Geometry Study Guide for the Math Basics

Page 4

Solids

Solids are 3-dimensional figures. The three dimensions are length, width, and height. In general, everything that exists (has substance) in the real world is a solid.

Common Solids

While human beings, animals, trees, and cars are solids, they are pretty complicated. In geometry, you’ll learn about some of the common (and simpler) 3-D solids.

Prism

A prism is composed of two parallel congruent faces called the bases. The other faces are rectangles. They are called lateral faces. You name a prism by the shape of its base.

Triangular prism: m-b-geom-s-g-44.jpg

Hexagonal prism: m-b-geom-s-g-45.jpg

Rectangular Prism: m-b-geom-s-g-46.jpg

Then there is the special case of the prism that has all square faces. Pick any two of these sides that are opposite each other, call them its bases, and you have a cube.

Cube: m-b-geom-s-g-47.jpg

Pyramid

A pyramid has one polygonal base. From each vertex are segments that join together at the vertex. Each lateral face is a triangle. Name a pyramid by its base.

Square pyramid: m-b-geom-s-g-48.jpg

Cylinder

A cylinder can be thought of as a circular prism. Think of a soda can or a roll of paper towels. The bases are circles, but there is only one lateral face that unrolls to make a large rectangle.

m-b-geom-s-g-49.jpg

Sphere

A sphere has the same definition as a circle, but for three dimensions: the set of all points equidistant from a point called the center.

Sphere: m-b-geom-s-g-50.jpg

Measuring Solids

There are two main measurements when it comes to solids: surface area and volume.

Surface Area of Prisms, Cylinders, and Pyramids

The surface area is just the sum of areas of the faces. The best way to find the surface area is to draw the net (the two dimensional representation of the figure unfolded) and find the area of each polygon.

Rectangular prism: m-b-geom-s-g-51.jpg

So, the Surface Area =

\[SA=6+6+8+8+12+12\] \[52\;cm^2\]

For cylinders, you can do the same. The length of the rectangular face is just the circumference of the circular base. Remember \(A_\text{circle}=\pi \cdot r^2\) and \(C=2 \pi \cdot r\).

m-b-geom-s-g-52.jpg

So:

\[SA=4\pi + 4\pi + 28\pi\] \[36 \pi \;in^2\]

For pyramids, also use the net to find the area of each face. The net shows the base (square) and the lateral faces (4 congruent triangles).

m-b-geom-s-g-53.jpg

The Area of the Base is \(A=2 \cdot 2 = 4 m^2\).

The area of one of the triangular faces is \(A_\triangle=\frac{1}{2} \cdot 2 \cdot 3 = 3 m^2\).

Since there are 4 triangles, the total surface area is:

\[SA=4\;m^2 + 4\cdot3\;m^2\] \[4 \;m^2 +12\;m^2\] \[16\;m^2\]

Volume of Prisms, Cylinders, and Pyramids

The Volume of the solid is the 3 dimensional space it occupies.

For a prism or cylinder, \(V=B \cdot h\), where \(B= \text{area of the base}\), and \(h=\text{height}\).

m-b-geom-s-g-54.jpg

For this prism:

\(B=2\cdot3=6\) and \(h=4\),

so \(V=6 \cdot 4=24\;cm^3\)

Note: Volume always uses cubic units.

m-b-geom-s-g-55.jpg

For this cylinder:

\(B=\pi \cdot 2^2=4\pi\) and \(h=7\),

so \(V=4\pi \cdot 7 = 28\pi \;in^3\)

The Volume of a pyramid is \(V_\text{pyramid}=\frac{1}{3}B \cdot h\).

m-b-geom-s-g-56.jpg

For this pyramid:

\(B=2\cdot2=4\) and \(h=5\),

so \(V=\frac{1}{3} \cdot 4 \cdot 5 = \frac{20}{3} = 6 \frac{2}{3}\;m^3\).

Surface Area of and Volumes of Spheres

These are formulas worth memorizing:

\[SA_\text{sphere}=4\pi \cdot r^2\] \[V_\text{sphere}=\frac{4}{3}\pi \cdot r^3\]

Let’s calculate surface area and volume of the following sphere.

m-b-geom-s-g-57.jpg

Surface area:

\[SA = 4\pi \cdot 3^2\] \[4\pi \cdot 9 = 36 \pi \;in^2\]

Volume:

\[V=\frac{4}{3}\pi \cdot 3^3\] \[\frac{4}{3}\pi \cdot 27\] \[\frac{108}{3}\pi=36 \pi \;in^3\]

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