DSQ: Calculating the perimeter of an equilateral triangle - GMAT Quantitative
Card 1 of 104
Given three equilateral triangles 
, 
, and 
, which has the greatest perimeter?
Statement 1: A circle with diameter equal to the length of 
can be circumscribed about 
.
Statement 2: A circle with diameter equal to the length of 
can be circumscribed about 
.
Given three equilateral triangles
Statement 1: A circle with diameter equal to the length of
Statement 2: A circle with diameter equal to the length of
Tap to reveal answer
Assume Statement 1 alone, and examine the diagram below, which shows a circle circumscribed about 
:
The diameter, which is equal to 
as given by Statement 1, is greater in length than any chord which is not a diameter - and all sides of 
are non-diameter chords. Therefore, 
has sides of greater length than 
, and its perimeter is therefore greater. However, nothing is given about 
.
If Statement 2 alone is assumed, then, similarly, 
can be shown to have perimeter greater than that of 
. But nothing can be determined about 
.
From the two statements together, however, 
has a perimeter greater than those of the other two triangles.
Assume Statement 1 alone, and examine the diagram below, which shows a circle circumscribed about
The diameter, which is equal to
If Statement 2 alone is assumed, then, similarly,
From the two statements together, however,
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Given three equilateral triangles 
, 
, and 
, which has the greatest perimeter?
Statement 1: 
Statement 2: 
Given three equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true. From Statement 1, since 
, it follows that 
; since the perimeter of an equilateral triangle is three times the length of one side, it follows that 
has perimeter greater than 
. Similarly, from Statement 2, it follows that 
has perimeter greater than 
. However, there is no way to determine whether 
or 
has the greater perimeter of the two.
Assume both statements to be true. From Statement 1, since
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What is the perimeter of 
?
(1) The area of the triangle is 
.
(2) 
is an equilateral triangle.
What is the perimeter of
(1) The area of the triangle is
(2)
Tap to reveal answer
To find the perimeter we should be able to calculate each sides of the triangle.
Statement 1 tells us the area of the triangle. From this we can't calculate anything else, since we don't know whether the triangle is of a special type.
Statement 2 tells us that the triangle is equilateral. Again This information alone is not sufficient.
Taken together these statements allow us to find the sides of the equilateral triangle ABC. Indeed, the area of an equilateral triangle is given by the following formula: 
. Where 
is the area and 
the length of the side.
Therefore both statements are sufficient.
To find the perimeter we should be able to calculate each sides of the triangle.
Statement 1 tells us the area of the triangle. From this we can't calculate anything else, since we don't know whether the triangle is of a special type.
Statement 2 tells us that the triangle is equilateral. Again This information alone is not sufficient.
Taken together these statements allow us to find the sides of the equilateral triangle ABC. Indeed, the area of an equilateral triangle is given by the following formula:
Therefore both statements are sufficient.
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Find the perimeter of 
given the following:
I) 
.
II) Side 
.
Find the perimeter of
I)
II) Side
Tap to reveal answer
To find perimeter, we need the side lengths.
I) Gives us the measure of two angles. The given measurement is equal to 60 degrees. This means the last angle is also 60 degrees.
II) Gives us one side length, but because we know from I) that this is an equilateral triangle, we know that all the sides have the same length.
Add up all the sides to get the perimeter.
We need I) and II) to find the perimeter
To find perimeter, we need the side lengths.
I) Gives us the measure of two angles. The given measurement is equal to 60 degrees. This means the last angle is also 60 degrees.
II) Gives us one side length, but because we know from I) that this is an equilateral triangle, we know that all the sides have the same length.
Add up all the sides to get the perimeter.
We need I) and II) to find the perimeter
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Find the perimeter of the equilateral triangle.
- A side measures
.
- The area is
.
Find the perimeter of the equilateral triangle.
- A side measures
- The area is
Tap to reveal answer
Recall the formula for perimeter of a triangle.
where 
represents the side length of the triangle
Statement 1: We're given the length of the side so all we need to do is plug this value into the equation. 
Statement 2: We're given the area so we first need to solve for the side length.
Now we can plug the value into the equation, like we did in Statement 1.
Recall the formula for perimeter of a triangle.
where
Statement 1: We're given the length of the side so all we need to do is plug this value into the equation.
Statement 2: We're given the area so we first need to solve for the side length.
Now we can plug the value into the equation, like we did in Statement 1.
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Given an equilateral triangle 
and a right triangle 
with right angle 
, which has the greater perimeter?
