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Area of an Equilateral Triangle

 

Triangles are classified into three types: scalene triangles, equilateral triangles, and isosceles triangles. Every three sides of an equilateral triangle are identical, and all angles are equivalent to 60°. Thus, the angles or degrees of an equilateral triangle are congruent to each other.

Some Other Properties of Equilateral Triangle

An equilateral triangle has several distinct but important features, including:

  • An equilateral triangle is provided only when the circumcentres of three or more of the smaller triangles are within the same distance of the centre.
  • A triangle is equilateral only when a triangle has the same perimeter or the same radius of any three of the smaller ones.
  • The median, angle bisector, and perpendicular are the same in an equilateral triangle and are simply referred to as the perpendicular bisector because of the requirements of congruence.
  • The triangle's ortho-centre and centroid are in the same place.
  • The median, angle bisector, and altitude for all sides of an equilateral triangle are identical and can be considered the equilateral triangle's lines of symmetry.
  • 3s is the perimeter of an equilateral triangle. Where 's' is the side of the equilateral triangle.

Area and Formula for the Area of a Sphere

An equilateral triangle's area is the total room it takes up on a two-dimensional plane. To review, an equilateral triangle is one with all of its sides identical and all of its interior angles measuring 60°. If the length of an equilateral triangle's side is known, the area of the triangle may be computed.
The equation for determining the area of an equilateral triangle is as follows:
Area of an Equilateral Triangle (S) = (√3/4) s2
Where 's' is the value of the length of each dimension.

You may also want to check the Area of Triangle Calculator.

Derivation for Area of Equilateral Triangle

Utilizing Basic Area of Triangle Formula to Derive the Equilateral Triangle's Formula

Consider an equilateral triangle with side measuring “s” units. Then, to the base of height "d", draw a perpendicular bisector.

Now we know,
Area of Triangle = ½ × height of the triangle × base of the triangle
Here, base = s, and height = d

Now, registering the Pythagoras Theorem in the triangle.

s2 = d2 + (s/2)2

⇒ d2 = s2 – (s2/4)

⇒ d2 = (3s2)/4

Or, d = ½(√3s)

Now, placing the value of “d” in the equation of the area of a triangle.
Area of Triangle = ½ × height of the triangle × base of the triangle

⇒ S = ½ × s × ½(√3s)

Or, Area of Equilateral Triangle = ¼(√3s2)

Deriving Area of Equilateral Triangle Using Trigonometry

If two lengths of a triangle are provided, then the height can be determined using trigonometric identities. Now, the height of a triangle KLM shall be- height = h, Sin M = m, and Sin k = k. Sin L

Now, the area of ABC = ½ × k × (h. sin M) = ½ × h × (m. sin K) = ½ × m (k. sin L)

Now, as it is an equilateral triangle, the interior angles measures, K = L = M = 60°, and k = h = m

Area = ½ × k × (k. Sin 60°) = ½ × k2 × Sin 60° = ½ × k2 × √3/2 So, Area of an Equilateral Triangle = (√3/4) k2

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