Each side of an equilateral triangle is 8 cm. find a) Area of triangl

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1. Given:

   An equilateral triangle with –

   Each side = 8 cm

   What To Find:

   We have to –

   • a. Find the area of the triangle.

   • b. Find the height of the triangle.

   Formulas Needed:

   The formulas are –

   [tex]\bf \to Area = \sqrt{s(s-a)(s-b)(s-c)} \: \: … (Heron’s \: Formula)\\ \\ \bf {\to Area = \dfrac{1}{2} \times b \times h \: \: … (Simple \: Formula)}[/tex]

   Abbreviations Used:

   For Heron’s Formula –

   • s = semi-perimeter
   • a = 1st side
   • b = 2nd side
   • c = 3rd side

   For Simple Formula –

   • b = base
   • h = height

   Solution:

   • Finding the area using Heron’s formula.

   First, find the s.

   [tex]\sf \to S = \dfrac{Sum \: of \: all \: sides}{2}[/tex]

   [tex]\sf \to S = \dfrac{8+8+8}{2}[/tex]

   [tex]\sf \to S = \dfrac{24}{2}[/tex]

   [tex]\sf \to S = 12 \: cm[/tex]

   Next, find the area.

   [tex]\sf \to Area = \sqrt{s(s-a)(s-b)(s-c)}[/tex]

   [tex]\sf \to Area = \sqrt{12(12-8)(12-8)(12-8)}[/tex]

   [tex]\sf \to Area = \sqrt{12(4)(4)(4)}[/tex]

   [tex]\sf \to Area = \sqrt{768}[/tex]

   [tex]\sf \to Area = 27.7 \: cm^2 \: approx[/tex]

   • Finding the height.

   [tex]\sf \to Area = \dfrac{1}{2} \times b \times h[/tex]

   [tex]\sf \to 27.7 = \dfrac{1}{2} \times 8 \times h[/tex]

   [tex]\sf \to 27.7 = 4 \times h[/tex]

   [tex]\sf \to \dfrac{27.7}{4} = h[/tex]

   [tex]\sf \to 6.925 \: cm = h[/tex]

   Final Answer:

   • a. ∴ Thus, the area of a triangle is 27.7 cm² approx.
   • b. ∴ Thus, the height of a triangle is 6.925 cm.

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