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