DSQ: Calculating the area of an equilateral triangle - GMAT Quantitative
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What is the area of 
?
(1) The height 
is 5.
(2) The base 
is 4.
What is the area of
(1) The height
(2) The base
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To find an area of a triangle we need the length of the height and the length of the corresponding basis.
Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.
Hence, both statements together are sufficient.
To find an area of a triangle we need the length of the height and the length of the corresponding basis.
Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.
Hence, both statements together are sufficient.
is an equilateral triangle inscribed in the circle. What is the area of triangle 
?
(1) The area of the circle is 
.
(2) The area of the circle minus the area of the triangle is 
.
(1) The area of the circle is
(2) The area of the circle minus the area of the triangle is
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To answer this question, we should be able to have information about any lengths in the circle or in the triangle, or also about the areas of the different regions.
Statement 1 alone is sufficient, since from the area of the circle we can find the length of the radius, which allows us to calculate the length of the height of the equilateral triangle. This would allow us to find the length of a side, giving us all the necessary information to calculate the area of the equilateral triangle.
Statement 2, although giving us information about the difference of the circle and the triangle, we cannot conclude anything because we don't know the proportions of the respective geometric figures. Therefore statement 2 is insufficient.
To conclude, statement one alone is sufficient.
To answer this question, we should be able to have information about any lengths in the circle or in the triangle, or also about the areas of the different regions.
Statement 1 alone is sufficient, since from the area of the circle we can find the length of the radius, which allows us to calculate the length of the height of the equilateral triangle. This would allow us to find the length of a side, giving us all the necessary information to calculate the area of the equilateral triangle.
Statement 2, although giving us information about the difference of the circle and the triangle, we cannot conclude anything because we don't know the proportions of the respective geometric figures. Therefore statement 2 is insufficient.
To conclude, statement one alone is sufficient.
Consider 
.
I) 
.
II) Side 
light years long.
What is the area of 
?
Consider
I)
II) Side
What is the area of
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To find the area of a triangle, we need the base and the height. Taking both statements together we can solve this problem.
I) tells us that OHT is an equilateral triangle.
II) gives us one side length. Which means we really know all the side lengths.
We can use either Pythagorean Theorem or 30/60/90 triangle ratios to find the height of OHT.
From there we can find the area.
Thus, both are needed. I) Tells us we have an equilateral, II) Lets us find the height. Both allow us to find the area.
To find the area of a triangle, we need the base and the height. Taking both statements together we can solve this problem.
I) tells us that OHT is an equilateral triangle.
II) gives us one side length. Which means we really know all the side lengths.
We can use either Pythagorean Theorem or 30/60/90 triangle ratios to find the height of OHT.
From there we can find the area.
Thus, both are needed. I) Tells us we have an equilateral, II) Lets us find the height. Both allow us to find the area.
Find the area of an equilateral triangle.
- A side measures
.
- An angle measures
.
Find the area of an equilateral triangle.
- A side measures
- An angle measures
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In an equilateral triangle all sides are of the same length and all internal angles measure to 
.
Statement 1: 
Where 
represents the length of the side.
If we're given the side, we can calculate the area:
Statement 2: We don't need the angle to find the area.
In an equilateral triangle all sides are of the same length and all internal angles measure to
Statement 1:
Where
If we're given the side, we can calculate the area:
Statement 2: We don't need the angle to find the area.
Given an equilateral triangle 
and Square 
, which, if either, has the greater area?
Statement 1: 
and 
are located on the same circle.
Statement 2: Each diagonal of Square 
has length 
times the height of .
Given an equilateral triangle
Statement 1:
Statement 2: Each diagonal of Square
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Assume Statement 1 alone. Since the vertices of both the triangle and the square are on the same circle, then the same circle is circumscribed about both polygons. For the sake of simplicity, we will assume the radius of that circle to be 1 (and the diameter to be 2); this argument will work regardless of the size of the circle.
A diagonal of a square doubles as a diameter of the circumscribed circle, so each diagonal of Square 
is 2; since a Square is a rhombus, the area is one half the product of the lengths of the diagonals, or 
.
Now examine the figure below, which shows , its altitudes, and its circumscribed circle:
The three altitudes of an equilateral triangle meet at the center of the circumscribed circle, or circumcenter, so 
; they also divide one another into segments such that one has twice the length of the other, so 
. Therefore, 
. Also, the six smaller triangles are all 30-60-90 by symmetry, so by the 30-60-90 Theorem, 
, and 
.
Therefore, the area of the triangle is
,
which can be shown to be less than the square's area of 2.
Assume Statement 2 alone. Again, for simplicity's sake, we will use 
, so we can keep the area of the equilateral triangle as before, 
; this argument works regardless of the dimensions. The length of a diagonal of the square will as before be 
, and its area will again be 2.
