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Here are a few recommended readings before getting started with this lesson.
Prisms can be categorized by the shape of their bases. For example, a prism with a triangular base is called a triangular prism. A rectangular prism, on the other hand, has a rectangular base.
Determine whether the given solid is a triangular prism, a rectangular prism, or neither.
There are two important characteristics that give information about 3D objects.
The volume of a solid is the measure of the amount of space inside the solid. It is the three-dimensional equivalent of the area of the figure. Volume is measured using cubic units, such as cubic meters, or m3. The applet below illustrates the volume of some solids. Move the slider to fill the solids.
The surface area SA of a three-dimensional shape is the total area of all the surfaces of the shape. The lateral area LA is the total area of all lateral surfaces, the surfaces of the shape that are not bases. Lateral area can be defined for polyhedrons, cones, and cylinders. In these instances, the surface area of the shape can be calculated by the formula below.
The net of a solid can be helpful to visualize the surfaces of a three-dimensional figure.
Consider a prism with a base area B and height h.
The volume of the prism is calculated by multiplying the base's area by its height.
V=Bh
By Cavalieri's principle, this formula holds true for all types of prisms, including skewed ones.
Recall that the volume of a solid is the measure of the amount of space inside the solid. Note that the top and bottom faces of the prism are always the same and parallel to each other.
Additionally, the prism is, so to speak, filled
with identical base areas that are stacked on top of each other to the height of the prism.
Consider a prism with a height h, base area B, and base perimeter P.
The surface area of the prism is the sum of the two base areas and the lateral area, which can be calculated as the product of the base perimeter and the height of the prism.
SA=2B+Ph
Note that although this proof is written for a regular prism, it is also true for a non-regular prism.
It is possible to derive two specific formulas for the volume and surface area of a rectangular prism.
Consider a rectangular prism with a width w, length ℓ, and height h.
The volume of the prism is calculated by multiplying the area of the base by the height of the prism.
V=wℓh
Consider a rectangular prism with a height h, base area B, and base perimeter P.
The surface area of the rectangular prism can be found using the following formula.
SA=2(wℓ+hℓ+hw)
P=2(w+ℓ), B=wℓ
Distribute h
Factor out 2
Consider a cube with a side length s.
The volume V of the cube can be calculated by raising the side length s to the power of 3, or cubing
it.
V=s3
Consider a cube with a side length s.
The surface area of the cube is given by the following formula.
SA=6s2
Substitute values
a⋅a=a2
Add terms
Substitute values
Multiply
Add terms
Multiply
Find the width of the prism by using the known volume of the prism and the formula for the volume of a rectangular prism. Then, use the formula for the surface area of a rectangular prism.
Substitute values
Multiply
Commutative Property of Multiplication
Rearrange equation
LHS/2.8=RHS/2.8
Use a calculator
Substitute values
Multiply
Add terms
Multiply
b=7, h=2.3
Multiply
b1⋅a=ba
Calculate quotient
Substitute values
Multiply
Add terms
Use the formula for the surface area of a prism and the given surface area to find the height of the prism. Then, find the volume of the prism.
b=4, h=2.7
Multiply
b1⋅a=ba
Calculate quotient
Substitute values
Multiply
LHS−10.8=RHS−10.8
Rearrange equation
LHS/11.5=RHS/11.5
Calculate the volume or surface area of the triangular or rectangular prism given the base area B, the height, and the side length of the base. The base is always a regular polygon. Round the answer to one decimal place if necessary.