![]() ![]() Heron's formula does not use trigonometric functions directly, but trigonometric functions were used in the development and proof of the formula. The trigonometric solution yields the same answer.Įxample 4: (SSS) Find the area of a triangle if its sides measure 31, 44, and 60. The nontrigonometric solution of this problem yields an answer of If the perimeter of an equilateral triangle is 78, then the measure of each side is 26. ![]() The measure of the included angle is 51° Find the area of the triangle.Įxample 2: (AAS and ASA) Find the area of the triangle shown in Figure 3 .įirst find the measure of the third angle of the triangle since all three angles are used in the area formula.Įxample 3: (AAS orASA) Find the area of an equilateral triangle with a perimeter of 78. One of many proofs of Heron's formula starts out with the Law of Cosines:Įxample 1: (SAS) As shown in Figure 2 , two sides of a triangle have measures of 25 and 12. If a, b, and c are the lengths of the three sides of a triangle, and s is the semiperimeter, then Three sides (SSS): A famous Greek philosopher and mathematician, Heron (or Hero), developed a formula that calculates the area of triangles given only the lengths of the three sides. The geometric mean of Y from a sample is computed as where is the sum of the weights over all observations in the data set. Two angles and a side (AAS) or (ASA): Using the Law of Sines and substituting in the preceding three formulas leads to the following formulas: Two sides and the included angle (SAS): Given Δ ABC (Figure ), the height is given by h = c sinA. (The letter K is used for the area of the triangle to avoid confusion when using the letter A to name an angle of a triangle.) Three additional categories of area formulas are useful. The most common formula for finding the area of a triangle is K = ½ bh, where K is the area of the triangle, b is the base of the triangle, and h is the height. Graphs: Special Trigonometric Functions. ![]()
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