Question

The Area of a triangle where sides are 4cm, 13cm, 15cm

$\bigstar$ Use Herons's formula

Answer 1

★ Assumption Needed:-    

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Let the sides be,

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• ➾  Side a = 4 cm
• ➾  Side b = 13 cm
• ➾  Side c = 15 cm

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★ To Calculate:-    

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• ➾  Area of the triangle

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★ To Use:-    

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~Heron's formula

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◼ Semi-perimeter

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• $\Large\boxed{\sf{Semi\:-\:Perimeter\:=\:\left(\:\dfrac{a\:+\:b\:+\:c\:}{2}\:\right)}}$

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Where,

• ➾  a = 4 cm = Length of side a
• ➾  b = 13 cm = Length of side b
• ➾  c = 15 cm = Length of side c

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◼ Area of the triangle

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• $\Large\boxed{\sf{Area_{(\:TRIANGLE\:)}\:=\:\sqrt{s\:(\:s\:-\:a\:)\:(\:s\:-\:b\:)\:(\:s\:-\:c\:)} }}$

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Where,

• ➾  s = semi-perimeter
• ➾  a = 4 cm = Length of side a
• ➾  b = 13 cm = Length of side b
• ➾  c = 15 cm = Length of side c

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★ Calculating:-      

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Step1: First we need to find the semi-perimeter so we will substitute the values in the formula. [ Semi-perimeter = a + b + c/2 ]

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$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:\left(\:\dfrac{a\:+\:b\:+\:c\:}{2}\:\right)}$

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$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:\left(\:\dfrac{4\:+\:13\:+\:15\:}{2}\:\right)}$

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$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:\left(\:\dfrac{32}{2}\:\right)}$

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$\qquad\sf{:\implies\:Semi\:-\:Perimeter\:=\:16\:cm}$

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$\therefore\underline{\sf{The\:Semi\:-\:perimeter\:of\:the\:triangle\:is\:16\:cm}}$

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Step2: Now to calculate the triangle area, we will again use the formula and substitute the values. [ Area of triangle = √s ( s - a ) ( s -b ) ( s - c )]

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

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$\qquad\sf{:\implies\:Area_{(\:TRIANGLE\:)}\:=\:\sqrt{16\:(\:16\:-\:4\:)\:(\:16\:-\:13\:)\:(\:16\:-\:15\:)} }$

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$\qquad\sf{:\implies\:Area_{(\:TRIANGLE\:)}\:=\:\sqrt{16\:(\:12\:)\:(\:3\:)\:(\:1\:)} }$

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$\qquad\sf{:\implies\:Area_{(\:TRIANGLE\:)}\:=\:\sqrt{16\:\times\:12\:\times\:3\:\times\:1\:} }$

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$\qquad\sf{:\implies\:Area_{(\:TRIANGLE\:)}\:=\:\sqrt{576} }$

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$\qquad\sf{:\implies\:Area_{(\:TRIANGLE\:)}\:=\:24\:cm^{2}}$

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$\therefore\underline{\sf{The\:area\:of\:the\:triangle\:is\:24\:cm^{2}.}}$

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★ Answer:-        

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• The area of the triangle is 24 cm².

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Answer 2

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