Finding Common Area of Two Circles

In summary, the problem involves finding the area common to two circles with radii 12 cm and 10 cm, whose centers are 14 cm apart. This can be done by decomposing the problem into finding the central angles using the cosine law and then using the formula for the area of a sector. A sketch of the intersecting circles and a triangle with sides 12, 10, and 14 cm can help in determining the central angles.
  • #1
z.js
57
0

Homework Statement


Two circles with radii 12 cm and 10 cm respectively have their centers 14 cm apart, find the area common to both circles. (note : This is in radians.)

Homework Equations


Area of a sector = 0.5r2θ - 0.5r2sin θ

The Attempt at a Solution


None. :confused:
 
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  • #2
Just decompose the problem. The formula you have in 2 is the area of a SEGMENT when the central angle is θ. Split the common area as the sum of segments of cycles. So the only thing remaining is to find the central angles. This is to be done using the radius and the distance between the two cycles using the cosine law. These for you to get started. I'll make now a figure to show you exactly what i mean. Sorry but this is my first post ;)
 
  • #3
z.js said:
Two circles with radii 12 cm and 10 cm respectively have their centers 14 cm apart, find the area common to both circles. (note : This is in radians.)
Hi z.js,

ditto to LeonhardEu :smile:

I'm sure you are not trying to do this without drawing a large diagram. But it's difficult to give hints when you haven't provided your sketch. On the intersecting circles diagram, you can draw in a triangle of sides 12, 10 and 14 cm. First step, determine two of its angles (yes, in radians).
 
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FAQ: Finding Common Area of Two Circles

1. How do you find the common area of two circles?

The common area of two circles can be found by using the formula A = πr2 - (πd2)/4, where r is the radius of the circles and d is the distance between their centers.

2. Can the common area of two circles be negative?

No, the common area of two circles cannot be negative. It represents the overlapping area between the two circles and is always a positive value.

3. What is the significance of finding the common area of two circles?

Finding the common area of two circles is useful in various real-life applications, such as calculating the overlapping area between two objects or determining the amount of space needed for a new construction project.

4. Is there a different formula for finding the common area of two circles if they have different radii?

Yes, if the two circles have different radii, the formula for finding the common area will be A = πr12 - πr22 + (πd2)/4, where r1 and r2 are the radii of the circles and d is the distance between their centers.

5. Can the common area of two circles be larger than the area of one of the circles?

Yes, it is possible for the common area of two circles to be larger than the area of one of the circles if they have a large overlapping area. This can occur when the two circles have a significant distance between their centers and/or similar radii.

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