- 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 $unit^{2}$.
- 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 = $\frac{\sqrt{3}}{4}$$a^{2}$
where, “a” denoted the sides of an Equilateral Triangle
Proof:
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 = \(\frac{1}{2} \times base \times height\)
To find the height, consider Triangle ABC,
Applying Pythagoras Theorem we know,
\(AB^{2} = AD^{2}+BD^{2}\)
\(a^{2} = h^{2} + \left ( \frac{a}{2} \right )^{2}\)
\(h^{2} = a^{2} – \frac{a^{2}}{4}\)
\(h^{2} = \frac{3a^{2}}{4}\)
\(h = \frac{\sqrt{3}a}{2}\)
Thus, we can calculate area by the basic equation,
A = \(\frac{1}{2} \times b \times h = \frac{1}{2} \times a \times \frac{\sqrt{3}a}{2}\)
Therefore, A = \(= \frac{\sqrt{3}a^{2}}{4} \;\; unit^{2}\)
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 = $\frac{\sqrt{3}}{4}$ $a^{2}$ = $\frac{\sqrt{3}}{4}$$\times$$7^{2}$ $cm^{2}$ = $\frac{\sqrt{3}}{4}$$\times$49 $cm^{2}$ = 21.21762 $cm^{2}$
Example 2: Find the area of an equilateral triangle whose side is 28 cm ? Solution: Given, Side of the equilateral triangle (a) = 28 cm We know, Area of an equilateral triangle = $\frac{\sqrt{3}}{4}$ $a^{2}$ = $\frac{\sqrt{3}}{4}$$\times$$28^{2}$ $cm^{2}$ = $\frac{\sqrt{3}}{4}$$\times$784 $cm^{2}$ = 339.48196 $cm^{2}$ |
For more, Formulas of triangles.