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==Problem 15==
==Problem==
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of <math>1</math> square unit, then what is the area of the shaded region, in square units?
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of <math>1</math> square unit, then what is the area of the shaded region, in square units?


Line 15: Line 15:
==Solution 1==
==Solution 1==


Let the radius of the large circle be <math>R</math>. Then the radii of the smaller circles are <math>\frac R2</math>. The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is <math>\frac 14</math>. This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is <math>\boxed{\textbf{(D) } 1}</math>
Let the radius of the large circle be <math>R</math>. Then, the radius of the smaller circles are <math>\frac R2</math>. The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is <math>\frac 14</math> (<math>\frac {R^2}{4}</math> is <math>\frac 14</math> of <math>R^2</math>.) This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is <math>\boxed{\textbf{(D) } 1}</math>.


==Solution 2==
==Solution 2==


Let the radius of the two smaller circles be <math>r</math>. It follows that the area of one of the
Let the radius of the two smaller circles be <math>r</math>. It follows that the area of one of the smaller circles is <math>{\pi}r^2</math>. Thus, the area of the two inner circles combined would evaluate to <math>2{\pi}r^2</math> which is <math>1</math>. Since the radius of the bigger circle is two times that of the smaller circles (the diameter), the radius of the larger circle in terms of <math>r</math> would be <math>2r</math>. The area of the larger circle would come to <math>(2r)^2{\pi} = 4{\pi}r^2</math>.


smaller circles is <math>{\pi}r^2</math>. Thus, the area of the two inner circles combined would
Subtracting the area of the smaller circles from that of the larger circle (since that would be the shaded region), we have <cmath>4{\pi}r^2 - 2{\pi}r^2 = 2{\pi}r^2 = 1.</cmath>


evaluate to <math>2{\pi}r^2</math> which is <math>1</math>. Since the radius of the bigger circle is two times
Therefore, the area of the shaded region is <math>\boxed{\textbf{(D) } 1}</math>.


that of the smaller circles(the diameter), the radius of the larger circle in terms of <math>r</math>  
==Solution 3==
The area of a small circle is <math>\frac{1}{2}= \pi r^2</math>. Solving, we get <math>r = \sqrt{\frac{1}{2\pi}}</math>.


would be <math>2r</math>. The area of the larger circle would come to <math>(2r)^2{\pi} = 4{\pi}r^2</math>.  
The radius of the large circle is <math>R=2r</math>. The area of the large circle is <math>{\pi}R^2={\pi}(2r)^2=4{\pi}r^2=4{\pi}\frac{1}{2\pi}=2</math>.


Subtracting the area of the smaller circles from that of the larger circle(since that would
Subtract the area of the small circles from the area of the large circle to get the area of the shaded region: <math>2-1=</math> <math>\boxed{\textbf{(D) } 1}</math>.


be the shaded region), we have <cmath>4{\pi}r^2 - 2{\pi}r^2 = 2{\pi}r^2 = 1.</cmath>
==Solution 4 (Similar to Solution 3)==
To get the area of the small circles, we can get the equation <math>\frac{1}{2}= \pi r^2</math>. Solving for <math>r</math>, we get <math>r = \sqrt{\frac{1}{2\pi}}</math>. Then, we can get the radius of the big circle by doubling the small circle's radius, and that gives
<math>2\sqrt{\frac{1}{2\pi}}</math>. Because the area of a circle is <math>\pi r^2</math>, you'll have to square the answer and multiple it by <math>\pi</math>. After you square it, you'll get <math>4 \cdot \frac{1}{2\pi}</math>, which equals to <math>\frac{4}{2\pi}</math>. Since the area of a circle is <math>\pi r^2</math>, we get <math>\pi \cdot \frac{4}{2\pi}</math>, and that equals <math>\frac{4\pi}{2\pi}</math>, which equals 2. Since each of the circles' area is <math>\frac{1}{2}</math>, the combined area of the small circles is <math>1</math>. Since <math>2-1=1</math>, the area of the shaded region is <math>\boxed{\textbf{(D) } 1}</math>.
 
== Solution 5 ==
Since the smaller circles are equivalent, the area of each of them is <math>\frac{1}{2}</math> square unit. Since the diameter of each of the two smaller circles is a radius of the larger circle, their areas must be in a ratio of <math>1:2</math>. This tells us that the area of the larger circle is double the area of the smaller circle, giving us <math>\frac{1}{2} \cdot 2 = \boxed{\textbf{(D) } 1}</math>.
 
