calculate the area and perimeter of an equilateral triangle with height of root of six leaving your answers in the surd form
You should make a point to memorize the ratio of sides of the 30-60-90° triangle, which is
1 : √3 : 2
so for yours:
√3/1 = √6/x
√3x = √6
x = √2
then the hypotenuse = 2√2
area = (1/2)base x height =(1/2)(√2)(√6)
= (1/2)√12
= (1/2)2√3
= √3
perimeter = √2 + √6 + 2√2
= 3√2 + √6
To calculate the area and perimeter of an equilateral triangle, we need to know either the length of one side or the height. In this case, you have provided the height of √6.
To find the side length of the equilateral triangle, we can use the formula:
side length = (√3 * height) / 2
Given that the height is √6, we can substitute this value into the formula:
side length = (√3 * √6) / 2
= √18 / 2
= √9 * √2 / 2
= 3√2 / 2
The perimeter of an equilateral triangle is given by multiplying the side length by 3 (since all three sides are equal):
perimeter = 3 * side length
= 3 * (3√2 / 2)
= 9√2 / 2
To calculate the area of an equilateral triangle, we can use the formula:
area = (side length^2 * √3) / 4
Substituting the value of the side length into the formula, we get:
area = ((3√2 / 2)^2 * √3) / 4
= (9*2 / 4) * √3
= 9/2 * √3
= (9√3) / 2
Therefore, the area of the equilateral triangle is (9√3) / 2 and the perimeter is 9√2 / 2.