Statement 1: 
Statement 2: 
Given an equilateral triangle
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. Since in 
, 
is the right angle, 
is the hypotenuse, which is longer than either of the two legs - 
and 
. Since, in equilateral triangle 
, 
, it follows that 
and 
. Consequently
,
and 
has the greater perimeter.
Statement 2 alone gives insufficient information, since we know nothing about the second leg or the hypotenuse of . We see this by examining these two cases.
Case 1: Let be an equilateral triangle of sidelength 6, and 
in . If second leg 
of has length 4, then the triangle is a 3-4-5 triangle, with hypotenuse having length 5. The perimeter of is 
, and the perimeter of is 
. has the greater perimeter.
Case 2: Let be an equilateral triangle of sidelength 6, and 
in . If second leg 
of has length 20, has the greater perimeter on the basis of one side alone.
Assume Statement 1 alone. Since in
and
Statement 2 alone gives insufficient information, since we know nothing about the second leg or the hypotenuse of
Case 1: Let
Case 2: Let
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Given two equilateral triangles 
and 
, which has the greater perimeter?
Statement 1: 
is the midpoint of 
.
Statement 2: 
is the midpoint of 
.
Given two equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
Neither statement alone is enough to determine which triangle has the greater perimeter, as each statement gives information about only one point.
Assume both statements to be true. Since 
is the segment that connects the endpoints of two sides of 
, it is a midsegment of the triangle, whose length is half the length of the side of 
to which it is parallel. Therefore, the sidelength of 
is half that of 
, and their perimeters are similarly related. This makes 
the triangle with the greater perimeter.
Neither statement alone is enough to determine which triangle has the greater perimeter, as each statement gives information about only one point.
Assume both statements to be true. Since
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Given two equilateral triangles 
and 
, which, if either, has the greater perimeter?
Statement 1: 
Statement 2: 
has greater area than 
.
Given two equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
Since an equilateral triangle has three sides of equal measure, the perimeter of an equilateral triangle is three times its sidelength, so the triangle with the greater common sidelength has the greater perimeter.
Statement 1 gives precisely this information; since one side of 
is longer than one side of 
, it follows that 
has the longer perimeter.
Statement 2 gives that 
has the greater area. Since the area of an equilateral triangle depends only on the common length of its sides, the triangle with the greater area, 
, must also have the greater sidelength and, consequently, the greater perimeter.
Since an equilateral triangle has three sides of equal measure, the perimeter of an equilateral triangle is three times its sidelength, so the triangle with the greater common sidelength has the greater perimeter.
Statement 1 gives precisely this information; since one side of
Statement 2 gives that
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Which, if either, is greater: the perimeter of equilateral triangle 
or the circumference of a given circle with center 
?
Statement 1: The midpoint of 
is inside the circle.
Statement 2: The midpoint of 
is on the circle.
Which, if either, is greater: the perimeter of equilateral triangle
Statement 1: The midpoint of
Statement 2: The midpoint of
Tap to reveal answer
For simplicity's sake, we will assume that 
has sidelength 1, and, consequently, perimeter 3; these arguments work regardless of the size of the triangle.
We will also need the circumference formula 
.
Assume Statement 1 alone. Since the midpoint of 
, which we will call 
, is inside the circle, the radius of the circle must be greater than 
. This makes the circumference at least 
times this, or 
, which is greater than 3.
Assume Statment 2 alone. Since the circle has as a radius the segment from 
to the midpoint of the opposite side, it is an altitude of 
, and the radius is the height of the triangle. By way of the 30-60-90 Theorem, this height is 
, and the circumference of the circle is 
times this, or 
. This is greater than 3.
Either statement alone establishes that the circumference of the circle is greater than 3, the perimeter of .
For simplicity's sake, we will assume that
We will also need the circumference formula
Assume Statement 1 alone. Since the midpoint of
Assume Statment 2 alone. Since the circle has as a radius the segment from
Either statement alone establishes that the circumference of the circle is greater than 3, the perimeter of
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Which, if either, of equilateral triangles 
and 
, has the greater perimeter?
Statement 1: 
Statement 2: 
Which, if either, of equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
If we let 
and 
be the sidelengths of and , respectively; the statements can be rewritten as:
Statement 1: 
Statement 2: 
Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter. The question can therefore be reduced to asking which of 
and 
, if either, is greater.