~ [[User:Aoum|aoum]]
 
==Video Solution (CREATIVE ANALYSIS!!!)==
https://youtu.be/tYfMj2SSVJc
 
~Education, the Study of Everything
 
==Video Solutions ==
https://youtu.be/-3WEf3EjGu0
 
https://youtu.be/-JR7R0PyU-w
 
~savannahsolver
 
==Video Solution by OmegaLearn==
https://youtu.be/51K3uCzntWs?t=1474
 
~ pi_is_3.14


Therefore, the area of the shaded region is <math>\boxed{\textbf{(D) } 1}</math>


==See Also==
==See Also==
Line 39: Line 63:


{{MAA Notice}}
{{MAA Notice}}
[[Category:Introductory Geometry Problems]]

Latest revision as of 11:41, 7 August 2025

Problem

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

[asy] size(4cm); filldraw(scale(2)*unitcircle,gray,black); filldraw(shift(-1,0)*unitcircle,white,black); filldraw(shift(1,0)*unitcircle,white,black); [/asy]


$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$

Solution 1

Let the radius of the large circle be $R$. Then, the radius of the smaller circles are $\frac R2$. The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is $\frac 14$ ($\frac {R^2}{4}$ is $\frac 14$ of $R^2$.) This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is $\boxed{\textbf{(D) } 1}$.

Solution 2

Let the radius of the two smaller circles be $r$. It follows that the area of one of the smaller circles is ${\pi}r^2$. Thus, the area of the two inner circles combined would evaluate to $2{\pi}r^2$ which is $1$. Since the radius of the bigger circle is two times that of the smaller circles (the diameter), the radius of the larger circle in terms of $r$ would be $2r$. The area of the larger circle would come to $(2r)^2{\pi} = 4{\pi}r^2$.

Subtracting the area of the smaller circles from that of the larger circle (since that would be the shaded region), we have \[4{\pi}r^2 - 2{\pi}r^2 = 2{\pi}r^2 = 1.\]

Therefore, the area of the shaded region is $\boxed{\textbf{(D) } 1}$.

Solution 3

The area of a small circle is $\frac{1}{2}= \pi r^2$. Solving, we get $r = \sqrt{\frac{1}{2\pi}}$.

The radius of the large circle is $R=2r$. The area of the large circle is ${\pi}R^2={\pi}(2r)^2=4{\pi}r^2=4{\pi}\frac{1}{2\pi}=2$.

Subtract the area of the small circles from the area of the large circle to get the area of the shaded region: $2-1=$ $\boxed{\textbf{(D) } 1}$.

Solution 4 (Similar to Solution 3)

To get the area of the small circles, we can get the equation $\frac{1}{2}= \pi r^2$. Solving for $r$, we get $r = \sqrt{\frac{1}{2\pi}}$. Then, we can get the radius of the big circle by doubling the small circle's radius, and that gives $2\sqrt{\frac{1}{2\pi}}$. Because the area of a circle is $\pi r^2$, you'll have to square the answer and multiple it by $\pi$. After you square it, you'll get $4 \cdot \frac{1}{2\pi}$, which equals to $\frac{4}{2\pi}$. Since the area of a circle is $\pi r^2$, we get $\pi \cdot \frac{4}{2\pi}$, and that equals $\frac{4\pi}{2\pi}$, which equals 2. Since each of the circles' area is $\frac{1}{2}$, the combined area of the small circles is $1$. Since $2-1=1$, the area of the shaded region is $\boxed{\textbf{(D) } 1}$.

Solution 5

Since the smaller circles are equivalent, the area of each of them is $\frac{1}{2}$ square unit. Since the diameter of each of the two smaller circles is a radius of the larger circle, their areas must be in a ratio of $1:2$. This tells us that the area of the larger circle is double the area of the smaller circle, giving us $\frac{1}{2} \cdot 2 = \boxed{\textbf{(D) } 1}$.

~ aoum

Video Solution (CREATIVE ANALYSIS!!!)

https://youtu.be/tYfMj2SSVJc

~Education, the Study of Everything

Video Solutions

https://youtu.be/-3WEf3EjGu0

https://youtu.be/-JR7R0PyU-w

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/51K3uCzntWs?t=1474

~ pi_is_3.14


See Also

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

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