Question

find the area of the triangle with sides 13cm,14cm,15cm​

Answer 1

Answer:

Semi-perimeter of triangle = 13+14+15/2 = 42/2 =21 cm

By Heron's formula-

Area of triangle =

$\sqrt{21(21 - 13)(21 - 14)(21 - 15)}$

》 $\sqrt{21 \times 7 \times 6 \times 5}$

》under root ($3 \times 7 \times 7 \times 2 \times 3 \times 5$)

》$21 \sqrt{10}$

Hence, Area of triangle = 21 root 10 cm-square

Answer 2

Answer:

Given :

»» The sides of triangle are 13cm, 14cm, 15cm.

To Find :

»» Area of triangle.

Concept :

»» Heron's formula is used to find the area of a triangle when the length of the 3 sides of the triangle is known. To use this formula, we need to know the perimeter of the triangle which is the distance covered around the triangle and is calculated by adding the length of all three sides.

Using Formulas :

${\longrightarrow{\small{\underline{\boxed{\sf{ Semi \: perimeter = \dfrac{1}{2} \big(a + b + c \big)}}}}}}$

$\longrightarrow{\small{\underline{\boxed{\sf{ Area=\sqrt{s(s - a)(s - b)(s - c)}}}}}}$

»» s = semi perimeter

»» a, b, c = sides of triangle

Solution :

★ Before finding the area of triangle we need to calculate the semi perimeter of triangle :

${\implies{\sf{ Semi \: perimeter = \dfrac{1}{2} \bigg(a + b + c \bigg)}}}$

${\implies{\sf{ Semi \: perimeter = \dfrac{1}{2} \bigg(13 + 14 + 15 \bigg)}}}$

${\implies{\sf{ Semi \: perimeter = \dfrac{1}{2} \bigg( \: 42 \: \bigg)}}}$

${\implies{\sf{ Semi \: perimeter = \dfrac{1}{2} \times 42}}}$

${\implies{\sf{ Semi \: perimeter = \dfrac{1}{\cancel{2}} \times \cancel{42}}}}$

${\implies{\sf{ Semi \: perimeter = 21 \: cm}}}$

Hence, the semi perimeter of triangle is 21 cm.

★ Now, calculating the area of triangle by heron's formula :

${\implies{\sf{ Area_{(\triangle)} =\sqrt{s(s - a)(s - b)(s - c)}}}}$

${\implies{\sf{ Area_{(\triangle)} =\sqrt{21(21 - 13)(21 - 14)(21 - 15)}}}}$

${\implies{\sf{ Area_{(\triangle)} =\sqrt{21(8)(7)(6)}}}}$

${\implies{\sf{ Area_{(\triangle)} =\sqrt{21 \times 8 \times 7 \times 6}}}}$

${\implies{\sf{ Area_{(\triangle)} =\sqrt{7056}}}}$

${\implies{\sf{ Area_{(\triangle)} =84 \: {cm}^{2} }}}$

Hence, the area of triangle is 84 cm².

Learn More :

$\begin{gathered}\begin{gathered}\boxed{\begin{minipage}{9cm}\\ \dag\quad \Large\underline{\bf Formulas\:of\:Triangle:-}\\ \\ \star\sf Triangle \:area = \dfrac{1}{2}\times b \times h\\ \\ \star\sf Triangle \: perimeter=a+b+c\\\\ \star\sf Scalene\:\triangle=\sqrt{s (s-a)(s-b)(s-c)}\\\\\star\sf Equilateral\: \triangle\:area = \dfrac{\sqrt{3}}{4}\times{side}^{2}\\\\\star\sf Equilateral \:\triangle\:perimeter = 3 \times side\\\\\star\sf Isosceles\: \triangle\:area= \dfrac{3}{4}\sqrt{{4b}^{2}-{a}^{2}}\\\\\star\sf Isosceles\:\triangle\:perimeter=a+2b\end{minipage}}\end{gathered}\end{gathered}$

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