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Volume of Geometrical Shapes

Video Lesson on Geometry

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Rectangular Solids

A rectangular solid is a three-dimensional object composed of rectangular sides. Each of the six rectangular sides is a face. The solid and dotted lines are the edges. Opposing faces are parallel and have the same length and width. If all of the six sides are of equal length, the rectangular solid is called a cube.

Volume = (length)(width)(height)

Volume = lwh

Consider the following example:

What is the volume of the figure above if DJ = 2, GJ = 4, and DB = 3?

Volume = lwh

V = 2 × 4 × 3 = 24

Surface Area

Surface Area = 2(length)(width) + 2(height)(width) + 2(length)(height)

SA = 2lw + 2hw + 2lh

In other words, the surface area is equal to the summation of the area of each of the faces. For example, if a rectangular solid had a length of 2, height of 4, and width of 3, then SA = 2(2)(3) + 2(4)(3) + 2(2)(4)

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Right Cylinders

A right circular cylinder has two circular bases of equal size whose centers are both perpendicularly intersected by the same line. In the diagram below, the surface with point P and the surface with point Q are both the same size. Surface P and surface Q are both perpendicularly intersected by line PQ, making these surfaces parallel.

Volume = (area of base) × (height)

V = pr2h where r is the radius, which is QR in the above diagram

In other words, the volume is equal to the area of the base multiplied by the height.

Consider the following example:

What is the area of the figure above if PQ = 5 and QR = 3?

V = (p32)(5) = 9p(5) = 45p

Surface Area

Surface Area = 2(pi)(radius)2 + 2(pi)(radius)(height)

SA = 2(pr2) + 2prh

In other words, the surface area is equal to the summation of the area of each of the faces. For example, if a right circular cylinder had a radius of 3 and height of 5, then SA = 2(p × 9) + 2p × 3 × 5 = 18p + 30p = 48p

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