 Geometry is a branch of mathematics that relates various known characteristics of a figure to other characteristics of the same or similar figures. These characteristics may be size, shape, angles, and so on. If several characteristics are known, they can be used to calculate other characteristics that are unknown. The Pythagorean theorem is an equation that describes the relationship between the three sides of a right triangle. The theorem states that the sum of the square of the two leg sides is equal to the square of the hypotenuse. The legs are the sides next to the right angle, and the hypotenuse is the side opposite the right angle. When a triangle has one side that is equal to 90 degrees, it's called a right triangle and the Pythagorean theorem can be applied. For example, to solve for c, simply take the square root of both sides. This leaves the equation c equals the square root of a squared plus b squared. If side a equals 3 and side b equals 4, then c will equal 5. The equation can also be rearranged to find either of the other two sides. Here's a good example of the benefit of the Pythagorean theorem. A carpenter is installing new windows in a home. The windows measure 54 inches by 72 inches. The carpenter wants to ensure that the windows are square before securing them in place. To do so, he'll need to measure the diagonal of the window to confirm that it is the right distance and that the window is square. What should the diagonal measure? Let's find out. First, isolate the variable c by taking the square root of both sides. Then substitute in the values for a and b. Finally, reduce the expression and simplify. In this example, the hypotenuse of the triangle should be 90 inches. If the diagonal of the window measures 90, then the carpenter knows the window will be square. In addition to the Pythagorean theorem, triangles have several more unique characteristics that are useful in everyday math. For instance, the three angles of the triangle will always add to be 180 degrees. So, if two of the angles are known, it's easy to calculate the third angle. Another unique characteristic is that the area of a triangle can be calculated by multiplying one half times the base times the height. The base of the triangle can be any one of the three sides, usually drawn at the bottom. The height of the triangle is the length perpendicular to the base. If the base and height are known, the area can be calculated. Finally, for all triangles, if two sides are equal, the angles opposite those sides are also equal. Similarly, if all three sides are equal, all three angles are equal. In addition to the characteristics that all triangles share, right triangles have several characteristics that are quite unique. Before we cover the details of those characteristics, it helps to identify the various parts of a right triangle. A right triangle is any triangle that has one angle equal to 90 degrees, which is called the right angle. The hypotenuse is always the side across from the right angle. When discussing either of the remaining two angles in the triangle, you'll often hear the terms opposite and adjacent applied to the sides of the triangle. The term adjacent identifies the side next to the angle that's not the hypotenuse. The term opposite identifies the side across or opposite from the angle. The terms opposite and adjacent can apply to different sides of the triangle depending upon the angle in question. Sine, cosine, and tangent are the three main functions in trigonometry and are often shortened to sine, cosine, and tan. These functions relate the ratio of the sides of a triangle to its angles. For instance, the sine of an angle is equal to the ratio of its opposite side and hypotenuse. Sine is equal to the ratio of the adjacent side and hypotenuse. And tangent is equal to the opposite side and the adjacent side. Since the sine, cosine, and tangent functions are ratios, the actual length of the sides can vary and produce similar angles. For instance, in this triangle, the sine of angle A is equal to three-fifths. If we doubled or even tripled the size of the triangle, the ratio would still simplify down to three-fifths. Since the value of the ratio is the same for a given angle, regardless of the size of the triangle, these functions are easily determined with a calculator. If any one variable in the equation is unknown, we can use the sine, cosine, and tangent equations to solve for the missing value. For example, if we know that angle A is 35 and the opposite side is 7, we can solve for the hypotenuse using the sine equation. First, rearrange the equation to solve for the hypotenuse. Then, using a calculator, enter 35 degrees into the sine function. Substitute the value into the equation and calculate the results. Similarly, we could solve for the adjacent side using the tangent equation. Since not all triangles are right triangles, it's useful to know how the trigonometric functions sine, cosine, and tangent relate to regular triangles. The law of sines states that the side of a triangle divided by its opposite angle is equal to any other side divided by its opposite angle. So for any given triangle, side A divided by the sine of angle A is equal to side B divided by the sine of angle B, which is equal to side C divided by the sine of angle C. Once again, if several