# Sss triangle

Calculate the area and heights in the triangle ABC by sides a = 8cm, b = 11cm, c = 12cm

Result

S =  42.789 cm2
h1 =  10.697 cm
h2 =  7.78 cm
h3 =  7.132 cm

#### Solution:

$a = 8 \ cm \ \\ b = 11 \ cm \ \\ c = 12 \ cm \ \\ \ \\ s = (a+b+c)/2 = (8+11+12)/2 = \dfrac{ 31 }{ 2 } = 15.5 \ cm \ \\ S = \sqrt{ s \cdot \ (s-a) \cdot \ (s-b) \cdot \ (s-c) } = \sqrt{ 15.5 \cdot \ (15.5-8) \cdot \ (15.5-11) \cdot \ (15.5-12) } \doteq 42.7895 = 42.789 \ cm^2$
$S = \dfrac{ a \cdot \ h_{ 1 } }{ 2 } \ \\ h_{ 1 } = 2 \cdot \ S / a = 2 \cdot \ 42.7895 / 8 \doteq 10.6973 = 10.697 \ \text{ cm }$
$h_{ 2 } = 2 \cdot \ S / b = 2 \cdot \ 42.7895 / 11 \doteq 7.7798 = 7.78 \ \text{ cm }$
$h_{ 3 } = 2 \cdot \ S / c = 2 \cdot \ 42.7895 / 12 = 7.1315 = 7.132 \ \text{ cm }$

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