In a certain circle, the length of a radius is equal to the reciprocal of the length of a diameter. What is the area of this circle?
r = 1/d
r = 1/(2r)
r^2 = 1/2
pi r^2 = area = pi/2
The length of a radius is equal to half the length of a diameter in a circle. In this case, the length of the radius is equal to the reciprocal of the length of a diameter. Let's represent the length of the diameter as d. Therefore, the length of the radius can be written as 1/d.
According to the given information, we can write the equation as follows:
1/d = d/2
To solve for d, we can cross-multiply:
2 = d^2
Next, we can take the square root of both sides to find the value of d:
√2 = d
Now that we know the value of d, we can calculate the area of the circle using the formula: A = πr^2. Since the radius is half the diameter, the radius is √2/2.
Plugging the value of the radius into the area formula:
A = π(√2/2)^2
= π(2/4)
= π/2
Therefore, the area of the circle is π/2 square units.
To find the area of a circle, we can use the formula A = πr^2, where A represents the area and r represents the radius of the circle.
In this case, we are given that the length of the radius is equal to the reciprocal of the length of the diameter. Let's call the length of the diameter "d."
If the length of the radius is equal to the reciprocal of the length of the diameter, we can express it as follows: r = 1 / d.
Now we are given the relationship between the radius and diameter. In a circle, the radius is half the length of the diameter. Therefore, we can write an equation as follows: r = d / 2.
Substituting the given value of the radius into our equation, we have d / 2 = 1 / d.
To solve this equation, we can multiply both sides by 2d to eliminate the denominators: d^2 = 2.
Taking the square root of both sides, we get d = √2.
Now, we can find the radius by dividing the diameter by 2: r = √2 / 2.
Finally, substituting the value of the radius into the area formula, we get A = π(√2/2)^2.
Calculating the area, we have A = π * (√2/2) * (√2/2) = π * (2/4) = π/2.
Therefore, the area of the circle is π/2.