Three identical circles are tangent to each other externally. if the area of the curvilinear formed by the point of tangency of the three circles is 142 cm2, compute the radius of each circle.
Let the radius of each of the equal circles be r
Joining the 3 centres will produce an equilateral triangle
The area we are given as 142 cm^2 would be
the area of that equilateral triangle - the area of the 3 sectors of
the circle contained within that triangle, so .....
area of triangle = (1/2)(2r)(2r) sin 60° = 2r^2(√3/2) = √3 r^2
the area of one of the sectors = (1/6)π r^2
so we have 3 of these , area of sectors = (1/2)π r^2
√3 r^2 - (1/2)π r^2 = 142
r^2 (√3 - π/2) = 142
you take over, let me know what you get
Why did the circle go on a diet? It wanted to be a little bit more tangential!
Now, to answer your question: Let's call the radius of each circle "r". Since the circles are externally tangent to each other, the distance between their centers is equal to the sum of their radii.
We can draw a triangle connecting the centers of the circles. This triangle is an equilateral triangle, as all three sides are equal to 2r (the radius of one circle) and all angles are 60 degrees.
Now, let's find the area of the curvilinear formed by the point of tangency. This is the area of a sector minus the area of an equilateral triangle. The sector has a central angle of 120 degrees (360 degrees divided by 3).
The area of the sector is given by (120/360) * πr^2 = (1/3) * πr^2.
The area of the equilateral triangle is (√3/4) * (2r)^2 = √3 * r^2.
So, the area of the curvilinear is [(1/3) * πr^2] - [√3 * r^2].
We are given that this area is 142 cm^2. Setting up the equation:
[(1/3) * πr^2] - [√3 * r^2] = 142.
Simplifying, we get [(π/3) - √3] * r^2 = 142.
Therefore, the radius of each circle is given by r = √(142 / [(π/3) - √3]).
Plug in the value of π into the equation and you'll get your answer. But remember, it might be irrational!
To solve this problem, we can use the fact that the curvilinear area formed by the point of tangency is simply the sum of three circular segments.
Let's denote the radius of each circle as "r".
First, we need to find the area of each circular segment.
The formula to calculate the area of a circular segment is:
A = (θ/360) * π * r^2 - 0.5 * r^2 * sin(θ),
where θ is the central angle of the segment.
Since the circles are identical and tangent to each other externally, we can see that the central angle of each segment is 120 degrees (360 degrees divided by 3).
Using this information, we can calculate the area of each segment:
A_segment = (120/360) * π * r^2 - 0.5 * r^2 * sin(120)
= (1/3) * π * r^2 - 0.5 * r^2 * √3/2
= (π/3 - (r^2 * √3)/4) * r^2.
Since there are three identical segments, the total curvilinear area is:
Total_area = 3 * A_segment
= 3 * (π/3 - (r^2 * √3)/4) * r^2
= π * r^2 - (3 * r^2 * √3)/4.
According to the problem, the total curvilinear area is 142 cm^2. Thus, we have the equation:
π * r^2 - (3 * r^2 * √3)/4 = 142.
Now, let's solve this equation to find the value of "r".
π * r^2 - (3 * r^2 * √3)/4 = 142.
Multiplying both sides by 4 to eliminate the fraction:
4 * π * r^2 - 3 * r^2 * √3 = 568.
Rearranging the terms:
4 * π * r^2 = 3 * r^2 * √3 + 568.
Dividing both sides by r^2 to isolate the variable:
4 * π = 3 * √3 + 568/r^2.
Now, we can solve for "r".
Subtracting 3 * √3 from both sides:
4 * π - 3 * √3 = 568/r^2.
Multiplying both sides by r^2:
r^2 * (4 * π - 3 * √3) = 568.
Finally, we can solve for "r" by taking the square root of both sides:
r = √(568 / (4 * π - 3 * √3)).
Using a calculator, we can calculate the value of "r".
r ≈ 5.8 cm.
Therefore, the radius of each circle is approximately 5.8 cm.
To solve this problem, we can use the properties of circles and their tangents. Here's how you can approach it:
Step 1: Draw a diagram
Start by drawing the three identical circles tangent to each other externally. Label the centers of the circles as A, B, and C, and let the radius of each circle be r.
Step 2: Identify the curvilinear area
The curvilinear area is formed by the point of tangency between the circles. In our diagram, it is the region enclosed by the three tangents, as shown:
_____
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_|___._|_____
| | | |
| | | |
|_____|___|__|
Step 3: Calculate the area of the curvilinear
The curvilinear area can be divided into three sectors, each corresponding to one circle. The area of a circle sector can be found using the formula:
Area = (θ/360) * π * r^2
Since the three circles are identical, each sector has the same angle θ. Therefore, the area of each sector is the same.
Let's assume the angle θ is equal to α. Then, the area of each sector is:
Area_sector = (α/360) * π * r^2
Since we have three identical sectors, the total area of the curvilinear can be expressed as:
Total_area = 3 * Area_sector
= 3 * (α/360) * π * r^2
Given that the total area is 142 cm^2, we can now set up the equation:
142 = 3 * (α/360) * π * r^2
Step 4: Solve for the radius
To find the radius, we need to solve the equation for r. Rearranging the equation, we have:
r^2 = (142 * 360) / (3 * π * α)
Given that π is approximately 3.14159, we can substitute the values and solve for r.
Step 5: Calculate the radius
Finally, calculate the radius using the equation from Step 4:
r = sqrt((142 * 360) / (3 * π * α))
When you have the value of α, substitute it into the equation to find the radius.