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Example Questions
Example Question #2 : Solid Geometry
In cubic inches, find the volume of a tetrahedron that has a surface area of .
First, we will need to find the length of a side of the tetrahedron.
We can use the surface area to find the lengh of a side. Recall that the formula to find the surface of a tetrahedron:
, where is the side length.
Now, recall the formula to find the volume of a tetrahedron:
Example Question #5 : Solid Geometry
Susan bought a chocolate bar that came in a container shaped like a triangular prism shown below. If the container is completely filled with chocolate, in cubic inches, what volume of chocoate did Susan buy?
The volume of a right triangular prism is given by the following equation:
Example Question #2 : Other Polyhedrons
Troy's company manufactures dice that are shaped like cubes and have side lengths of . If the plastic needed to make the dice costs per cubic centimeter, how much does it cost Troy to make one die?
First, find the volume of the die. For a cube, the volume has the following formula:
Because it costs for each cubic centimeter, you will need to multiply this number by to get the cost of each die.
Example Question #1 : How To Find The Volume Of A Polyhedron
A pyramid is placed inside a cube so that they share a base and height. If the surface area of the cube is , what is the volume of the pyramid, in square feet?
First, we need to find the length of a side for the cube.
Recall that the surface area of the cube is given by the following equation:
, where is the length of a side.
Plugging in the surface area given by the equation, we can then find the side length of the cube.
Now, because the pyramid and the cube share a base, we know that the pyramid must be a square pryamid.
Recall how to find the volume of a pyramid:
Now, since the pyramid is the same height as the cube, the height of the pyramid is also .
Example Question #11 : Solid Geometry
In cubic feet, find the volume of the pentagonal prism illustrated below. The pentagon has an area of , and the prism has a height of .
For any prism, the volume is given by the following equation:
The question gives us the area of the base and the height.
Example Question #641 : Geometry
If the side lengths of a cube are tripled, what effect will it have on the volume?
The volume will be times as large.
The volume will be times as large.
The volume will be times as large.
The volume will be times as large.
The volume will be times as large.
Start by taking a cube that is . The volume of this cube is .
Next, triple the sides of this cube so that it becomes . The volume of this cube is .
The volume of the new cube is times as large as the original.
Example Question #641 : Geometry
The surface area of a cylinder is given by , where is the radius and is the height. If a cylinder has a surface area of and a height of , what is its radius?
The fastest way to solve this problem is to plug all of the answer choices in for and look for an output of .
Alternatively, one could set the surface area formula equal to (knowing that ), but this would require solving a quadratic.
Rearranging this we get
Now factoring out a we get:
Since we cannot have a negative length our answer is .
Example Question #1 : How To Find The Surface Area Of A Polyhedron
If a half cylinder with a height of 5 and semicircular bases with a radius of 2, what is the surface area?
The surface area of the half cylinder will consist of the lateral area and the base area.
There are three parts to the surface area:
Rectangular region, semi-circular bases of the half cylinder, and outer face of the cylinder.
The cross section of the half cylinder is a rectangle. Find the area of the rectangle. The diameter of the semicircle represents the width of the rectangle, which is double the radius. The length of the rectangle is the height of the cylinder.
Next, find the area of a semicircular bases.
Since there are two semicircular bases of the semi-cylinder, the total area of the semicircular bases is .
Find the area of the outer region. The area of the outer region is the half circumference multiplied by the height of the cylinder.
Sum the areas of the rectangle, the two circular bases, and the outer region to find the lateral area.
Example Question #561 : Geometry
Our backyard pool holds 10,000 gallons. Its average depth is 4 feet deep and it is 10 feet long. If there are 7.48 gallons in a cubic foot, how wide is the pool?
133 ft
33 ft
7.48 ft
100 ft
30 ft
33 ft
There are 7.48 gallons in cubic foot. Set up a ratio:
1 ft3 / 7.48 gallons = x cubic feet / 10,000 gallons
Pool Volume = 10,000 gallons = 10,000 gallons * (1 ft3/ 7.48 gallons) = 1336.9 ft3
Pool Volume = 4ft x 10 ft x WIDTH = 1336.9 cubic feet
Solve for WIDTH:
4 ft x 10 ft x WIDTH = 1336.9 cubic feet
WIDTH = 1336.9 / (4 x 10) = 33.4 ft
Example Question #562 : Geometry
A cube has a volume of 64cm3. What is the area of one side of the cube?
16cm2
16cm
4cm2
16cm3
4cm
16cm2
The cube has a volume of 64cm3, making the length of one edge 4cm (4 * 4 * 4 = 64).
So the area of one side is 4 * 4 = 16cm2
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