# How do you use Heron's formula to find the area of a triangle with sides of lengths 7 , 4 , and 9 ?

Feb 7, 2016

$A r e a = 13.416$ square units

#### Explanation:

Heron's formula for finding area of the triangle is given by
$A r e a = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

Where $s$ is the semi perimeter and is defined as
$s = \frac{a + b + c}{2}$

and $a , b , c$ are the lengths of the three sides of the triangle.

Here let $a = 7 , b = 4$ and $c = 9$

$\implies s = \frac{7 + 4 + 9}{2} = \frac{20}{2} = 10$

$\implies s = 10$

$\implies s - a = 10 - 7 = 3 , s - b = 10 - 4 = 6 \mathmr{and} s - c = 10 - 9 = 1$
$\implies s - a = 3 , s - b = 6 \mathmr{and} s - c = 1$

$\implies A r e a = \sqrt{10 \cdot 3 \cdot 6 \cdot 1} = \sqrt{180} = 13.416$ square units

$\implies A r e a = 13.416$ square units

Feb 23, 2016

$13.416 . u n i t s$

#### Explanation:

Use Heron's formula:

Heron's formula:

color(blue)(Area=sqrt(s(s-a)(s-b)(s-c))

Where,

color(brown)(a-b-c=sides,s=(a+b+c)/2=semiperimeter color(brown)(of color (brown)(triangle

So,

color(red)(a=7

color(red)(b=4

color(red)(c=9

color(red)(s=(7+4+9)/2=20/2=10

Substitute the values

$\rightarrow A r e a = \sqrt{10 \left(10 - 7\right) \left(10 - 4\right) \left(10 - 9\right)}$

$\rightarrow = \sqrt{10 \left(3\right) \left(6\right) \left(1\right)}$

$\rightarrow = \sqrt{10 \left(18\right)}$

$\rightarrow = \sqrt{180}$

We can further simplify that,

color(green)(sqrt180=sqrt(36*5)=6sqrt5~~13.416.units