Question

9. Calculate the area and the height of an equilateral triangle whose perimeter is 60 cm.​

Answer 1

Question:

Calculate the area and the height of an equilateral triangle whose perimeter is 60 cm.

Given:

• perimeter = 60 cm

To find:

• Area
• height

Answer:

To find area first we have to find sides of triangle.

To find area first we have to find sides of triangle.So:

$perimeter = side \: 1 + side \: 2 + side \: 3$

As we know equilateral triangle has all sides equal:

$\therefore \: perimeter = 3 \times sides$

Put the value of perimeter:

$\implies 60 = 3 \times side$

$\implies side = \dfrac{60}{3}$

$\implies side = 20✓$

So we get the value of side ie 20

formula used to find area of equilateral triangle:

$area \: = \frac{\sqrt{3} {(side)}^{2} }{4}$

Explanation:

$\implies area = \frac{\sqrt{3} \times {20}^{2} }{4}$

$\implies area = \frac{ \sqrt{3} \times 400}{4}$

$\implies area = 10 0\sqrt{3} \: {cm}^{2}$

Can leave upto here but exactly area would be

$area = 10 0 \times 1732cm²$

$area = 17.32cm²$

Now to find height:

Formula used:

$height = \frac{ \sqrt{3} \times 20}{2}$

Explanation:

$\implies height = 10\sqrt{3 }$

Can leave upto here

Can leave upto hereBut exactly height would be:

$\implies height = 10 \times 1.732$

$\implies height = 17.32 \: cm✓$

Answer 2

$\bf{\underline{Given:-}}$

Perimeter of the equilateral triangle = 60 cm

$\bf{\underline{To\:Find:-}}$

• Height of the triangle
• Area of the triangle

$\bf{\underline{Solution:-}}$

We know,

$\sf{\underline{Perimeter\:of\:an\:equilateral\:triangle = 3a}}$

Therefore,

3a = 60

= $\sf{ a = \dfrac{60}{3}}$

$\sf{\implies a = 20\:cm}$

Now,

$\sf{S = \dfrac{a+b+c}{2}}$

We know all the sides of an equilateral triangle are equal.

Hence,

$\sf{S = \dfrac{20+20+20}{2}}$

= $\sf{S = \dfrac{60}{2}}$

$\sf{\implies S = 30\:cm}$

Now,

According to Heron's Formula

$\sf{Area = \sqrt{s(s-a)(s-b)(s-c)}\:\:sq.units}$

= $\sf{Area = \sqrt{30(30-20)(30-20)(30-20)}\:\:cm^2}$

= $\sf{Area = \sqrt{30\times10\times10\times10}\:\:cm^2}$

= $\sf{Area = 10\sqrt{10\times3\times10}\:\:cm^2}$

= $\sf{Area = 10\times10\sqrt{3}\:\:cm^2}$

= $\sf{Area = 100\sqrt{3}\:\:cm^2}$

$\sf{\implies Area = 100\times 1.732\:\:cm^2}$

$\sf{\implies Area = 173.2\:\: cm^2\:\:[Approx]}$

$\sf{\therefore The \:area\:of\: the\: triangle\: is\: 100\sqrt{3}\:cm^2 \:[or \:173.2\: cm^2\: (Approx)]}$

Now,

To find Height of the triangle

= $\sf{Area = \dfrac{1}{2}\times base \times height}$

$\sf{\implies 100\sqrt{3} = \dfrac{1}{2} \times 20 \times height}$

$\sf{\implies 100\sqrt{3} = 10\times height}$

$\sf{\implies \dfrac{100\sqrt{3}}{10} = height}$

$\sf{\implies height = 10\sqrt{3}\:\:cm}$

$\sf{\implies height = 10\times 1.732\:\:cm}$

$\sf{\implies height = 17.32\:\:cm\:\:[Approx]}$

$\sf{\therefore The \:height \:of\: the\: triangle\: is\: 10\sqrt{3}\:cm\:\:[or\: 17.32 \:cm \:(Approx)]}$

$\bf{\underline{Additional\:Information}}$

Area of equilateral triangle can also be found using formula:-

$\sf{\dfrac{\sqrt{3}}{4}a^2}$