All SSAT Upper Level Math Resources
Example Questions
Example Question #1 : How To Find The Volume Of A Tetrahedron
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
Give its volume.
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 10 and its height is 18, so its area is
The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its
-coordinate, 8, so this is the height of the pyramid. The volume of the pyramid is
Example Question #2 : How To Find The Volume Of A 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?
The correct answer is not among the other responses.
The tetrahedron looks like this:
is the origin and
are the other three points, which are fifteen units away from the origin on each of the three (perpendicular) axes.
This is a triangular pyramid, and we can consider the 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 15. Therefore,
.
Example Question #3 : How To Find The Volume Of A Tetrahedron
Above is the base of a triangular pyramid, which is equilateral. The height of the pyramid is equal to the perimeter of its base. In terms of , give the volume of the pyramid.
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 height of the pyramid is equal to the perimeter, so it will be three times , or
The volume of the pyramid is one third the product of this area and the height of the pyramid:
Example Question #1 : How To Find The Volume Of A Tetrahedron
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 .
The tetrahedron looks like this:
is the origin and
are the other three points.
This is a triangular pyramid, and we can consider the 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. Therefore,
After some rearrangement:
Example Question #5 : How To Find The Volume Of A Tetrahedron
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 .
The tetrahedron looks like this:
is the origin and
are the other three points, each of which lies along one of the three (mutually perpendicular) axes.
This is a triangular pyramid, and we can consider the base; its area is half the product of its legs, or
.
The volume of the tetrahedron is one third the product of its base area and its height
. Therefore, the volume is
Example Question #1 : How To Find The Volume Of A Tetrahedron
Find the volume of a regular tetrahedron that has a side length of .
Use the following formula to find the volume of a regular tetrahedron:
Now, plug in the given side length.
Example Question #7 : How To Find The Volume Of A Tetrahedron
Find the volume of a regular tetrahedron that has a side length of .
Use the following formula to find the volume of a regular tetrahedron:
Now, plug in the given side length.
Example Question #8 : How To Find The Volume Of A Tetrahedron
Find the volume of a regular tetrahedron with a side length of .
Use the following formula to find the volume of a regular tetrahedron:
Now, plug in the given side length.
Example Question #1 : How To Find The Volume Of A Tetrahedron
Find the volume of a regular tetrahedron with side lengths of .
Use the following formula to find the volume of a regular tetrahedron:
Now, plug in the given side length.
Example Question #10 : How To Find The Volume Of A Tetrahedron
Find the volume of a regular tetrahedron with side lengths of .
Use the following formula to find the volume of a regular tetrahedron:
Now, plug in the given side length.
All SSAT Upper Level Math Resources
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