This was voted on Quora as one of the most difficult SAT math problems.

In the figure, arc SBT is one quarter of a circle with center R and radius 6. If the length plus the width of rectangle ABCR is 8, then the perimeter of the shaded region is:

(A) 8 + 3π

(B) 10 + 3π

(C) 14 + 3π

(D) 1 + 6π

(E) 12 + 6π

Very important: An SAT Math section has something like 20 problems in 25 minutes, so you would have to be able to solve this problem in 1 to 2 minutes, which is quite the challenge.

Can you figure it out? Give it a try; then watch the video for a solution.

One of the Hardest SAT Math Problems – Can You Solve It?

Or keep reading for a text/image explanation. “All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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One of the Hardest SAT Math Problems

“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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(This was transcribed quickly after I made the video–please let me know if there are any typos/errors and I will correct them, thanks).

The perimeter of the shaded region is the length of:

SA + AC + CT + arc SBT

Let’s break it down into steps. First let’s solve for arc SBT.

It is given the arc is one quarter of the circle with radius 6, so the arc length is 1/4 of the circle’s circumference:

arc SBT = (2πr)/4 = 2π(6)/4 = 3π

Now what about the remaining length?

SA + AC + CT

The trick is seeing how to find SA and CT in terms of a radius of the circle.

Note that:

SA = SR – AR

CT = RT – RC

So we can re-write the length we want to find as:

SA + AC + CT

= (SR – AR) + AC + (RT – RC)

= SR + AC + RT – (AR + RC)

Now we note:

SR = RT = 6 (each is a radius of the circle)

AC = RB = 6 (the diagonals of a rectangle are equal, and RB is a radius of the circle)

AR + RC = 8 (this is the length plus width of the rectangle, which is given to be 8)

So magically the equation transforms into known lengths, and we have:

SA + AC + CT

= SR + AC + RT – (AR + RC)

= 6 + 6 + 6 – 8

= 10

Thus, we find the perimeter:

SA + AC + CT + arc SBT

= 10 + 3π

So that’s it! And thus (B) is the correct answer.

Source

Quora discussion most difficult SAT math problems

https://www.quora.com/What-are-some-of-the-most-difficult-SAT-math-problems

If you want a much, much harder challenge than this problem, try the following problem from the Korean SAT:

Video: https://youtu.be/Y6caQ_8_frU

https://mindyourdecisions.com/blog/2018/03/13/can-you-solve-a-hard-korean-test-question-cone-shaped-mountain-puzzle