How is your geometry skills?

Question

Two circles have equal radii and their centers are equidistant apart. Find the area of the region common to both circles.

There has been some confusion about what this problem is saying. I will restate it to make it clear. Thank you all for your input.

Restatement:

If two circles share a common radius, what is the area of the region common to both circles?

Answer 1

First off its how are your geometry skills.

They are fine though its a weak area for me.

Since the radi are equidescant and the radi are the same then the centers are also on the circumference of the other circle.

Connecting either intersection point to the two centers creates a equilateral triangle with sides r

The area of such an equilateral triangle is sqrt(3) r ^ 2 / 4 (skipping the calculation of that which is simple enough)

One can notice that there are 4 'wedges' and two triangles in the given region. The wedge is precisely the area that is the difference between 1/6th of the circle and the triangle.

Which is (pi/6 - sqrt(3)/4) r^2

We have 4 of these and 2 triangles so the area is

4 * (pi/6 - sqrt(3)/4) r^2 + 2 * sqrt(3)/4 r^2

= [2pi/3 -sqrt(3)/2] r^2

Answer 2

"Two circles have equal radii" - this means that they're the same size as each other. The radii is the measurement between the center of the circles and their outer edges.

"and their centers are equidistant apart" - they couldn't be anything else.

Equidistant means the same disrance apart. Try it with coins on a table - there's no way you can arrange 4 or more coins so they're all the same distance from each other. You could arrange 3 coins in an equilateral triangle and they'd be equidistant. With 2 coins no matter where you put them they're bound to be equidistant (the distance between coin 1 and coin 2 has to be the same as the distance between coin 2 and coin 1).

"Find the area of the region common to both circles." If the circles overlap then there will be a region common to both circles but there's nothing to indicate whether the circles overlap, are next to each other or on the other side of the world from each other.

Even if the circles do overlap and there is a region common to both of them you'd need to specify by how much they overlapped.

In short, there is no answer to your question.

Answer 3

I am not sure I understand the question. Is the question equivalent to "If two circles have equal radii and each contains the others' center, find the area of the region common to both circles."?

Let's start by drawing two circles that satisfy the above. First draw, x^2+y^2=r^2
and x^2+(y-r)^2=r^2. Observe that (0,0) is on the second circle and (0,r) is on the first.

Solving {x^2+y^2=r^2,x^2+(y-r)^2=r^2} for x and y gives us the intersection points of the two circles: x=-\sqrt{3}r/2 , y=r/2 and x=\sqrt{3}r/2 , y=r/2

Now, draw the line horizontal line y=r/2. Shade the region to the right of the y-axis, above the line y=r/2 and below the graph of the circle x^2+y^2=r^2. Notice that by symmetry, the area of the desired region is 4 times this shaded area. So, to find the area of the region common to both circles, we'll find the shaded area and multiply the result by 4.

Solving x^2+y^2=r^2 and taking the positive part gives us an equation of the top half of the circle x^2+y^2=r^2: y=\sqrt{r^2-x^2}.

Then, the area is
A=4*\int_{0}^{\sqrt{3} r/2}( \sqrt{r^2-x^2} -r/2) dx
= 2*(x \sqrt{r^2-x^2}+r^2 \arcsin(x/r) )-r x]_{0}^{\sqrt{3}r/2}
=2((\pi r^2/3-\sqrt{3}r^2/4)-0)
=r^2 (4 \pi -3 \sqrt{3})/6

Note: To find the antiderivative, we use a trig substitution (by hand) or a table of integrals or a computer algebra system.

Answer 4

My geometry skills are pretty good the last time I checked. But I can't answer your question, if I don't know HOW far apart the there centers. If the distant between the centers of the two circles is more than their radius then they will not have a common region and the area of the common region will be zero.

Answer 5

That's like no question. 2 points being equally apart is like a non statement. We have 2 equal circles. Tha is all we know. Unless equidistant refers to the radius but that is guessing and math is not guessing.

Answer 6

I'll get back to you when I figure out what you meant by "their centers are equidistant apart" (are you sure you didn't mean, "some distance apart")

Answer 7

I could answer Ur question if I knew from what the centres are equi - distant from.

Answer 8

How are your geometry skills is correct. Yours are ok but your English sucks.

Answer 9

im good at algebra....