Assume Statement 1 alone. Since the vertices of both the triangle and the square are on the same circle, then the same circle is circumscribed about both polygons. For the sake of simplicity, we will assume the radius of that circle to be 1 (and the diameter to be 2); this argument will work regardless of the size of the circle.
A diagonal of a square doubles as a diameter of the circumscribed circle, so each diagonal of Square
Now examine the figure below, which shows
The three altitudes of an equilateral triangle meet at the center of the circumscribed circle, or circumcenter, so
Therefore, the area of the triangle is
which can be shown to be less than the square's area of 2.
Assume Statement 2 alone. Again, for simplicity's sake, we will use
Give the area of equilateral triangle 
.
Statement 1: 
is a diameter of a circle with circumference 
.
Statement 2: 
is a side of a 45-45-90 triangle with area 
.
Give the area of equilateral triangle
Statement 1:
Statement 2:
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Assume Statement 1 alone. To find the diameter of a circle with circumference 
, divide the circumference by 
to get 
. This is also the length of each side of the triangle, so we can get the area using the area formula:
.
Assume Statement 2 alone. A 45-45-90 Triangle has congruent legs, and the area is half the product of their lengths, so if we let 
be the common sidelength,
By the 45-45-90 Theorem, the hypotenuse has length 
times this, or 
.
Since it is not given whether 
is a leg or the hypotenuse of a right triangle, however, the length of 
- and consequently, the area - is not clear.
Assume Statement 1 alone. To find the diameter of a circle with circumference
Assume Statement 2 alone. A 45-45-90 Triangle has congruent legs, and the area is half the product of their lengths, so if we let
By the 45-45-90 Theorem, the hypotenuse has length
Since it is not given whether
Given three equilateral triangles 
, 
, and 
, which has the greatest area?
Statement 1: 
Statement 2: 
Given three equilateral triangles
Statement 1:
Statement 2:
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Assume Statement 1 alone to be true. Since 
, it follows that the area of , which is 
, is greater than that of 
, which is 
. However, no information is given about . If Statement alone is assumed, it similarly follows that has greater area than , but nothing is known about the area of . If both Statements are given, however, then, by transitivity, has the greatest area of the three.
Assume Statement 1 alone to be true. Since
Given two equilateral triangles 
and 
, which has the greater area?
Statement 1: 
is the midpoint of 
.
Statement 2: 
is the midpoint of 
.
Given two equilateral triangles
Statement 1:
Statement 2:
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Neither statement alone is enough to determine which triangle has the greater area, 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 
; it follows that is the triangle with the greater area.
Neither statement alone is enough to determine which triangle has the greater area, as each statement gives information about only one point.
Assume both statements to be true. Since
Which figure, if either, has the greater area: equilateral triangle 
or a given circle with center 
?
Statement 1: The midpoint of 
is on the circle.
Statement 2: 
is outside the circle.
Which figure, if either, has the greater area: equilateral triangle
Statement 1: The midpoint of
Statement 2:
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The area formula for an equilateral triangle is 
; that of a circle is 
. Both will be used here.
Assume Statement 1 alone. If we let 
be the radius of the circle, then, since the points on the circle include the midpoint of 
, the distance from center 
to that midpoint is 
, and the length of 
is 
. The area of the circle is 
, and that of the triangle is
.
Since 
, the circle has the greater area.
Assume Statement 2 alone. For simplicity's sake, assume that the triangle has sidelength 1. Then its area is 
. Since we only know that 
is the center of the circle and 
is outside it, it follows that the radius must be less than 1. This means that the area of the circle must be less than
Since the area falls in the range 
, and 
, we cannot tell whether the circle or the triangle has the greater area.
The area formula for an equilateral triangle is
Assume Statement 1 alone. If we let
Since
Assume Statement 2 alone. For simplicity's sake, assume that the triangle has sidelength 1. Then its area is
Since the area falls in the range
Which, if either, of equilateral triangles 
and 
, has the greater area?
Statement 1: 
Statement 2: 
Which, if either, of equilateral triangles
Statement 1:
Statement 2:
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Since the area of an equilateral triangle is wholly dependent on its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, if either, has the greater area.
If we let 
and 
be the common sidelengths of and , respectively, Statements 1 and 2, respectively, can be rewritten as:
Statement 1: 
Statement 2: 
.
The question is now whether 
, 
, or 
.
Statement 1 alone is insufficient to determine which sidelength is greater; both 
and 
are easily seen to be solutions, with 
in the first case, and 
in the second. Consequently, it is possible for either triangle to have the greater sidelength and the greater area.
Statement 2 alone, however, is sufficient. If 
, if follows that
and
This means that 
. has the greater sidelength and, consequently, the greater area.
Since the area of an equilateral triangle is wholly dependent on its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, if either, has the greater area.
