How to find the volume of a tetrahedron - ISEE Upper Level: Quantitative Reasoning
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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
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - 
- are on the horizontal plane 
, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the 
-coordinates, 
, 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 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
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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 twelve units away from the origin, each on one of the three (mutually perpendicular) axes.
This is a triangular pyramid, so look at 
as its base; the area 
of the base is half the product of its legs, or
.
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, so look at
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
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Above is the base of a triangular pyramid, which is equilateral. 
, and the pyramid has height 30. What is the volume of the pyramid?
Above is the base of a triangular pyramid, which is equilateral.
Tap to reveal answer
Altitude 
divides 
into two 30-60-90 triangles.
By the 30-60-90 Theorem, 
, or
is the midpoint of 
, so 
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
Altitude
By the 30-60-90 Theorem,
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
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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 terms of .
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - 
- are on the horizontal plane 
, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles (drawing not to scale):
Its base is 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, 
, 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 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← 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 and height are both 18, so its area is
The fourth vertex is off the 
-plane; its perpendicular distance to the aforementioned face is its -coordinate, 9, 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 and height are both 18, 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
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , 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 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← 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 twelve units away from the origin, each on one of the three (mutually perpendicular) axes.
This is a triangular pyramid, so look at as its base; the area of the base is half the product of its legs, or
.
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, so look at
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
← Didn't Know|Knew It →
Above is the base of a triangular pyramid, which is equilateral. , and the pyramid has height 30. What is the volume of the pyramid?
Above is the base of a triangular pyramid, which is equilateral.
Tap to reveal answer
Altitude divides into two 30-60-90 triangles.
By the 30-60-90 Theorem, , or
is the midpoint of , so
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
Altitude
By the 30-60-90 Theorem,
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
← 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 terms of .
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles (drawing not to scale):
Its base is 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , 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 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← 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 and height are both 18, so its area is
The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 9, 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 and height are both 18, 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 .
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 twelve units away from the origin, each on one of the three (mutually perpendicular) axes.
This is a triangular pyramid, so look at as its base; the area of the base is half the product of its legs, or
.
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, so look at
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
← Didn't Know|Knew It →
Above is the base of a triangular pyramid, which is equilateral. , and the pyramid has height 30. What is the volume of the pyramid?
Above is the base of a triangular pyramid, which is equilateral.
Tap to reveal answer
Altitude divides into two 30-60-90 triangles.
By the 30-60-90 Theorem, , or
is the midpoint of , so
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
Altitude
By the 30-60-90 Theorem,
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
← 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 terms of .
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles (drawing not to scale):
Its base is 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , 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 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← 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 and height are both 18, so its area is
The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 9, 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 and height are both 18, 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
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , 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 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← 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 twelve units away from the origin, each on one of the three (mutually perpendicular) axes.
This is a triangular pyramid, so look at as its base; the area of the base is half the product of its legs, or
.
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
.
The tetrahedron looks like this:
This is a triangular pyramid, so look at
The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,
← Didn't Know|Knew It →
Above is the base of a triangular pyramid, which is equilateral. , and the pyramid has height 30. What is the volume of the pyramid?
Above is the base of a triangular pyramid, which is equilateral.
Tap to reveal answer
Altitude divides into two 30-60-90 triangles.
By the 30-60-90 Theorem, , or
is the midpoint of , so
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
Altitude
By the 30-60-90 Theorem,
The area of the triangular base is half the product of its base and its height:
The volume of the pyramid is one third the product of this area and the height of the pyramid:
← 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 terms of .
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles (drawing not to scale):
Its base is 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , 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 20 and its height is 9, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← 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 and height are both 18, so its area is
The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 9, 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 and height are both 18, 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
Give its volume in terms of
Tap to reveal answer
A tetrahedron is a triangular pyramid and can be looked at as such.
Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:
Its base is 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , 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 12 and its height is 15, so its area is
The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the
← Didn't Know|Knew It →