All GMAT Math Resources
Example Question #1 : Geometry
How many degrees does the hour hand on a clock move between 3 PM and 7:30 PM the same day?
An hour hand rotates 360 degrees for every 12 hours, so the hour hand would move
The hour hand moves four and a half hours along the clock
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?
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?
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 ?
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
Note: Figure NOT drawn to scale.
Order the degree measures of the arcs from least to greatest.
, 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 #6 : Geometry
A circle is inscribed in a square with area 100. What is the area of the circle?
Not enough information.
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 #7 : 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?
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 #8 : 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?
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 #9 : Geometry
A square has the same area as a circle with radius 12 inches. What is the sidelength of that square, in terms of ?
The area of a circle with radius 12 inches is
This is also the area of the square, so the sidelength of that square is the square root of that area:
Example Question #10 : Geometry
What is the area of a circle which goes through the points ?
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: