a Use the fact that the height of an equilateral triangle can be calculated using the formula h= sqrt(3)2s.
B
b Find the apothem of the triangle using the fact that it is three times shorter than the triangle's height.
A
a See solution.
B
b See solution.
Practice makes perfect
a We are asked to find the area of the triangle shown below using the formula A= 12bh, where b is the base and h is the height of the triangle.
It is given that the triangle is an equilateral triangle with side s. Hence, b=s. As for h, the length of the height of an equilateral triangle is determined by the following formula.
h=sqrt(3)/2s
Let's now substitute these values into the given formula and simplify it.
This way we verified that formula A= sqrt(3)4s^2 can be indeed used to calculate the area of an equilateral triangle.
b Let's now verify the formula for the area of an equilateral triangle A= sqrt(3)4s^2, using the formula for the area of a regular polygon A= 12ap.
Let's recall that an apothem of a regular polygon is a line segment from the center of the polygon to any of its sides. On the diagram, the center of the triangle is represented by O. Hence, segment OK is its apothem.
As shown above, it's also a radius of the inscribed circle, which is three times shorter than the triangle's height.
h=sqrt(3)/2s ⇒ OK=sqrt(3)/6s
Next, we will find the value of p, or the perimeter of the triangle. From the diagram, we know that the length of the side is s. There are three sides of equal length, so the perimeter of the triangle is as follows.
p=3s
Now, we have all the needed information. Let's substitute a with sqrt(3)6s and p with 3s into the formula and then simplify.