If we let
Statement 1:
Statement 2:
The question is now whether
Statement 1 alone is insufficient to determine which sidelength is greater; both
Statement 2 alone, however, is sufficient. If
and
This means that
Given two equilateral triangles 
and 
, which has the greater area?
Statement 1: 
Statement 2: 
Given two equilateral triangles
Statement 1:
Statement 2:
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Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area.
Statement 1 alone gives that the length of one side of is greater than that on one side of .It follows that has the greater area.
Statement 2 alone gives that 
, from which it follows that
and
.
Again, this shows that has the greater sidelength and the greater area.
Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area.
Statement 1 alone gives that the length of one side of
Statement 2 alone gives that
and
Again, this shows that
Given two equilateral triangles and , which, if either, has the greater area?
Statement 1: 
Statement 2: 
Given two equilateral triangles
Statement 1:
Statement 2:
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Let 
and 
be the sidelengths of and , respectively; the statements can be rewritten as:
Statement 1: 
Statement 2: 
Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area. The question can therefore be reduced to asking which of 
and 
, if either, is greater.
Assume Statement 1 alone. Then
,
and has the greater sidelength. It follows that its area is the greater as well.
Assume Statement 2 alone. Then
or
.
However, this is not enough to prove which triangle, if either, has the greater sidelength; if 
, for example, 
or 
would make this inequality true. Therefore, it is not clear whether 
or 
, if either, is greater, and it is not clear which triangle has the greater sidelength - and area.
Let
Statement 1:
Statement 2:
Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area. The question can therefore be reduced to asking which of
Assume Statement 1 alone. Then
and
Assume Statement 2 alone. Then
or
However, this is not enough to prove which triangle, if either, has the greater sidelength; if
Which, if either, of equilateral triangles 
and 
, has the greater area?
Statement 1: 
Statement 2: 
Which, if either, of equilateral triangles
Statement 1:
Statement 2:
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If we let and be the sidelengths of and , respectively; the statements can be rewritten as:
Statement 1: 
Statement 2: 
Since the area of an equilateral triangle is dependent solely upon the length of each side, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater area. 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, as we see by examining these two scenarios:
Case 1: 
Case 2: 
Both cases satisfy Statement 1, but in the first case, 
, meaning that has greater sidelength and area 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 second equation by 
and add to the first:
Now substitute back:
The sidelengths are the same, and, consequently, so are the areas of the triangles.
If we let
Statement 1:
Statement 2:
Since the area of an equilateral triangle is dependent solely upon the length of each side, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater area. The question can therefore be reduced to asking which of
Statement 1 alone is not sufficient to yield an answer, as we see by examining these two scenarios:
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 second equation by
Now substitute back:
The sidelengths are the same, and, consequently, so are the areas of the triangles.
What is the area of ?
(1) The height is 5.
(2) The base is 4.
What is the area of
(1) The height
(2) The base
Tap to see back →
To find an area of a triangle we need the length of the height and the length of the corresponding basis.
Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.
Hence, both statements together are sufficient.
To find an area of a triangle we need the length of the height and the length of the corresponding basis.
Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.
Hence, both statements together are sufficient.
is an equilateral triangle inscribed in the circle. What is the area of triangle ?
(1) The area of the circle is .
(2) The area of the circle minus the area of the triangle is .
(1) The area of the circle is
(2) The area of the circle minus the area of the triangle is
Tap to see back →
To answer this question, we should be able to have information about any lengths in the circle or in the triangle, or also about the areas of the different regions.
Statement 1 alone is sufficient, since from the area of the circle we can find the length of the radius, which allows us to calculate the length of the height of the equilateral triangle. This would allow us to find the length of a side, giving us all the necessary information to calculate the area of the equilateral triangle.
Statement 2, although giving us information about the difference of the circle and the triangle, we cannot conclude anything because we don't know the proportions of the respective geometric figures. Therefore statement 2 is insufficient.
To conclude, statement one alone is sufficient.
To answer this question, we should be able to have information about any lengths in the circle or in the triangle, or also about the areas of the different regions.
Statement 1 alone is sufficient, since from the area of the circle we can find the length of the radius, which allows us to calculate the length of the height of the equilateral triangle. This would allow us to find the length of a side, giving us all the necessary information to calculate the area of the equilateral triangle.
Statement 2, although giving us information about the difference of the circle and the triangle, we cannot conclude anything because we don't know the proportions of the respective geometric figures. Therefore statement 2 is insufficient.
To conclude, statement one alone is sufficient.
Consider .
I) .
II) Side light years long.
What is the area of ?
Consider
I)
II) Side
What is the area of
Tap to see back →
To find the area of a triangle, we need the base and the height. Taking both statements together we can solve this problem.
I) tells us that OHT is an equilateral triangle.