of the values are known, we can rearrange the equations to determine the remaining values that are unknown. For example, if angle A, side B, and angle B are known, what is side A? Rearranging the equations and substituting the values will allow us to solve for side A. Circles also have unique characteristics. Circles have a total of 360 degrees. Their area can be found by the formula A equals pi times the radius squared. Many times, circles are defined by their diameter. In this case, the area is equal to pi times the diameter squared divided by 4. In manufacturing, it's important to know how changes to the diameter affect the area of a circle. If the diameter of the circle is doubled, what will that do to the area? By substituting 2 times D in for the diameter, we can see that when the diameter of a circle is doubled, its area is quadrupled. This is because both the old diameter and its multiplier 2 are being squared. Similarly, if we triple the diameter, the multiplier becomes 3 and the new area will be 9 times larger. When we double the diameter, the multiplier is 2. When we triple the diameter, the multiplier is 3. In general, an increase in the diameter of a circle will result in an increase in the area equivalent to the multiplier squared. A painter was hired to paint the cement floor in a basement. The basement is rectangular in shape but has several large circular posts and a large triangular built-in cabinet. He needs to determine the total square footage he's required to paint so that he can purchase materials. First, let's determine a general formula for the area the painter must paint. The paint area is equal to the total area of the rectangular basement minus the triangular built-in and minus the posts, there are 4. Substituting the formula for the area of a rectangle, triangle and circle gives a final equation of length times width minus one-half base times height minus four times pi times diameter squared divided by four. The painter measures the appropriate distances and now we can substitute the correct numbers and solve for the area. A bolt circle occurs when bolt holes are periodically spaced in a circular pattern. Bolts arranged in this pattern offer the most stability and strength for machined parts and are often used in flanges and wheels. Bolt circle patterns typically have three to eight bolts equally spaced along the circumference of a circle forming a triangle, square, pentagon or other equilateral polygon. The lines that connect two of the holes and the center of the bolt hole circle form a triangle. The center angle of each triangle can be found by dividing the total number of degrees in a circle 360 by the number of bolt holes. For instance, a three bolt circle has a center angle of 360 divided by three which is 120 degrees. A four bolt circle has a center angle of 360 divided by four which is 90 degrees. One of the most common bolt circle patterns is the five bolt pattern found most often on wheels. In manufacturing, it's useful to determine the location of the holes relative to the coordinate axes. To do so, let's first determine the value of the angle between each hole. 360 divided by five equals 72. So each triangle made by two adjoining bolt holes is 72 degrees. Next, we know that the first bolt hole is exactly aligned with the Y axis. Additionally, the value of the angle between the X and Y axis is 90 degrees. The second bolt hole is 72 degrees from the first hole. 90 minus 72 is 18. So the angle between the second hole and the X axis is 18 degrees. The third bolt hole is 72 degrees from the second. However, we've already determined that the angle between the X axis and the second hole is 18. Therefore, the angle between the X axis and the third hole is 18 minus 72 which is negative 54. So the third hole is at an angle of negative 54 from the X axis. The value is negative because it goes below the X axis. Since the holes are equally spaced, we know that the second and fifth holes are at the same angle above the X axis. So the fifth hole is also 18 degrees above the X axis. Similarly, the fourth hole mirrors the third hole and is negative 54 degrees from the X axis. If the radius of the bolt circle and the interior angle between two adjacent bolts is known, then the distance between each bolt can be calculated using the law of signs. If we look closely at the two adjacent bolts, we can see that they form a triangle with the center. We already know the center angle. It's 360 divided by the total number of bolts, 5, which equals 72. We also know the value for two of the triangle sides. Since each bolt hole is on the bolt circle, those sides are equal to the radius of that circle. Next, since the two sides are equal, we know that the angles opposite the two sides are also equal. And finally, since there are 180 degrees in a triangle, one of the angles is 72, and the other two are equal, we can solve for the other two angles. Now that we've collected all the facts about the triangle, we can use the information to solve for the third side, which is the distance between the two bolt holes. Using the law of signs, we can rearrange the equation to solve for the unknown side. Substituting in the values we know, we can then calculate the value of the third side. The value of the third side is equal to the distance between the two bolt holes.