I have a question my brother asked me but I need a math expert.

You have two interlocking circles and the radius of circle B goes through the center of circle A and of course the radius of circle A goes through the center of circle B. The radius of each circle is 30 feet. Now, if you draw a line C from the top center of circle A across to the top center of circle B, the line would be 60 feet long and would leave an area below it created by the line C and part of an arc of circle A and part of an arc of circle B. Please give me the area of that space in square feet or the formula for how it's worked. Thanks.

We will describe the problem as follows.

Two circles A,B of equal radii (r=30') are centred at P, Q, distance r apart.

Points R and S on circles A, B are such that RS form a common tangent to both circles. Hence PRSQ form a square of side r.

Arcs are drawn with centre P and Q, radius r, which intersect at point D inside the square.

Hence Δ PDQ is an equilateral triangle of side r.

The required area bounded by the side RS , arcs RD and DS will be equal to

Area of square PRSQ - Area of ΔADQ - area of sector RPD - Area of sector SQD.

Note that sectors RPD and SQD have centrai angles of (90-60)=30°.

Typo corrections:

Area of square PRSQ - Area of ΔPDQ - area of sector RPD - Area of sector SQD.

Note that sectors RPD and SQD have central angles of (90-60)=30°.

Many thanks.

You're welcome!

To find the area of the space created by line C and the arcs of circles A and B, we need to consider the difference between the areas of the two circular segments (the areas bounded by the chord and the arcs of the circles).

To visualize the situation, imagine placing the two circles side by side, such that they overlap horizontally. The length of line C would be equal to the diameter of the smaller circle (circle B) since it passes through the center of both circles. Given that the radius of each circle is 30 feet, the diameter of circle B is 60 feet.

Now, we can calculate the areas of the two circular segments and subtract them to find the desired area:

1. Calculate the area of the circular segment of circle A:
- Since line C, the chord, is horizontal, it will divide the segment symmetrically, creating two equal segments.
- The height of each segment (h) can be found using the Pythagorean theorem. In this case, h = 30 - r, where r is the height of the smaller circular segment (i.e., the radius of circle B).
- To find the angle (θ) of the circular segment, we can use the inverse cosine function. Since the width of the segment is equal to the diameter of circle B, θ = 2 * cos^(-1)(30/r).
- The area of each circular segment can be calculated using the formula: A = (r^2 / 2) * (θ - sin(θ)).

2. Multiply the area of one circular segment by 2 to account for both segments.

3. Subtract the area of the circular segment from the area of the sector formed by the circle B:
- The area of the sector is simply (πr^2) * (θ / 360), where θ is the angle of the sector.

4. Finally, subtract the area of the circular segment from the area of the sector to get the desired area.

To summarize, the formula for the area of the space created by line C and the arcs of the circles is:
Area = [2 * (r^2 / 2) * (θ - sin(θ))] + [(πr^2) * (θ / 360)] - [2 * (r^2 / 2) * (θ - sin(θ))].

Plug in the values r = 30 and θ = 2 * cos^(-1)(30/r), then calculate to find the area in square feet.