Question

Derivation of formula of altitude of equilateral triangle

Answer 1

Step-by-step explanation:

An equilateral triangle is a triangle in which all three sides are equal.

Equilateral triangles are also equi-angular, which means, all three internal angles are also equal to each other and the only value possible is 60° each.

The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle.

Area of a triangle is measured in unit2.

In an equilateral triangle, the median, angle bisector and perpendicular are all the same and can be simply termed as the perpendicular bisector due to congruence conditions.

A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.

A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.

Area of Equilateral Triangle Formula: A = 3√4a2

where, “a” denoted the sides of an Equilateral Triangle

Proof:

Area of an Equilatereal triangle

In the figure above, the sides of an equilateral triangle are equal to “a” units.

We know that the area of Triangle is given by;

A = 12×base×height

To find the height, consider Triangle ABC,

Applying Pythagoras Theorem we know,

AB2=AD2+BD2

a2=h2+(a2)2

h2=a2–a24

h2=3a24

h=3√a2

Thus, we can calculate area by the basic equation,

A = 12×b×h=12×a×3√a2 Therefore, A = =3√a24unit2

Lets work out a few examples:-

Example 1: Find the area of an equilateral triangle whose side is 7 cm ?

Solution:

Given,

Side of the equilateral triangle = a = 7 cm

Area of an equilateral triangle = 3√4 a2

= 3√4×72 cm2

= 3√4×49 cm2

= 21.21762 cm2

Example 2: Find the area of an equilateral triangle whose side is 28 cm ?

Solution:

Given,

Side of the equilateral triangle (a) = 28 cm

We know, Area of an equilateral triangle = 3√4 a2

= 3√4×282 cm2

= 3√4×784 cm2

= 339.48196 cm2