Download Discuss on Areas of Triangles

Discuss on Areas of Triangles
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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. (The letterK 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.
Two sides and the included angle (SAS): Given Δ ABC (Figure ), the height is
given by h = c sinA. Therefore,
Figure 1
Reference triangles for area formulas.
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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:
Similarly,
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. This is known as Heron's formula. If a, b, and c are
the lengths of the three sides of a triangle, and s is the semiperimeter, then
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and
One of many proofs of Heron's formula starts out with the Law of Cosines:
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Example 1: (SAS) As shown in Figure 2 , two sides of a triangle have measures
of 25 and 12. The measure of the included angle is 51° Find the area of the
triangle.
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Figure 2
Drawing for Example 1.
Use the SAS formula:
Example 2: (AAS and ASA) Find the area of the triangle shown in Figure 3 .
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Figure 3
Drawing for Example 2.
First find the measure of the third angle of the triangle since all three angles are
used in the area formula.
Example 3: (AAS orASA) Find the area of an equilateral triangle with a
perimeter of 78.
If the perimeter of an equilateral triangle is 78, then the measure of each side is
26. The nontrigonometric solution of this problem yields an answer of
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The trigonometric solution yields the same answer.
Example 4: (SSS) Find the area of a triangle if its sides measure 31, 44, and
60.
Use Heron's formula:
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Heron's formula does not use trigonometric functions directly, but trigonometric
functions were used in the development and proof of the formula.
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