Question

The area of an equilateral triangle is numerically equal to its perimeter. Find its perimeter correct to 2 decimal places.

Answer 1

Let each side of the equilateral triangle = x

∴ Its area = `sqrt(3)/4 x^2`

Area perimeter = 3x

By the given condition = `sqrt(3)/4 x^2 = 3x`

`x^2 = 3x xx 4/sqrt(3)`

`x^2 = (3x xx 4 xx sqrt(3))/(sqrt(3) xx sqrt(3)) = (3x xx 4 xx sqrt(3))/3 = 4xsqrt(3)`

⇒ `x^2 = sqrt(3) (4x) ⇒ x = 4sqrt(3)`   [∵ x ≠ 0]

∴ Perimeter = `12sqrt(3)` units

= 12 (1.732) = 20.784 = 20.78 units

Answer 2

Let the side length of the equilateral triangle be a.

Step 1: Write the formula for the area and the perimeter

1. Area of an equilateral triangle: Area = `sqrt3/4 a^2`
2. Perimeter of the triangle: Perimeter = 3a

Step 2: Given that the area equals the perimeter

`sqrt3/4 a^2 = 3a`

Step 3: Simplify the equation

Divide through by a (assuming a ≠ 0): `sqrt3/4 a = 3`

Multiply through by 4: `sqrt3 a = 12`

Divide by `sqrt3`: `a = 12/sqrt3`

Rationalize the denominator: `a = (12sqrt3)/3 = 4sqrt3`

Step 4: Calculate the perimeter

Perimeter = 3a

`= 3(4sqrt3) = 12sqrt3`

Step 5: Approximate to 2 decimal places

Using `sqrt3 = 1.732`

12√3 = 12 × 1.732 = 20.784

= 20.784