A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?

A. π∗r^2
B. π∗r^2 + 10
C. π∗r^2 + 1/4∗π^2∗r^2
D. π∗r^2 + (40−2π∗r)^2
E. π∗r^2 + (10−1/2π∗r)^2

Here’s another tough geometry question from the GMAT Prep tests 1 and 2. This one comes up a lot in GMAT tutoring sessions even with people excelling on the quant side. Why? People speed through the initial read, don’t define the question, and rush the setup. Then the follow through is hopeless. So in sessions we work on slowing down and getting things defined carefully before doing the actual solving.

Let’s define the question: Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?

We need the total area of the circle and the square. So: Area of Square + Area of Circle. Easy! Let’s get each of those things defined using the information we have.

Setup/Solve

We know that the radius of the circle is r (one piece is used to form a circle with radius r). Great – define the area of a circle with radius r. π∗r^2.

Half done! Now we need the area of the square. Is there any way to define the area of the square using r? There must because that’s the only variable in the answer choices.

What information is left in the question? A thin piece of wire 40 meters long is cut into two pieces

The 40 meters. Right. So, how can we use that?

The 40 meters is the circumference of the circle + the perimeter of the square.

How do we separate out the perimeter of the square?

Circle + Square = 40

then

Square = 40 – circle

So, Perimeter of the square = 40 – 2π*r

How do you get from the perimeter of a square to the area? Divide it by 4 to get a side. And then square that side.

Here’s the side:

(40 – 2π*r)/4 = 10 – 1/2π*r

Now let’s square it:

(10 – 1/2π*r)^2

Now we have the area of the square.

Last step: add them up!

π∗r^2 + (10−1/2π∗r)^2

Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r? π∗r^2 + (10−1/2π∗r)^2

Correct Answer: E

More GMAT Geometry Practice Questions

Here’s a tough GMAT geometry question that’s similar but different. I find it similar because like the “wire question” it focuses a bit more on the algebra than the geometry.

The perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse of the triangle? 

Same deal with this one. The geometry aspect is very simple. You don’t need to make a lot of inferences. The challenge is in the algebra.

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

Here’s very tough GMAT geometry question from GMAT question of the day that also requires solid algebra skills. This one has a “measurement error” angle that I’ve seen on real GMATs that adds a level of subtlety.