Question

abc is an equilateral triangle of side a. Its area will be
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Answer 1

Area of equilateral triangle with side a is $Area=\frac{\sqrt{3}a^{2} }{4}$

Definition:

• Equilateral triangle is a triangle in which all three sides are the same lenght.
• The three angles opposite the equal side are equal in measure since the three sides are equal.
• Therefore,it is also called an equiangular triangle,since each angle measures 60 degrees.
• Just like other types of triangles,an equilateral triangle also has its area,perimeter.

Step-by-step explanation:

Attached is the figure of the equilateral triangle ,if we see the figure the area of a triangle is given by,

$Area=\frac{1}{2} *b*h$

where base =a,and height=h

Therefore,$Area=\frac{1}{2} *a*h$     (1)

Now,form the attached figure,the altitude h bisects the base into equal halves,such as $\frac{a}{2}$ and $\frac{a}{2}$.It also forms two equivalent right-angled triangles.

So,for a right triangle,using pythagoras theorem,we can write as follows,

$a^{2}=h^{2} +(\frac{a}{2}) ^{2}$   or $h^{2}=a^{2} -(\frac{a}{2}) ^{2}$

$h^{2} =3\frac{a^{2} }{4} \\h=\frac{\sqrt{3} a}{2}$

Subtitute the value of h in equation (1)

$Area=\frac{1}{2} *a*h\\Area=\frac{1}{2} *a*\frac{\sqrt{3}a }{2} \\\\Area=\frac{\sqrt{3}a^{2} }{4}$

Therefore ,area of the equilateral triangle with side a is $Area=\frac{\sqrt{3}a^{2} }{4}$