A triangle has sides with lengths: 7, 6, and 8. How do you find the area of the triangle using Heron's formula?

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A triangle has sides with lengths: 7, 6, and 8. How do you find the area of the triangle using Heron's formula?

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Answer 1 · 2015-12-21T13:23:27

Substitute the lengths into the formula and calculate.

Explanation:

Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is
$A = {\sqrt{s (s - a) (s - b) (s - c)}}$ $A={\sqrt {s(s-a)(s-b)(s-c)}}$
where s is the semiperimeter of the triangle; that is,
$s = {\frac{a + b + c}{2}}$ $s={\frac {a+b+c}{2}}$
In this example $s = \frac{7 + 6 + 8}{2} = \frac{21}{2}$ $s = (7 + 6 + 8)/2 = 21/2$
$s - a = \frac{21}{2} - 7 = \frac{21 - 7 \cdot 2}{2} = \frac{7}{2}$ $s - a = 21/2 - 7 = (21- 7*2)/2 = 7/2$
$s - b = \frac{21}{2} - 6 = \frac{21 - 12}{2} - \frac{9}{2}$ $s - b = 21/2 - 6 = (21 - 12)/2 - 9/2$
$s - c = \frac{21}{2} - 8 = \frac{5}{2}$ $s - c = 21/2 - 8 = 5/2$
$A = \sqrt{\frac{21}{2} \cdot \frac{7}{2} \cdot \frac{9}{2} \cdot \frac{5}{2}}$ $A = sqrt(21/2 * 7/2 * 9/2 * 5/2)$
$A = \sqrt{\frac{21 \cdot 7 \cdot 9 \cdot 5}{16}}$ $A = sqrt((21*7*9*5)/16)$
$A = \frac{\sqrt{3 \cdot 7 \cdot 7 \cdot 3 \cdot 3 \cdot 5}}{4}$ $A= sqrt(3*7*7*3*3*5)/4$
$A = 7 \cdot 3 \cdot \frac{\sqrt{3 \cdot 5}}{4}$ $A=7*3*sqrt(3*5)/4$
$A = 21 \cdot \frac{\sqrt{15}}{4}$ $A=21*sqrt(15)/4$

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A triangle has sides with lengths: 7, 6, and 8. How do you find the area of the triangle using Heron's formula?

1 Answer

Explanation:

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