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Mathematics I (Math 1132)

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Durham College, Mathematics
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Mathematics I (Math 1132)

Calculate the areas and volumes of different solids

Prisms

A prism is a polyhedron with two parallel, identical faces (called bases), and whose remaining faces (called lateral faces) are parallelograms.

Examples of Prisms

The lateral faces are formed by joining corresponding vertices of the bases. The altitude (height) of a prism is the perpendicular distance between the bases. To find the volume of any prism, you use the following general formula (below).

Volume=(area of base)(altitude)
  • Depending on the prism’s base, the “area of base” part will change.
  • It’s best to use the word altitude to designate the prism’s overall length because using the word height, for instance, could be confused with a base dimension, especially if it’s a triangular prism. Therefore, to prevent ambiguity, altitude is used.

Rectangular Prism

Let’s take a look at the rectangular prism:

 

Specific formulas for its volume and surface area (SA) are shown below. The volume formula can derived by replacing the well-known area of a rectangle formula (length × width) into where you see area of base. A rectangular prism is the only prism where any one of its faces can be considered the prism’s base. In addition, the SA formula is derived by adding up the area of each face. The formula below is currently in factored form, but it’s simply twice the base area plus twice the side areas plus twice the other sides areas.

1   Volume=(area of base)(altitude)Volume=(l×w)(h)V=l·w·h2   Surface area=2lw+hw+lh

Let’s look at two examples where volume and surface area are calculated:

Cubes

If you’re dealing with a cube − technically a square prism – deriving its formula is relatively simple. Remember that a square is an equilateral quadrilateral, where all sides and angles are the same. Therefore, we can assign any letter for one side – the letter a has been chosen below:

1   Volume=(area of base)(altitude)Volume=a·aaV=a32 Surface area=6a2
Surface area of a Cube Explained

Triangle Prism

If the prism base were a triangle, the volume formula would change to the following:

1   Volume=(area of base)(altitude)Volume=b·h2lV=b·h·l22 Surface area=bh+2ls+2bl
  • Recall that the area of triangle is A = ½ × b × h.
  • s stands for slant height (the slant height is the hypotenuse of the right triangle formed by the height and half the base length – shown in the video below).

Lets review an example where these two features are calculated.

Pyramids

Here’s the basic anatomy of a pyramid:

A summary of some important formulas related to pyramids are listed below. The term lateral area describes the surface area of all the faces except for the base. Therefore, if you’re asked to find the surface area, you first calculate the lateral area, and add it to the area of the base that you find individually. The slant height (s) – as opposed to the actual height of the solid – is also used to find the area of each individual face. Sometimes you’ll be forced to find the slant height using the Pythagorean theorem just to find the SA if it’s not provided.

On a similar note, the volume of any pyramid is always 1/3 the volume of its equivalent base prism. For example, if you have a square-based pyramid, its volume will be one third of a prism having the same base and altitude.

A frustum is a portion of a solid (usually a pyramid or a cone) which remains after its upper part has been cut off by a plane parallel to its base.

Let’s review an important problem pertaining to frustum pyramids.

Cylinder, Cone, and Sphere

Cylinder

The volume and surface area formulas of a cylinder are written below:

1   Volume=area of circle basealtitudeV=πr2h2   SA=2πr2+2πr·h

A cylinder is made up of two circle bases and a rectangle. Therefore, to find its SA, the area of its circular base is multiplied by 2 (parts 1 and 3 in the diagram), and the circumference acts as the width of its rectangular sleeve that you use to find its area, after which you sum the three parts.

Cone

A cone is a solid bounded by a plane region (its circular base) and the surface formed by line segments joining a point (the vertex) to points on the boundary of the base. If a perpendicular line can be extended from the center of its base and passes through the vertex, you have a right circular cone (left in diagram); otherwise it is called oblique cone. For the sake of simplicity, we’ll only focus on the former.

Just like with pyramids, cones are always 1/3 the volume of their equivalent cylinder having the same base and altitude.

The volume and lateral area formulas of a cone are provided below. Recall that s stands for slant height and is NOT equivalent to the actual height of the solid.

Sphere

A sphere is a three-dimensional version of a circle; think of it as a solid ball a marble. The two key formulas you need to know are shown below:

Let’s watch a sample problem where these two formulas are used and manipulated to find unknown values.

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