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# GMAT Math : Geometry

## Example Questions

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### Example Question #1 : Geometry

How many degrees does the hour hand on a clock move between 3 PM and 7:30 PM?

Explanation:

An hour hand rotates 360 degrees for every 12 hours, so the hour hand moves .

There are 4.5 hours between 3 PM and 7:30 PM, so the total degree measure is

.

### Example Question #2 : Geometry

A sector of a circle has a central angle equal to 45 degrees. What percentage of the circle is comprised by the sector?

Explanation:

The entire circle is 360 degrees, therefore we can set up proportions and cross multiply.

### Example Question #3 : Geometry

A teacher buys a supersized pizza for his after-school club. The super-pizza has a diameter of 18 inches. If the teacher is able to perfectly cut from the center a 36 degree sector for himself, what is the area of his slice of pizza, rounded to the nearest square inch?

27

26

25

24

28

25

Explanation:

First we calculate the area of the pizza. The area of a circle is defined as . Since our diameter is 18 inches, our radius is 18/2 = 9 inches. So the total area of the pizza is  square inches.

Since the sector of the pie he cut for himself is 36 degrees, we can set up a ratio to find how much of the pizza he cut for himself. Let x be the area of the pizza he cut for himself. Then we know,

Solving for x, we get x=25.45 square inches, which rounds down to 25.

### Example Question #4 : Geometry

Note: Figure NOT drawn to scale

Refer to the above diagram.

What is  ?

Explanation:

The degree measure of  is half the degree measure of the arc it intercepts, which is . We can use the measures of the two given major arcs to find , then take half of this:

### Example Question #5 : Geometry

A giant clock has a minute hand that is eight feet long. The time is now 2:40 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Explanation:

Between noon and 2:40 PM, two hours and forty minutes have elapsed, or, equivalently, two and two-thirds hours. This means that the minute hand has made   revolutions.

In one revolution, the tip of an eight-foot minute hand moves  feet, or  inches.

After   revolutions, the tip of the minute hand has moved  inches.

### Example Question #6 : Geometry

Note: Figure NOT drawn to scale.

.

Order the degree measures of the arcs  from least to greatest.

Explanation:

, so, by the Multiplication Property of Inequality,

.

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as

.

### Example Question #7 : Geometry

A circle is inscribed in a square with area 100.  What is the area of the circle?

Not enough information.

Explanation:

A square with area 100 would have a side length of 10, which is the diameter of the circle.  The area of a circle is , so the answer is .

### Example Question #8 : Geometry

The above figure shows a square inscribed inside a circle. What is the ratio of the area of the circle to that of the square?

Explanation:

Let  be the radius of the circle. Its area is

The diagonal of the square is equal to the diameter of the circle, or . The area of the square is half the product of its (congruent) diagonals:

This makes the ratio of the area of the circle to that of the square .

### Example Question #9 : Geometry

Tom has a rope that is 60 feet long.  Which of the following is closest to the largest area that Tom could enclose with this rope?

Explanation:

The largest square you could make would be  with an area of .  However, the largest region that can be enclosed will be accomplished with a circle (so you don't lose distance creating the angles).  This circle will have a circumference of 60 ft.  This gives a radius of

Then the area will be

This is closer to 280 than to 300

### Example Question #10 : Geometry

What is the area of a circle which goes through the points  ?

Explanation:

As can be seen in this diagram, the three points form a right triangle with legs of length 5 and 12.

A circle through these three points circumscribes this right triangle.

An inscribed right, or , angle intercepts a  arc, or a semicircle, making the hypotenuse a diameter of the circle. The diameter of the circle is therefore the hypotenuse of the right triangle, which we can find via the Pythagorean Theorem:

The radius of the circle is half this, or

The area of the circle is therefore:

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