Statement 1 alone is not sufficient to yield an answer:
Case 1: 
Case 2: 
Both cases satisfy Statement 1, but in the first case, 
, meaning that has greater sidelength and perimeter than 
, and in the second case, 
, meaning the reverse. By a similar argument, Statement 2 is insufficient.
Now assume both statements to be true. The two equations together comprise a system of equations:
Multiply the first equation by 3 and the second by 
, then add:
Now substitute back:
has the greater sidelength and, consequently, the greater perimeter.
If we let
Statement 1:
Statement 2:
Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter. The question can therefore be reduced to asking which of
Statement 1 alone is not sufficient to yield an answer:
Case 1:
Case 2:
Both cases satisfy Statement 1, but in the first case,
Now assume both statements to be true. The two equations together comprise a system of equations:
Multiply the first equation by 3 and the second by
Now substitute back:
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Given two equilateral triangles and , which, if either, has the greater perimeter?
Statement 1: 
Statement 2: 
Given two equilateral triangles
Statement 1:
Statement 2:
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The area of an equilateral triangle is given by the formula
,
where 
is its common sidelength. It follows that the triangle with the greater sidelength has the greater area.
We will let 
and 
stand for the common sidelength of and , respectively. The question becomes which, if either, of 
and 
is the greater.
Statement 1 alone can be rewritten by multiplying:
Therefore, 
.
Therefore, 
, the length of one side of is less than 
, the length of one side of .
Statement 2 alone can be rewritten as 
. Again, it follows that 
.
From either statement alone, it follows that 
. has the greater sidelength, and, consequently, the greater area.
The area of an equilateral triangle is given by the formula
where
We will let
Statement 1 alone can be rewritten by multiplying:
Therefore,
Therefore,
Statement 2 alone can be rewritten as
From either statement alone, it follows that
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Which, if either, of equilateral triangles and , has the greater perimeter?
Statement 1: 
Statement 2: 
Which, if either, of equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter.
If we let 
and 
be the common sidelengths of and , respectively, Statement 1 can be rewritten as the equation 
. This can be expressed as follows:
Therefore, 
.
Statement 2 can be rewritten as
Once again,
Since either statement alone establishes that , it follows that has the longer sides and, consequently, the greater perimeter of the triangles.
Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter.
If we let
Therefore,
Statement 2 can be rewritten as
Once again,
Since either statement alone establishes that
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Give the perimeter of equilateral triangle 
.
Statement 1: 
is a radius of a circle with area 
.
Statement 2: 
is the hypotenuse of a 30-60-90 triangle with area 
.
Give the perimeter of equilateral triangle
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. To find the radius of a circle with area 
, use the area formula:
This is also the length of each side of , so its perimeter is three times this, or 24.
Assume Statement 2 alone. If we let 
be the length of 
, then, since this the hypotenuse of a 30-60-90 triangle, by the 30-60-90 Theorem, the legs measure 
and 
. Half the product of their lengths is equal to area 
, so
.
As before, the sidelength of 
is 8 and the perimeter is 24.
Assume Statement 1 alone. To find the radius of a circle with area
This is also the length of each side of
Assume Statement 2 alone. If we let
As before, the sidelength of
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Given three equilateral triangles , , and , which has the greatest perimeter?
Statement 1:
Statement 2:
Given three equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true. From Statement 1, since , it follows that ; since the perimeter of an equilateral triangle is three times the length of one side, it follows that has perimeter greater than . Similarly, from Statement 2, it follows that has perimeter greater than . However, there is no way to determine whether or has the greater perimeter of the two.
Assume both statements to be true. From Statement 1, since
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What is the perimeter of ?
(1) The area of the triangle is .
(2) is an equilateral triangle.
What is the perimeter of
(1) The area of the triangle is
(2)
Tap to reveal answer
To find the perimeter we should be able to calculate each sides of the triangle.
Statement 1 tells us the area of the triangle. From this we can't calculate anything else, since we don't know whether the triangle is of a special type.
Statement 2 tells us that the triangle is equilateral. Again This information alone is not sufficient.
Taken together these statements allow us to find the sides of the equilateral triangle ABC. Indeed, the area of an equilateral triangle is given by the following formula: . Where is the area and the length of the side.
Therefore both statements are sufficient.
To find the perimeter we should be able to calculate each sides of the triangle.
Statement 1 tells us the area of the triangle. From this we can't calculate anything else, since we don't know whether the triangle is of a special type.
Statement 2 tells us that the triangle is equilateral. Again This information alone is not sufficient.
