{2 \pi} \times \text{Circumference of Circle}$

Let us now tackle the problem!

This problem is about calculating area of square and circle.

Let me assume that the diagram of the problem is as in the attachment.

Let : radius of the circle = R

$\text{Area of Square} = (\text{length of one side})^2$

$\text{Area of Square} = (2R)^2$

$\text{Area of Square} = A_1 = 4R^2$

$\text{Area of Circle} = \pi \times (\text{radius})^2$

$\text{Area of Square} = \pi (R)^2$

$\text{Area inside the square but outside the circle } = \text{Area of Square} - \text{Area of Circle}$

$\text{Area inside the square but outside the circle } = 4R^2 - \pi R^2$

$\text{Area inside the square but outside the circle } = A_2 = (4 - \pi)R^2$

The ratio of the area inside the square but outside the circle to the area of the square:

$A_2 : A_1 = (4 - \pi)R^2 : 4R^2$

$A_2 : A_1 = (4 - \pi) : 4$

$A_2 : A_1 = 1 - \frac{1}{4}\pi$

$A_2 : A_1 \approx 0.2146$

Learn more

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Answer details

Grade: College

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse, Circle , Arc , Sector , Area , Radian , Degree , Unit , Conversion

Answer 2

Area of the square:  A s = s².
Area of the circle: A c = ( 1/2 s )² π = 1/4 s² · 3.14 = 0.785 s²
Area inside the square but outside the circle: s² - 0.785 s² = 0.215 s²
The ratio:
0.215 s² : s² = 0.215 : 1 = 215 : 1000 = 43 : 200 .