Question

define write the heron's formula, for finding the area of a triangle​

Answer 1

$\huge\sf{answer}$

$\color{gold}area = \green{\sqrt{s(s - a)(s - b)(s - c)}}$

$\color{blue}\dagger \: where \: \pink{s} \: donates \: semi \: perimeter \:$

$\red{\dagger \: semi \: perimeter \: is \:obtained \: by \: adding \: sides }\\ \star \: side \mapsto \: a \\ \star \: side \: \mapsto \: b \\ \star \: side \: \mapsto \: c \\ \hookrightarrow \: and \: then \: divinding \: the \: result \: by \: 2$

$\purple{\circ \: for \: example}$

$1. \: determine \: the \: area \: of \: a \: triangle \: whose \: sides \: are - \\ \star \: 5 \: cm \: \\ \star \: 13 \: cm \\ \star \: 12 \: cm$

$\huge \: \bold{solution}$

The sides of a triangle are :- a = 5 cm , b = 13 cm , c = 12 cm

s = a + b + c / 2

s = 5 + 12 + 13 / 2 = 15 cm

$\bold{area = \sqrt{s(s - a)(s - b)(s - c)}}$

$= \sqrt{15 \times 10 \times 3 \times 2} \\ = \sqrt{900 } \\ \bold{ = {30 \: cm}^{2}}$

________________________________

Answer 2

$\rightarrow \bf \underline{Heron's \: Formula :}$

• The heron's formula was created by Heron of Alexandria. He was a Greek Engineer and Mathematician in 10-18 AD.
• Heron's formula is used for finding the Area of Triangles, with 3 sides.
• Heron's Formula is applicable for all triangles.
• The only requirement for this Formula is that 3 sides are required for calculation.

↪ Formula :

• For finding the Area, we must first find the Semi-Perimeter of the Triangle. And to find the Semi-Perimeter, we must use the formula of :

$\boxed{\bf Semi-Perimeter = \dfrac{a + b + c}{2} }$

• Next, to find the Area, we have to use the formula of :

$\boxed{\bf Area = \sqrt{s (s-a) (s-b) (s-c) } }$

⇒ Now, an example is solved below to find the Area of the Triangle.

_____________________

Question :

• Find the area of the triangle whose sides measure 10 cm, 17 cm, and 21 cm.

Answer :

Now,

${\sf Semi-Perimeter = \dfrac{a + b + c}{2} }$

                           ${ \sf = \dfrac{10 + 17 + 21}{2} }$

                           ${ \sf = \dfrac{48}{2} }$

                           $\sf = 24$

${ \sf Area = \sqrt{s (s-a) (s-b) (s-c) } }$

       $\sf { = \sqrt{24 (24-10) (24-17) (27-21) } }$

       $\sf { = \sqrt{(24) \times (14) \times (7) \times (3) } }$

       $\sf { = \sqrt{7056} }$

       $\sf = 84$

Therefore, the Area of the Triangle is 84 cm².