Triangle which has all its sides equal in length and the internal angles, all of the three, is 60°, is called an equilateral triangle. The amount of space inside an 2 dimensional equilateral figure is the area of the equilateral triangle. So, if the length of the sides of an equilateral triangle are known, the after can be easily calculated.
To find the area of an equilateral triangle, the formula is given below:
Area of equilateral triangle (A) = (√¾)a²
Here, a = length on one side of the equilateral triangle
HOW WAS IT DERIVED?
Given below are three methods from which the formula for the area of an equilateral triangle is derived :
• With the use of trigonometry
• With the use of basics of triangle formula
• By using construction of rectangle
1) HOW TO DERIVE THE AREA OF AN EQUILATERAL TRIANGLE BY USING TRIGONOMETRY
The height of a triangle can be calculated, if two of its sides are given by using the properties of trigonometry.
Height= (a×sinB) = (b×sinC) = (c×sinA)
Therefore, area of the triangle = ½×a×(b×sinC) = ½×b(c×sinA)= ½×c×(a×sinB)
As we all know that, a=b=c=60°in an equilateral triangle,
Area= ½×a×(a×sin60°) = ½×a²×sin60° = ½×a²×√¾
According to the equation, area of an equilateral triangle = (√¾)a²
2) HOW TO DERIVE THE AREA OF AN EQUILATERAL TRIANGLE BY USING THE FORMULA IS BASIC TRIANGLE
Area of a triangle = ½ × base × height
Let base=a and height=h
According to the Pythagoras theory of a triangle,
a² = h² + (a/2)²
=> h² = a² – a²/4
=> h² = (3a²)/4
Or, h = ½ (√3a)
Now if we put the value of h in the formula of area of a triangle we get,
A = ½ × a × ½(√3a)
Or, area of an equilateral triangle = ¼ × (√3a²)
3) HOW TO DERIVE THE AREA OF AN EQUILATERAL TRIANGLE BY USING RECTANGLE CONSTRUCTION
Let the sides of the triangle be a
• Divide the triangle into two halves, by drawing a straight line from the top vertex of the triangle to the base, at the mid-point
• By cutting one half, along the line, and join it with the other half of the triangle, this forming a rectangle.
Let the sides of the triangle be ‘a’ and the height of the triangle be ‘ ‘b’.
Therefore, the area of an equilateral triangle = ½×a×h ….. (i)
Half of the rectangle that has been joined to form so, is a right angled triangle.
Therefore, by applying the theorem if Pythagoras,
=> a² = h² + (a/2)²
=> h² = (¾)a²
=> h = (√3/2)a ……. (ii)
From the equation,
Area of an equilateral triangle = (½)×a×(√3/2)a
Or, the area of an equilateral triangle = (¾)a²
HERE ARE SOME QUESTIONS FOR YOU TO SOLVE
1) What would be the area of an equilateral triangle who’s sides are 3cm?
2) Measure the area of an equilateral triangle who’s sides are equal to 32cm.
3) What would be the length of the sides of an equilateral triangle whose area of 678cm²
4) If the area of an equilateral triangle is 5367cm², what will be the measure of its sides?
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