Question

Find the area of an equilateral triangle with side √3/4. please give the correct answer and the first correct answer will get the brainliest !!!

Answer 1

$\huge{\boxed{\red{\rm{Given:-}}}}$

The side of an equilateral triangle = $\bf \frac{\sqrt{3}}{4}$ unit.

$\huge{\boxed{\purple{\sf{To\ be\ found}}}}$

The area of the equilateral triangle.

Here is a formula for finding the area of an equilateral triangle

$= \boxed{ \dfrac{\sqrt{3}}{4}(\rm side)^{2} }\ unit\ sq.$

So, area =

$= \frac{\sqrt{3}}{4} \times \bigg( \frac{\sqrt{3}}{4} \bigg)^{2}$

$= \frac{\sqrt{3}}{4} \times \frac{\sqrt{3}}{4} \times \frac{\sqrt{3}}{4}$

$= \frac{3 \sqrt{3}}{64}\ unit\ sq.$

Hence,

The area of the equilateral triangle whose side = $\frac{\sqrt{3}}{4}$ unit. = $\frac{3 \sqrt{3}}{64}\ unit\ sq.$

$\underline{\tt \red{More}\ \green{Inform}\blue{ation}}$

Here is the derivation of the Area of an equilateral triangle with sides.

[Look at the attached image]

ΔABC is an equilateral triangle.

All side = a

AO is the perpendicular bisector to BC (In an equilateral triangle the median is perpendicular bisector itself.)

∠AOB = 90° [ΔAOB is right angled triangle]

So,

BO = CO = $\frac{a}{2}$

• We know that area of a triangle = $\boxed{\red \sf \frac{1}{2} \times base \times height}$ ---(i)

In triangle AOB we can use the Pythagoras theorem to get the value of height.

$\boxed{\tt (Hypotenuse)^{2} = (Perpendicular)^{2} +(base)^{2}}$

Here, base = BO = $\frac{a}{2}$, perpendicular = AO = h, and Hypoteneus = AB = a

Now,

(AB)² = (AO)² + (BO)²

⇒ $a^{2} = h^{2} + (\frac{a}{2})^{2}$

⇒ $a^{2}-(\frac{a}{2})^{2} = h^{2}$

⇒ $\sqrt{a^{2}-(\frac{a}{2})^{2} } = h$

⇒ $\sqrt{a^{2}-\frac{a^{2}}{4} } = h$

⇒ $\sqrt{\frac{4a^{2}-a^{2}}{4} } = h$

⇒ $\sqrt{\frac{3a^{2}}{4} } = h$

∴ $\frac{\sqrt{3}a}{2} = h$ ____(ii)

Now, put the value of height in the area of the triangle.

$\tt = \frac{1}{2} \times base \times height$

$\tt = \frac{1}{2} \times a \times \frac{\sqrt{3}a}{2}$

$\tt = \frac{a \times \sqrt{3}a}{4}$

$\boxed{\tt = \frac{\sqrt{3}a^{2}}{4}}\ i.e. \boxed{\tt = \frac{\sqrt{3}}{4}(side)^{2}}$

As

Side = a

Hence,

Area of an equilateral triangle = $\boxed{\purple{\frac{\sqrt{3}}{4}(side)^{2}}}$

Answer 2

What is an equilateral triangle?

Equilateral triangle is a triangle whose all three sides are equal i.e. same in length.

There is one formula used for finding the area of an equilateral triangle = (√3)/(4) × (side)² unit²

Now,

Given that side of the equilateral triangle = (√3)/(4)

so,

area = (√3)/(4) × [(√3)/(4)]²

       = (√3)/(4) × (√3)/(4) × (√3)/(4)

       = (3√3)/(64) unit ²