Question

The sides of a triangle are 3 cm, 5 cm and 6 cm, the area is :: Answer will be in square root​

Answer 1

Given that, the sides of the triangle are 3 cm, 5 cm & 6 cm. We've to find out the area.

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀

⠀

⠀⠀⠀$\sf\underline{\bigstar\;Using\; Heron's\: formula\;:}$

⠀⠀

$\star\;{\boxed{\sf{\pink{Area_{\;(triangle)} = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

⠀⠀

$\sf Here \begin{cases}\sf{a = \bf{3\:}} \\\sf{b = \bf{5}} \\\sf{c = \bf{6}} \end{cases}$

⠀⠀

$:\implies\sf S_{(\: Semiperimeter)} = \dfrac{a + b + c}{2} \\\\\\:\implies\sf s_{(\: Semiperimeter)} = \dfrac{3 + 5 + 6}{2} \\\\\\:\implies\sf s_{(\: Semiperimeter)} = \dfrac{\cancel{14}}{\cancel{\:2}}\\\\\\:\implies{\underline{\boxed{\frak{\pink{S_{(\: Semiperimeter)} = 7}}}}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀

⠀⠀

$\bf{\dag}\;{\underline{\frak{Now,\:Putting\:values\;in\;formula}}}$

⠀⠀⠀

$:\implies\sf Area_{\triangle} = \sqrt{7(7 -3) (7 - 5) (7 - 6)}\\\\\\:\implies\sf Area_{\triangle} = \sqrt{7 \times 4 \times 2 \times 1} \\\\\\:\implies{\underline{\boxed{\frak{\purple{Area_{\triangle} = 2\sqrt{14} cm^2}}}}}$

⠀⠀⠀

$\therefore\:{\underline{\sf{Area\:of\:triangle\:is\: \bf{2 \sqrt{14}\:cm^2}.}}}$

Answer 2

Given:-

$\bigstar$The sides of a triangle are 3 cm , 5 cm and 6 cm.

To Find:-

$\bigstar$Area of triangle..

Solution:-

$\bigstar$To find the area of triangle, we area using herons formula,

Let,

• a = 3

• b = 5

• c = 6

$\bigstar$First we have known the semi perimeter to Substitute the value given in the herons formula.

$\longrightarrow{ \underline{ \boxed{ \sf{Semi \: perimeter \: = \frac{a + b + c}{2} }}}}$

$\\ \longrightarrow \sf \: Semi \: perimeter = \frac{3 + 5 + 6}{2}$

$\\ \longrightarrow \sf \: Semi \: perimeter = \frac{14}{2}$

$\\\longrightarrow \sf \pink \: Semi \: perimeter = 7$

$\bigstar$Using herons formula,

$\longrightarrow{ \underline{ \boxed{ \sf{Herons \: formula = \sqrt{s \times(s - a) \: (s - b) \: (s - c)} }}}}$

$\\\longrightarrow \: \sf \: = \sqrt{7 \times (7 - 3) \: (7 - 5) \: (7 - 6)}$

$\\\longrightarrow \: \sf \: = \sqrt{7 \times (4) \: (2) \: (1)}$

$\\ \longrightarrow \: \sf \: = \sqrt{7 \times 8 \times 1}$

$\\\longrightarrow \: \sf \: = \sqrt{7 \times 2 \times 2 \times 2 \times 1}$

$\\\longrightarrow \: \sf \: = \sqrt[2] {7 \times 2}$

$\\ \longrightarrow \: \sf \: \sqrt[2]{14} \: {cm}^{2}$

$\bigstar{ \bold{ \: Area \: of \: the \: triangle \: is \: \sqrt[2]{14} \: cm {}^{2} }}$