II) gives us one side length. Which means we really know all the side lengths.
We can use either Pythagorean Theorem or 30/60/90 triangle ratios to find the height of OHT.
From there we can find the area.
Thus, both are needed. I) Tells us we have an equilateral, II) Lets us find the height. Both allow us to find the area.
To find the area of a triangle, we need the base and the height. Taking both statements together we can solve this problem.
I) tells us that OHT is an equilateral triangle.
II) gives us one side length. Which means we really know all the side lengths.
We can use either Pythagorean Theorem or 30/60/90 triangle ratios to find the height of OHT.
From there we can find the area.
Thus, both are needed. I) Tells us we have an equilateral, II) Lets us find the height. Both allow us to find the area.
Find the area of an equilateral triangle.
- A side measures .
- An angle measures .
Find the area of an equilateral triangle.
- A side measures
- An angle measures
Tap to see back →
In an equilateral triangle all sides are of the same length and all internal angles measure to .
Statement 1:
Where represents the length of the side.
If we're given the side, we can calculate the area:
Statement 2: We don't need the angle to find the area.
In an equilateral triangle all sides are of the same length and all internal angles measure to
Statement 1:
Where
If we're given the side, we can calculate the area:
Statement 2: We don't need the angle to find the area.
Given an equilateral triangle and Square , which, if either, has the greater area?
Statement 1: and are located on the same circle.
Statement 2: Each diagonal of Square has length times the height of .
Given an equilateral triangle
Statement 1:
Statement 2: Each diagonal of Square
Tap to see back →
Assume Statement 1 alone. Since the vertices of both the triangle and the square are on the same circle, then the same circle is circumscribed about both polygons. For the sake of simplicity, we will assume the radius of that circle to be 1 (and the diameter to be 2); this argument will work regardless of the size of the circle.
A diagonal of a square doubles as a diameter of the circumscribed circle, so each diagonal of Square is 2; since a Square is a rhombus, the area is one half the product of the lengths of the diagonals, or .
Now examine the figure below, which shows , its altitudes, and its circumscribed circle:
The three altitudes of an equilateral triangle meet at the center of the circumscribed circle, or circumcenter, so ; they also divide one another into segments such that one has twice the length of the other, so . Therefore, . Also, the six smaller triangles are all 30-60-90 by symmetry, so by the 30-60-90 Theorem, , and .
Therefore, the area of the triangle is
,
which can be shown to be less than the square's area of 2.
Assume Statement 2 alone. Again, for simplicity's sake, we will use , so we can keep the area of the equilateral triangle as before, ; this argument works regardless of the dimensions. The length of a diagonal of the square will as before be , and its area will again be 2.
Assume Statement 1 alone. Since the vertices of both the triangle and the square are on the same circle, then the same circle is circumscribed about both polygons. For the sake of simplicity, we will assume the radius of that circle to be 1 (and the diameter to be 2); this argument will work regardless of the size of the circle.
A diagonal of a square doubles as a diameter of the circumscribed circle, so each diagonal of Square
Now examine the figure below, which shows
The three altitudes of an equilateral triangle meet at the center of the circumscribed circle, or circumcenter, so
Therefore, the area of the triangle is
which can be shown to be less than the square's area of 2.
Assume Statement 2 alone. Again, for simplicity's sake, we will use
Give the area of equilateral triangle .
Statement 1: is a diameter of a circle with circumference .
Statement 2: is a side of a 45-45-90 triangle with area .
Give the area of equilateral triangle
Statement 1:
Statement 2:
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Assume Statement 1 alone. To find the diameter of a circle with circumference , divide the circumference by to get . This is also the length of each side of the triangle, so we can get the area using the area formula:
.
Assume Statement 2 alone. A 45-45-90 Triangle has congruent legs, and the area is half the product of their lengths, so if we let be the common sidelength,
By the 45-45-90 Theorem, the hypotenuse has length times this, or .
Since it is not given whether is a leg or the hypotenuse of a right triangle, however, the length of - and consequently, the area - is not clear.
Assume Statement 1 alone. To find the diameter of a circle with circumference
Assume Statement 2 alone. A 45-45-90 Triangle has congruent legs, and the area is half the product of their lengths, so if we let
By the 45-45-90 Theorem, the hypotenuse has length
Since it is not given whether
Given three equilateral triangles , , and , which has the greatest area?
Statement 1:
Statement 2:
Given three equilateral triangles
Statement 1:
Statement 2:
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Assume Statement 1 alone to be true. Since , it follows that the area of , which is , is greater than that of , which is . However, no information is given about . If Statement alone is assumed, it similarly follows that has greater area than , but nothing is known about the area of . If both Statements are given, however, then, by transitivity, has the greatest area of the three.
Assume Statement 1 alone to be true. Since