A circle is inscribed in a square. The number of square units in the area and the number of units in the perimeter of the square are the same. What is the area of the circle in terms of pi?
Explain your thought process.
I completely don't understand this question. Can you help me? Thank you! :)
I have to be honest, this makes little sense to me also.
see
http://www.jiskha.com/display.cgi?id=1258421966
Sure! Let's break down the question step by step.
First, the question states that a circle is inscribed in a square. This means that the circle is perfectly enclosed within the square, and its circumference (perimeter) touches all four sides of the square.
Next, the question tells us that the area of the square is equal to its perimeter. In other words, the number of square units in the area of the square is equal to the number of units in its perimeter.
To solve this problem, we need to understand the formulas for the area and the perimeter of both a square and a circle.
The area (A) of a square is given by the formula:
A = side^2
The perimeter (P) of a square is given by the formula:
P = 4 * side
The circumference (C) of a circle is given by the formula:
C = 2 * π * radius
The area (A) of a circle is given by the formula:
A = π * radius^2
Now, let's assume that the side of the square is "s".
According to the problem statement, the area of the square is equal to the perimeter:
s^2 = 4s
To solve this equation, we can subtract 4s from both sides:
s^2 - 4s = 0
Now, we can factor out an "s" from the left side:
s(s - 4) = 0
From this equation, we can see that we have two possibilities:
1. s = 0
This means that the side of the square is zero, which is not possible.
2. s - 4 = 0
This means that the side of the square is 4 units.
Since we have determined the side of the square, we can now find the area of the square by substituting the value of "s" into the area formula:
A = side^2 = 4^2 = 16 square units
Finally, we need to find the area of the circle. Since the circle is inscribed in the square, its diameter is equal to the side of the square. Therefore, the radius of the circle is half of the side length of the square, which is 2 units.
Using the area formula for a circle, we can determine the area of the circle:
A = π * radius^2 = π * (2^2) = 4π square units
So, the area of the circle is 4π square units.