- In the figure above,
ABis the diameter of the circle, andAC=BC. What is the area of the shaded region?

(A) 4π – 2

(B) 2π – 1

(C) π

(D) π – 1

(E) π – 2

Answer and explanation after the jump…

As is usually the case with shaded region problems, the easiest way (and in this case, the only way) to solve for the shaded region is to solve for the regions around it first. The relationship we want to keep in mind is:

In this particular case, we’re going to have to do a little legwork to figure out what out “whole” is before we get down to business.

Let’s start by dropping a vertical from the top of our isosceles triangle (and noting that in doing so, we’re drawing a radius, so it’s got a length of 2):

That vertical is of course perpendicular to *AB*, and creates a right angle that nicely frames the area we’re looking to solve for. So the Area_{whole} we’re looking for here is actually only the part of the circle marked off by that right angle. Since a circle has 360 degrees of arc and we’re only dealing with 90 of them, we’re dealing with one fourth of the circle.

Area_{circle} = π*r*^{2
}Area_{circle} = π(2^{2})

Area_{circle} = 4π

So the area of the sector we care about is simply one fourth of that, or π:

**Area _{whole} = π**

Now we just need to find the area of the unshaded part (the right triangle we created, in red):

Area_{unshaded} = 1/2 (2)(2)

**Area _{unshaded} = 2**

So the area of our shaded region must be…

Area_{shaded} = π – 2

That’s answer choice (E).

One more note about this one: since the diagram is drawn to scale, it’s possible (and *wise*) to use your guesstimation skills once you’ve found an answer (or even to eliminate answers before you do much calculating). Which is to say: when you know the area of the triangle is 2, does it make sense that the shaded region is about 1.14159? I’d say, by eyeballing it, that yeah, it does. It wouldn’t have made sense, though, had we made a math mistake somehow and ended up with a different choice, like (D) π – 1. To pick that answer would be to say (insanely) that the shaded region is as big as the right triangle we made by dropping that vertical. That’s *crazy* talk.

what if you do this

sector ACB- area of triangle ACB

45/360 (4π) – o.5*4*2

the answer should be the same but isn’t??

The problem with that is that sector ACB isn’t 45 degrees! It’s 180!

Be careful with that, though, too. If you take the whole top half of the circle (sector ACB) then you’re not going to get only the shaded region when you take away the triangle…you’re also going to have the unshaded leftover bit on the upper left.

OMG! I can’t believe I missed that! I guess I was really sleep deprived when I did this question…! I’m usuallly awesome at maths in school but SAT math manages to screw with my head.. 🙁

Thanks a lot!! your blog is awesome..!

Great post.

Thankyakindly!

I originally thought that the answer was 4π -2 I don’t understand why it became π

if we were supposed to do Areawhole – Areaunshaded

Well, if you subtract the area of the triangle from the area of the whole circle, you’re left with much more than just the shaded region. So first we shrink the “whole” to be only 1/4 of the circle.

Any harder questions? Im in grade 7 and i found this pretty easy

This site’s full of harder questions. Try the diagnostic drills.

I answered it right, I’m so happy!! I just found the area of the quarter-arc and subtracted it by the area of the 2-2-sqrt(2) triangle.