Really majestic
Q.What is the area of a ring (an annulus), the region between two concentric circles? The line segment is tangent to the inner circle and has a length of 8.
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l
Answer:
The solutions will be slightly easier to see if we first rotate the shapes so the tangent line is horizontal.
Next draw the inner circle’s radius, with length b, vertically so it makes a right angle and bisects the tangent line (a chord of the outer circle) into two segments of length 4 = a. Finally draw the outer circle’s radius, with length c, to connect to the endpoint of the line segment, forming a hypotenuse of a right triangle.
Now we can solve the problem in several different ways.
Method 1: Make up numbers!
The area between the two circles is equal to the area of the outer circle minus the area of the inner circle. The radius of the outer circle is the hypotenuse of a triangle with one leg 4 and another leg equal to the radius of the inner circle.
Imagine this problem appeared on a multiple choice test or a math competition. If the answer is known to be a single number, then the answer must not depend on the size of the radii of the two circles. In that case, you might as well make up values for a specific case.
We need a right triangle with one leg equal to 4, so we might as well imagine it is a 3-4-5 right triangle. Then we get the area between the circles is:
Ï€(outer radius)2 – Ï€(inner radius)2
= Ï€(52) – Ï€(32)
= 16Ï€
And that’s it! If your only goal was to solve this problem quickly, you can correct answer and move on.
But why exactly does the area only depend on the length of the tangent line segment? Let’s do a little more investigation and discover a more general formula
Method 2: solve the problem generally
Let’s consider the diagram with the right triangle again.
The area of the region between the two circles can be written in terms of the radii of the two circles:
Ï€(outer radius)2 – Ï€(inner radius)2
= Ï€c2 – Ï€b2
= Ï€(c2 – b2)
Now we can use the fact that the two radii are part of a right triangle. By the Pythagorean theorem, we have:
b2 + 42 = c2
We can then re-arrange the equation to find:
42 = c2 – b2
Thus we can solve the area of the shaded region:
Ï€(c2 – b2)
= π(42)
= 16Ï€
Now we have proven the answer is true. Furthermore, if half of the tangent line segment has length a, then by the Pythagorean theorem, we would have:
a2 = c2 – b2
Thus, the area between the two circles is then:
Ï€(c2 – b2)
= a2Ï€
The area only depends on the length of the tangent line segment!
Method 3: Let the inner circle radius go to zero
In the first method we made up values for the radii of the circles using the 3-4-5 right triangle. But what if we shrunk the inner circle until its radius length was 0? In that case, the diagram shrinks to be only the outer circle, and that circle would have a radius equal to 4 because the tangent line would become the radius.
Thus, the area “between” the circles is 42Ï€ = 16Ï€.
This is the same answer!
Q.What is the area of a ring (an annulus), the region between two concentric circles? The line segment is tangent to the inner circle and has a length of 8.
.
.
..
l
Answer:
The solutions will be slightly easier to see if we first rotate the shapes so the tangent line is horizontal.
Next draw the inner circle’s radius, with length b, vertically so it makes a right angle and bisects the tangent line (a chord of the outer circle) into two segments of length 4 = a. Finally draw the outer circle’s radius, with length c, to connect to the endpoint of the line segment, forming a hypotenuse of a right triangle.
Now we can solve the problem in several different ways.
Method 1: Make up numbers!
The area between the two circles is equal to the area of the outer circle minus the area of the inner circle. The radius of the outer circle is the hypotenuse of a triangle with one leg 4 and another leg equal to the radius of the inner circle.
Imagine this problem appeared on a multiple choice test or a math competition. If the answer is known to be a single number, then the answer must not depend on the size of the radii of the two circles. In that case, you might as well make up values for a specific case.
We need a right triangle with one leg equal to 4, so we might as well imagine it is a 3-4-5 right triangle. Then we get the area between the circles is:
Ï€(outer radius)2 – Ï€(inner radius)2
= Ï€(52) – Ï€(32)
= 16Ï€
And that’s it! If your only goal was to solve this problem quickly, you can correct answer and move on.
But why exactly does the area only depend on the length of the tangent line segment? Let’s do a little more investigation and discover a more general formula
Method 2: solve the problem generally
Let’s consider the diagram with the right triangle again.
The area of the region between the two circles can be written in terms of the radii of the two circles:
Ï€(outer radius)2 – Ï€(inner radius)2
= Ï€c2 – Ï€b2
= Ï€(c2 – b2)
Now we can use the fact that the two radii are part of a right triangle. By the Pythagorean theorem, we have:
b2 + 42 = c2
We can then re-arrange the equation to find:
42 = c2 – b2
Thus we can solve the area of the shaded region:
Ï€(c2 – b2)
= π(42)
= 16Ï€
Now we have proven the answer is true. Furthermore, if half of the tangent line segment has length a, then by the Pythagorean theorem, we would have:
a2 = c2 – b2
Thus, the area between the two circles is then:
Ï€(c2 – b2)
= a2Ï€
The area only depends on the length of the tangent line segment!
Method 3: Let the inner circle radius go to zero
In the first method we made up values for the radii of the circles using the 3-4-5 right triangle. But what if we shrunk the inner circle until its radius length was 0? In that case, the diagram shrinks to be only the outer circle, and that circle would have a radius equal to 4 because the tangent line would become the radius.
Thus, the area “between” the circles is 42Ï€ = 16Ï€.
This is the same answer!
Visual calculus
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Reviewed by biharishayar
on
September 16, 2017
Rating:
Nice question
ReplyDeletegreat sir
ReplyDeleteGoods Sir..teaching
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