ISEE Upper Level Math : Geometry
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Example Questions
Example Question #52 : Solid Geometry
There is a sculpture in front of town hall which is shaped like a cube. If it has a volume of 
There is a sculpture in front of town hall which is shaped like a cube. If it has a volume of 
To find the side length of a cube from its volume, simply use the following formula:
Plug in what is known and use some algebra to get our answer:
Example Question #1 : How To Find The Length Of An Edge Of A Cube
You have a cube with a volume of 
You have a cube with a volume of 
If we begin with the formula for volume of a cube, we can work backwards to find the side length.
Making our answer:
Example Question #371 : Geometry
The above cube has surface area 486. Evaluate 
The surface area of a cube is six times the square of the length of each edge, which here is 
Substituting, then solving for 
Since the sidelength is positive,
Example Question #2 : How To Find The Length Of An Edge Of A Cube
You have a crate with equal dimensions (height, length and width). If the volume of the cube is 
Not enough information to solve the equation.
You have a crate with equal dimensions (height, length and width). If the volume of the cube is 
Let's begin with realizing that we are dealing with a cube. A crate with equal dimensions will have equal height, length, and width, so it must be a cube.
With that in mind, we can find our side length by starting with the volume and working backward.
So, to find our side length, we just need to take the cubed root of the volume.
So, our answer is 6 meters, a fairly large crate!
Example Question #1 : How To Find The Length Of An Edge Of A Cube
If the volume of a cube is 30, what must be the length of the edge of the cube?
Write the formula to find the volume of a cube.
Substitute the volume into the equation.
Cube root both sides.
The answer is:
Example Question #1 : How To Find The Length Of An Edge Of A Cube
You have a cube with a volume of
You have a cube with a volume of
If we begin with the formula for volume of a cube, we can work backwards to find the side length.
Making our answer:
Example Question #1 : Tetrahedrons
A triangular pyramid, or tetrahedron, with volume 100 has four vertices with Cartesian coordinates
where 
Evaluate 
The tetrahedron is as follows (figure not to scale):
This is a triangular pyramid with a right triangle with legs 10 and 
The height of the pyramid is 5, so
Set this equal to 100 to get 
Example Question #1 : Tetrahedrons
A triangular pyramid, or tetrahedron, with volume 1,000 has four vertices with Cartesian coordinates
where 
Evaluate 
The tetrahedron is as follows:
This is a triangular pyramid with a right triangle with two legs of measure 
Since the height of the pyramid is also 
Set this equal to 1,000:
Example Question #3 : Tetrahedrons
A triangular pyramid, or tetrahedron, with volume 240 has four vertices with Cartesian coordinates
where 
Evaluate 
The tetrahedron is as follows (figure not to scale):
This is a triangular pyramid with a right triangle with two legs of measure
The height of the pyramid is 24, so the volume is
Set this equal to 240 to get 
Example Question #1 : Tetrahedrons
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 - 
Its base and height are both 18, so its area is
The fourth vertex is off the 
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