Ambiguous Shaded Region

Geometry Level 2

The figure shows two semicircles whose diameters lie on a common line. The chord $DE$ of the larger circle is parallel to the diameter, tangent to the smaller circle, and has length $12$ .

If the purple area is $A \pi$ , what is $A$ ?

The answer is 18.

2 solutions

Ahmad Saad
Feb 17, 2016

From the diagram, $R^2 - r^2 = 6^2.$ Thus, the area of the purple shaded region $\begin{aligned} &= \frac12 \pi R^2 - \frac12 \pi r^2 \\ &= \frac12 \pi (R^2 - r^2) \\ &= \frac12 \pi \cdot 36 \\ &= 18 \pi. \end{aligned}$

Therefore, $A = 18.$

We don't know if these are semicircles. We don't know if the diameters (assuming they are semicircles) are collinear. So we assume also a point of tangency between the 2? Is this point at the endpoint of the assumed diameters?

The problem statment needs to be more clearly written.

Bradford Thompson - 5 years, 3 months ago

Thanks. I've clarified that these are semicircles whose diameters lie on a common line.

Note that we do not need a point of tangency, but just that one semicircle is contained within the other. The area of the shaded region doesn't change as we move the smaller semicircle about.

Calvin Lin Staff - 5 years, 3 months ago

DE should be parallel to the diameter? Or isn't that necessary?

Siva Bathula - 5 years, 3 months ago

@Siva Bathula – Yes that's needed too. Thanks!

Calvin Lin Staff - 5 years, 3 months ago

Awesome, thanks for fixing it!

Bradford Thompson - 5 years, 3 months ago

@Bradford Thompson – Np. Thanks for alerting us to the ambiguity :) That's how Brilliant gets great for everyone.
Siva's comment about "parallel line" is indeed important.

Calvin Lin Staff - 5 years, 3 months ago

Since 8 is possibly guessed with $DE = 8$ , I've edited the length of DE to be 12.

Calvin Lin Staff - 5 years, 3 months ago

Thanks for your correction. I've corrected the length of DE to be 12 instead of 8.

Ahmad Saad - 5 years, 3 months ago

Thanks! That's really helpful :)

Calvin Lin Staff - 5 years, 3 months ago

@Calvin Lin – But the diameter of the smaller circle isn't indicated as half of the diameter of the bigger circle

Caeo Tan - 5 years, 3 months ago

@Caeo Tan – the diameter of the smaller circle isn't equal to a half of the diameter of the bigger circle. It's diameter have any value less than the diameter of the bigger circle. (ie. r < R).

Note that : r = the distance between the chord and diameter of the bigger circle and they are parallel to each other.

Ahmad Saad - 5 years, 3 months ago

@Ahmad Saad – and after that the radii of the circles are: R=7.5 cm ,

r=4.5 cm

Safeen Ssm - 5 years, 2 months ago

can you explain what you did in this diagram? where does 36 come from?

Matthew Agona - 4 years, 6 months ago

Colin Saxelby
Dec 12, 2016

It doesn't seem like there's enough information, so I assumed there's a general result here that works irrespective of the size of the smaller circle. So I imagined the smaller one shrinking to 0, which put the chord on the diameter of the larger circle, which made the area simple - pi.6.6/2 - so A = 18. And I was right! Now I've had more time to think about it, you can do it with R^2-r^2 and Pythagoras, exactly as Ahmed Saad has set out.