Ask someone about trigonometry or geometry and you might get a blank stare or maybe even a nasty glare. For many, these two subjects are not fondly remembered from their youth. They can recognize a triangle, a rectangle, and even a parallelogram but mention the Pythagorean Theorem and their eyes begin to gloss over. Remembering geometry and trigonometry or even learning them to begin with can be a little challenging and sometimes confusing.
There are so many angles, formulas, rules, and theorems to consider and remember. Not to mention the fact that you have not used a protractor or maybe even a calculator for several years. It is no wonder that some of the information gets lost. But now you have a problem. You need to calculate the area of a triangle so that you know how much fabric to order. You remember the formula for the area (Area = ½ x Base x Altitude) but are a little fuzzy about how to determine the base and altitude.
Triangles and Altitudes
Every triangle (equilateral, isosceles, and scalene) has three bases or sides, three altitudes (heights) and as the name suggests three angles. All of the angles within the triangle add up to 180º and any two sides or bases when added together will be longer than the third side or base.
Each of the altitudes is formed by drawing a perpendicular (90º) line from a base to the vertex, or intersecting point, opposite the base. If all the angles in the triangle are less than 90º, an acute triangle, then each of the altitude lines will be within the triangle. However if one of the angles is greater than 90º, an obtuse triangle, two of the bases will have to be extended beyond the triangle in order to draw the altitude line to the opposite vertex. For a right angle triangle, two of the three altitude lines already exist at the 90º (right) angle formed by two of the bases. The third altitude can be drawn from the base opposite of the right angle, the hypotenuse.
The three altitude lines all intersect at the same point. In a right triangle, the three altitude lines intersect at the 90º corner of the triangle. For an acute triangle the intersection will be inside the triangle and for an obtuse triangle the intersection will be outside the triangle. This is a good check to see if the altitude lines have been drawn correctly.
Calculating an Altitude
At least two pieces of information are required to calculate an altitude. If you know the area and the length of the base, then the altitude can be calculated by rearranging the Area formula. If you are working with a right triangle and know the length of the hypotenuse and one of the bases then you can determine the altitude or length of the remaining base using The Pythagorean Theorem. Otherwise, you will need to know the length of one of the bases and the angle formed by that base and an adjacent base. Examples of each are below.
Let’s say that you know the area of the triangle is 36 square inches and one of the sides (bases) has a length of 6 inches. Plug this information into the area formula to get the following:
Area = ½ x BASE x HEIGHT
36 = ½ x 6 x HEIGHT = 3 x HEIGHT
Divide both sides by 3 to get…
HEIGHT (Altitude from base) = 12 inches
If you have a right triangle and know the length of the hypotenuse and one of the sides (bases), then you can find the altitude from that base using the Pythagorean Theorem. A common and very useful right angle triangle for remembering the Pythagorean Theorem is the 3-4-5 triangle in which the two bases that form the right angle have lengths of 3 and 4 and the hypotenuse has a length of 5. Assume you don’t already know the answer but instead have been given just the length of one of the sides, 3 inches, and the length of the hypotenuse, 5 inches. Plug this information into the theorem (formula).
c^2 = a^2 + b^2 (hypotenuse^2 = baseA^2 + baseB^2; ^2 denotes squared)
5^2 = 3^2 + b^2
25 = 9 + b^2
Subtract 9 from both sides to get…
16 = b^2
Take the square root of both sides to get…
b = 4 inches (Altitude from BaseA)
*Base and Angle (from Law of Sines, http://www.cliffsnotes.com/study_guide/Law-of-Sines.topicArticleId-11658,articleId-11575.html) Method*
In many cases you will not know the area and will not be working with a right triangle. Instead you will know the size of the angles and the lengths of the bases. For example, you may know that one of the bases has a length of 9 inches and there is a 60 degree angle between that base and an adjacent base. The altitude in this case is defined by the following:
Altitude (from Adjacent Base) = Base x sine(angle) = 9 x sine(60º)
Altitude (from Adjacent Base) = 9 x 0.866 = 7.794 inches
It is important to note the distinction of where the altitudes are calculated. In the area method, the calculated altitude is from the given base. This is also the case for the Pythagorean Theorem; the altitude is from the given base and not the hypotenuse. However for the base and angle, Law of Sines method, the altitude that is calculated is from the adjacent base to the opposite vertex.
Calculating the altitude of a triangle can be useful in a number of applications ranging from gardening and quilting to engineering and navigation. Each of these methods, when properly applied, will provide you the altitude you require.