Equilateral Triangle Definition, Area and Perimeter

Equilateral triangles

An equilatera triangle is an equiangular triangle with three equal sides. If a triangle is equilateral then it is a regular polygon with three vertices, three edges, and three internal angles each equal to 60°.

The angles of an equilateral triangle are each 60°.

The general formula of the area of a triangle is,
A =  ½bh
Where,
A = Area
b = base
h = height

If the lenth of the base of a triangle is 3cm and the height is 5cm, the area of the triangle is,
A = ½ × 3 × 5
    = 7.5cm²

In an equilateral triangle, the triangle is bisected by the height into two equal right triangles and the base is divided into two halves. Since base = b, half base = b/2.

The height of the equilateral triangle can be calculated from any of the right triangle using Pythagoras theorem which states that,
a² = b² + c²

Where, a = hypotenuse, b = opposite, and c = adjacent.

In each of the right triangle,
a² = (b/2)² + h²

Since the lengths of the three sides of the equilateral triangle are equal,
a = b/2 + b/2
   = b (base)

Substituting b for a,
b² = (b/2)² + h²
b² – (b/2)² = h²
h² = b² –  b²/4
      = 3b²/4
h = √3b²/4
   = √3×b/2

Substituting √3×b/2 into A = ½bh,
A = ½ × b × √3 × b/2
    = ½ × √3×b²/2
    = √3×b²/4

Hence, the formula of the area of equilateral triangle is √3×b²/4.

The perimeter of an equilateral triangle is the sum of the lenths of its three sides. Since a side is equal to b,

P = b + b + b
   = 3b

Where P = perimeter, and b = base.

Question: what is the perimeter and area of an equilateral triangle whose base is equal to 4cm?

Solution:
a, base (b) = 4cm
    Perimeter = 3b
    Substituting 4 for b,
    Perimeter = 3 × 4
                       = 12cm

b, base = 4cm
    Area = √3×b²/4.
              = √3 × 4²/4
              = √3 × 16/4
              = √3 × 4
              = 4√3 cm²

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