DSQ: Calculating the area of a circle - GMAT Quantitative Reasoning
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Data Sufficiency Question
Calculate the area of a circle.
1. The radius of the circle is 4.
2. The circumference of the circle is 24.
Data Sufficiency Question
Calculate the area of a circle.
1. The radius of the circle is 4.
2. The circumference of the circle is 24.
The area of a circle can be calcuated using the equation:
and the circumference calculated using:
The radius is the only information required for calculating the area of a circle and that can be obtained from the circumference, therefore, either statement is sufficient.
The area of a circle can be calcuated using the equation:
and the circumference calculated using:
The radius is the only information required for calculating the area of a circle and that can be obtained from the circumference, therefore, either statement is sufficient.
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You are given a circle and a square. Which one has the larger area?
Statement 1: The radius of the circle is two-thirds the sidelength of the square.
Statement 2: The circumference of the circle is 
times the perimeter of the square.
You are given a circle and a square. Which one has the larger area?
Statement 1: The radius of the circle is two-thirds the sidelength of the square.
Statement 2: The circumference of the circle is
Let 
be the sidelength of the square. Then its area is 
.
From Statement 1, it follows that the radius of the circle is 
.
From Statement 2, it follows that , since the perimeter of the square is 
, the circumference of the circle is 
, and the radius is 
- the same fact given in Statement 1.
Either way, it follows that the area of the circle in terms of 
is
,
so all we have to do is compare 
to 1 in order to determine whether the square or the circle is larger in area.
Let
From Statement 1, it follows that the radius of the circle is
From Statement 2, it follows that , since the perimeter of the square is
Either way, it follows that the area of the circle in terms of
so all we have to do is compare
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You are given a circle and an equilateral triangle. Which one has the greater area?
Statement 1: The sidelength of the triangle is three times the radius of the circle.
Statement 2: The perimeter of the triangle is 99 inches.
You are given a circle and an equilateral triangle. Which one has the greater area?
Statement 1: The sidelength of the triangle is three times the radius of the circle.
Statement 2: The perimeter of the triangle is 99 inches.
From Statement 2, we can calculate the area of the triangle, but we are given no clues about the area of the circle, actual or relative.
From Statement 1, we know that if we call the radius of the circle 
, we know the sidelength of the triangle is 
.
The area of the circle is 
.
The area of the triangle is 
.
All we have to do is compare 
to 
to determine whether the circle or the triangle has the greater area.
From Statement 2, we can calculate the area of the triangle, but we are given no clues about the area of the circle, actual or relative.
From Statement 1, we know that if we call the radius of the circle
The area of the circle is
The area of the triangle is
All we have to do is compare
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What is the area of the gray region in the above figure?
Statement 1: The diameter of the larger circle is one mile.
Statement 2: The radius of the smaller circle is 1,320 feet.
What is the area of the gray region in the above figure?
Statement 1: The diameter of the larger circle is one mile.
Statement 2: The radius of the smaller circle is 1,320 feet.
The radius of the larger circle is equal to the diameter of the smaller, and, subsequently, twice the radius of the smaller. If one radius is known, then the other can be calculated, and the areas of the two circles can be as well; the difference between their areas is the area of the gray region. Statement 1 tells us the diameter of the large circle, from which its radius can be determined by dividing by 2; Statement 2 tells us the radius of the radius of the smaller circle. From either, the radius of the other circle can be calculated.
The radius of the larger circle is equal to the diameter of the smaller, and, subsequently, twice the radius of the smaller. If one radius is known, then the other can be calculated, and the areas of the two circles can be as well; the difference between their areas is the area of the gray region. Statement 1 tells us the diameter of the large circle, from which its radius can be determined by dividing by 2; Statement 2 tells us the radius of the radius of the smaller circle. From either, the radius of the other circle can be calculated.
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Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.
What is the area of the inscribed circle?
Statement 1: The area of the circumscribed circle is 
Statement 2: The perimeter of the hexagon is 30.
Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.
What is the area of the inscribed circle?
Statement 1: The area of the circumscribed circle is
Statement 2: The perimeter of the hexagon is 30.
Examine the diagram below, which shows the hexagon, segments from its center to two vertices and the midpoint of a side, and the two circles.
From Statement 1 alone, the radius 
of the circumscribed circle can be calculated using the area formula. From Statement 2 alone, the perimeter can be divided by 6 to obtain 
; since two consecutive radii and one side of a hexagon can be proved to form an equilateral triangle, 
. As a consequence, from either statement alone, 
can be calculated.
can be proved a 30-60-90 triangle, so the 30-60-90 Theorem can be used to calculate 
, the radius of the inscribed circle. From this, the area of the inscribed circle can be calculated.
Examine the diagram below, which shows the hexagon, segments from its center to two vertices and the midpoint of a side, and the two circles.
From Statement 1 alone, the radius
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