Taken together these statements allow us to find the sides of the equilateral triangle ABC. Indeed, the area of an equilateral triangle is given by the following formula:
Therefore both statements are sufficient.
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Find the perimeter of given the following:
I) .
II) Side .
Find the perimeter of
I)
II) Side
Tap to reveal answer
To find perimeter, we need the side lengths.
I) Gives us the measure of two angles. The given measurement is equal to 60 degrees. This means the last angle is also 60 degrees.
II) Gives us one side length, but because we know from I) that this is an equilateral triangle, we know that all the sides have the same length.
Add up all the sides to get the perimeter.
We need I) and II) to find the perimeter
To find perimeter, we need the side lengths.
I) Gives us the measure of two angles. The given measurement is equal to 60 degrees. This means the last angle is also 60 degrees.
II) Gives us one side length, but because we know from I) that this is an equilateral triangle, we know that all the sides have the same length.
Add up all the sides to get the perimeter.
We need I) and II) to find the perimeter
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Find the perimeter of the equilateral triangle.
- A side measures .
- The area is .
Find the perimeter of the equilateral triangle.
- A side measures
- The area is
Tap to reveal answer
Recall the formula for perimeter of a triangle.
where represents the side length of the triangle
Statement 1: We're given the length of the side so all we need to do is plug this value into the equation.
Statement 2: We're given the area so we first need to solve for the side length.
Now we can plug the value into the equation, like we did in Statement 1.
Recall the formula for perimeter of a triangle.
where
Statement 1: We're given the length of the side so all we need to do is plug this value into the equation.
Statement 2: We're given the area so we first need to solve for the side length.
Now we can plug the value into the equation, like we did in Statement 1.
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Given an equilateral triangle and a right triangle with right angle , which has the greater perimeter?
Statement 1:
Statement 2:
Given an equilateral triangle
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. Since in , is the right angle, is the hypotenuse, which is longer than either of the two legs - and . Since, in equilateral triangle , , it follows that and . Consequently
,
and has the greater perimeter.
Statement 2 alone gives insufficient information, since we know nothing about the second leg or the hypotenuse of . We see this by examining these two cases.
Case 1: Let be an equilateral triangle of sidelength 6, and in . If second leg of has length 4, then the triangle is a 3-4-5 triangle, with hypotenuse having length 5. The perimeter of is , and the perimeter of is . has the greater perimeter.
Case 2: Let be an equilateral triangle of sidelength 6, and in . If second leg of has length 20, has the greater perimeter on the basis of one side alone.
Assume Statement 1 alone. Since in
and
Statement 2 alone gives insufficient information, since we know nothing about the second leg or the hypotenuse of
Case 1: Let
Case 2: Let
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Given three equilateral triangles , , and , which has the greatest perimeter?
Statement 1: A circle with diameter equal to the length of can be circumscribed about .
Statement 2: A circle with diameter equal to the length of can be circumscribed about .
Given three equilateral triangles
Statement 1: A circle with diameter equal to the length of
Statement 2: A circle with diameter equal to the length of
Tap to reveal answer
Assume Statement 1 alone, and examine the diagram below, which shows a circle circumscribed about :
The diameter, which is equal to as given by Statement 1, is greater in length than any chord which is not a diameter - and all sides of are non-diameter chords. Therefore, has sides of greater length than , and its perimeter is therefore greater. However, nothing is given about .
If Statement 2 alone is assumed, then, similarly, can be shown to have perimeter greater than that of . But nothing can be determined about .
From the two statements together, however, has a perimeter greater than those of the other two triangles.
Assume Statement 1 alone, and examine the diagram below, which shows a circle circumscribed about
The diameter, which is equal to
If Statement 2 alone is assumed, then, similarly,
From the two statements together, however,
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Given two equilateral triangles and , which has the greater perimeter?
Statement 1: is the midpoint of .
Statement 2: is the midpoint of .
Given two equilateral triangles
Statement 1:
Statement 2:
Tap to reveal answer
Neither statement alone is enough to determine which triangle has the greater perimeter, as each statement gives information about only one point.
Assume both statements to be true. Since is the segment that connects the endpoints of two sides of , it is a midsegment of the triangle, whose length is half the length of the side of to which it is parallel. Therefore, the sidelength of is half that of , and their perimeters are similarly related. This makes the triangle with the greater perimeter.
Neither statement alone is enough to determine which triangle has the greater perimeter, as each statement gives information about only one point.
Assume both statements to be true. Since
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