All GMAT Math Resources
Example Questions
Example Question #31 : Circles
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:
Example Question #5 : Calculating The Area Of A Circle
A circle on the coordinate plane has equation
Which of the following represents its area?
The equation of a circle centered at the origin is
where is the radius of the circle.
The area of a circle is ; since in the equation of the circle, , we can substitute to get the area .
Example Question #2 : Calculating The Area Of A Circle
Refer to the above figure, in which the dot indicates the center of the larger circle. The larger circle is one yard in diameter. Give the area of the gray region in square inches.
The larger circle has diameter one yard, or 36 inches; its radius is half that, or 18 inches, so its area is
square inches.
The diameter of the smaller circle is equal to the radius of the larger, or 18 inches; its radius is half that, or 9 inches, so its area is:
square inches.
The area of the gray region is the difference of the two:
square inches.
Example Question #2 : Radius
For $10, Brandon can order either a 12"-diameter pizza, two 6"-diameter pizzas, or three 4"-diameter pizzas. Which option is the best value, assuming all pizzas are the same thickness?
three 4" pizzas
one 12" pizza
two 6" pizzas
Cannot be determined
All three pizzas have the same value.
one 12" pizza
For a circle, .
Therefore, the area of the 12" pizza .
The area of the two 6" pizzas .
The area of the three 4" pizzas = .
The 12" pizza is the best option.
Example Question #3 : Radius
Refer to the above circle. The smaller circle has diameter feet. In terms of , give the area of the gray region in square inches.
Multiply the diameter of the smaller circle by 12 to convert it to inches - this will be , which is also the radius of the larger circle. The radius of the smaller circle will be half this, or .
Using the area formula for a circle , we can substitute these quantities for and subtract the area of the smaller circle from that of the larger:
square inches
square inches
square inches
Example Question #32 : Geometry
A circle on the coordinate plane has equation
What is its area?
The area of a circle is equal to , where is the radius.
The standard form of the area of a circle with radius and center is
Once we get the equation in standard form, we know , which can be multiplied by .
Complete the squares:
so can be rewritten as follows:
Therefore, and .
Example Question #4 : Radius
A square has the same area as a circle with a radius of 12 inches. What is the sidelength of that square, in terms of ?
The area of a circle with radius 12 is
.
This is also the area of the square, so the sidelength of that square is the square root of the area:
Example Question #271 : Problem Solving Questions
A circle on the coordinate plane has equation .
What is its area?
The equation of a circle centered at the origin is
,
where is the radius of the circle.
The area of a circle is .
From the equation given in the question stem, we know that , so we can plug this into the area formula:
Example Question #271 : Problem Solving Questions
Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.
The circumscribed circle has circumference . Give the area of the inscribed circle.
Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.
The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the inscribed circle has half the radius of the circumscribed circle.
The circumscribed circle has circumference , so its radius is
The inscribed circle has radius half this, or 5, so its area is
Example Question #41 : Geometry
Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.
The inscribed circle has circumference . Give the area of the circumscribed circle.
Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.
The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the circumscribed circle has twice the radius of the inscribed circle.
The inscribed circle has circumference , so its radius is
The circumscribed circle has radius twice this, or 20, so its area is