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:
[tex]\text{Area of Sector} = \frac{\text{Central Angle}}{2 \pi} \times \text{Area of Circle}[/tex]
The length of arc:
[tex]\text{Length of Arc} = \frac{\text{Central Angle}}{2 \pi} \times \text{Circumference of Circle}[/tex]
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
[tex]\text{Area of Square} = (\text{length of one side})^2[/tex]
[tex]\text{Area of Square} = (2R)^2[/tex]
[tex]\text{Area of Square} = A_1 = 4R^2[/tex]
[tex]\text{Area of Circle} = \pi \times (\text{radius})^2[/tex]
[tex]\text{Area of Square} = \pi (R)^2[/tex]
[tex]\text{Area inside the square but outside the circle } = \text{Area of Square} - \text{Area of Circle}[/tex]
[tex]\text{Area inside the square but outside the circle } = 4R^2 - \pi R^2[/tex]
[tex]\text{Area inside the square but outside the circle } = A_2 = (4 - \pi)R^2[/tex]
The ratio of the area inside the square but outside the circle to the area of the square:
[tex]A_2 : A_1 = (4 - \pi)R^2 : 4R^2[/tex]
[tex]A_2 : A_1 = (4 - \pi) : 4[/tex]
[tex]A_2 : A_1 = 1 - \frac{1}{4}\pi[/tex]
[tex]A_2 : A_1 \approx 0.2146[/tex]
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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
