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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