Calculating the volume of a tetrahedron - GMAT Quantitative
Card 1 of 56
What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?
What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?
Tap to reveal answer
The volume of a pyramid can be calculated using the formula
where 
is the height and 
is the area of the base.
Since the base is an equilateral triangle, its area can be calculated using the formula
Therefore, the volume can be rewritten as
Substitute 
:
The volume of a pyramid can be calculated using the formula
where
Since the base is an equilateral triangle, its area can be calculated using the formula
Therefore, the volume can be rewritten as
Substitute
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A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.
Give its volume.
A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.
Give its volume.
Tap to reveal answer
The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting 
:
Now, in the formula for the volume of a pyramid, substitute 
:
The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting
Now, in the formula for the volume of a pyramid, substitute
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The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.
The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.
Tap to reveal answer
The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is 
square inches. The volume of a pyramid is one-third the product of its height and the area of its base, so substitute 
in the following:
cubic inches.
The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is
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In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates 
.
What is the volume of this tetrahedron?
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
What is the volume of this tetrahedron?
Tap to reveal answer
The tetrahedron looks like this:
is the origin and 
are the other three points, which are 60 units away from the origin on each of the three (mutually perpendicular) axes.
This is a triangular pyramid, and we can consider 
the (right triangular) base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, and we can consider
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,
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In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates 
.
Give its volume.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume.
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - 
- are on the 
-plane, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 60 and its height is 40, so its area is
The fourth vertex is off the 
-plane; its perpendicular distance to the aforementioned face is its 
-coordinate, 20, so this is the height of the pyramid. The volume of the pyramid is
.
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices -
Its base is 60 and its height is 40, so its area is
The fourth vertex is off the
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In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
,
where 
Give its volume in terms of 
.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
where
Give its volume in terms of
Tap to reveal answer
The tetrahedron looks like this:
is the origin and are the other three points, whose distances away from the origin on each of the three (perpendicular) axes are shown.
This is a triangular pyramid, and we can consider 
the (right triangular) base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 
. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, and we can consider
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is
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A right pyramid has height 
; its base is a square with four sides of length 
each. What is the volume of this pyramid?
A right pyramid has height
Tap to reveal answer
The volume of a pyramid can be calculated using the formula
,
where 
is the area of the base and 
is the height. The base of the pyramid in question is a square; if we let its sidelength be 
, this base will be 
, and the volume formula will be
Setting 
and 
:
or
.
The volume of a pyramid can be calculated using the formula
where
Setting
or
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What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?
What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?
Tap to reveal answer
The volume of a pyramid can be calculated using the formula
where is the height and is the area of the base.
Since the base is an equilateral triangle, its area can be calculated using the formula
Therefore, the volume can be rewritten as
Substitute :
The volume of a pyramid can be calculated using the formula
where
Since the base is an equilateral triangle, its area can be calculated using the formula
Therefore, the volume can be rewritten as
Substitute
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A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.
Give its volume.
A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.
Give its volume.
Tap to reveal answer
The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting :
Now, in the formula for the volume of a pyramid, substitute :
The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting
Now, in the formula for the volume of a pyramid, substitute
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The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.
The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.
Tap to reveal answer
The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is square inches. The volume of a pyramid is one-third the product of its height and the area of its base, so substitute in the following:
cubic inches.
The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is
← Didn't Know|Knew It →
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
What is the volume of this tetrahedron?
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
What is the volume of this tetrahedron?
Tap to reveal answer
The tetrahedron looks like this:
is the origin and are the other three points, which are 60 units away from the origin on each of the three (mutually perpendicular) axes.
This is a triangular pyramid, and we can consider the (right triangular) base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, and we can consider
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,
← Didn't Know|Knew It →
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
Give its volume.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume.
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the -plane, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 60 and its height is 40, so its area is
The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 20, so this is the height of the pyramid. The volume of the pyramid is
.
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices -
Its base is 60 and its height is 40, so its area is
The fourth vertex is off the
← Didn't Know|Knew It →
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
,
where
Give its volume in terms of .
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
where
Give its volume in terms of
Tap to reveal answer
The tetrahedron looks like this:
is the origin and are the other three points, whose distances away from the origin on each of the three (perpendicular) axes are shown.
This is a triangular pyramid, and we can consider the (right triangular) base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is . Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, and we can consider
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is
← Didn't Know|Knew It →
A right pyramid has height ; its base is a square with four sides of length each. What is the volume of this pyramid?
A right pyramid has height
Tap to reveal answer
The volume of a pyramid can be calculated using the formula
,
where is the area of the base and is the height. The base of the pyramid in question is a square; if we let its sidelength be , this base will be , and the volume formula will be
Setting and :
or
.
The volume of a pyramid can be calculated using the formula
where
Setting
or
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What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?
What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?
Tap to reveal answer
The volume of a pyramid can be calculated using the formula
where is the height and is the area of the base.
Since the base is an equilateral triangle, its area can be calculated using the formula
Therefore, the volume can be rewritten as
Substitute :
The volume of a pyramid can be calculated using the formula
where
Since the base is an equilateral triangle, its area can be calculated using the formula
Therefore, the volume can be rewritten as
Substitute
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A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.
Give its volume.
A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.
Give its volume.
Tap to reveal answer
The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting :
Now, in the formula for the volume of a pyramid, substitute :
The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting
Now, in the formula for the volume of a pyramid, substitute
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The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.
The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.
Tap to reveal answer
The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is square inches. The volume of a pyramid is one-third the product of its height and the area of its base, so substitute in the following:
cubic inches.
The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is
← Didn't Know|Knew It →
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
What is the volume of this tetrahedron?
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
What is the volume of this tetrahedron?
Tap to reveal answer
The tetrahedron looks like this:
is the origin and are the other three points, which are 60 units away from the origin on each of the three (mutually perpendicular) axes.
This is a triangular pyramid, and we can consider the (right triangular) base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, and we can consider
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,
← Didn't Know|Knew It →
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
Give its volume.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume.
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the -plane, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 60 and its height is 40, so its area is
The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 20, so this is the height of the pyramid. The volume of the pyramid is
.
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices -
Its base is 60 and its height is 40, so its area is
The fourth vertex is off the
← Didn't Know|Knew It →
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
,
where
Give its volume in terms of .
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
where
Give its volume in terms of
Tap to reveal answer
The tetrahedron looks like this:
is the origin and are the other three points, whose distances away from the origin on each of the three (perpendicular) axes are shown.
This is a triangular pyramid, and we can consider the (right triangular) base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is . Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, and we can consider
The volume of the tetrahedron is one third the product of its base and its height, the latter of which is
← Didn't Know|Knew It →