An equilateral triangle has an apothem of 5 cm. Find the perimeter of the triangle to the nearest centimeter.
All of the angles are equal: A = B = C = 180/3 = 60o.
All of the sides are equal: a = b = c.
h = 5 cm, sin60 = h/a = 5/a, a = 5/sin60 = 5.77 cm.
P = a + b + c = 5.77 + 5.77 + 5.77 = 17 cm.
Correction: apothem = s/2*Tan(180/n).
5 = S/2*Tan(180/3), S = 17.32 cm.
P = 17.32 + 17.32 + 17.32 = 52 cm.
To find the perimeter of an equilateral triangle, we need to know the length of one side of the triangle. The apothem is the distance from the center of the triangle to the midpoint of one of its sides. In an equilateral triangle, the apothem is also the height of the triangle.
In an equilateral triangle, the apothem is related to the side length by the formula:
apothem = (√3 / 2) * side length
Since we know that the apothem is 5 cm, we can solve for the side length:
5 cm = (√3 / 2) * side length
Dividing both sides by √3 / 2, we get:
side length = (5 cm) * (2 / √3) = (10 / √3) cm
To find the perimeter of the triangle, we multiply the side length by 3:
perimeter = (10 / √3) cm * 3
Using a calculator, we can approximate the value of √3 to be about 1.732:
perimeter = (10 cm / 1.732) * 3 ≈ 17.32 cm
Rounding to the nearest centimeter, the perimeter of the equilateral triangle is approximately 17 cm.
To find the perimeter of an equilateral triangle, we need to know the length of one side. However, we are given the apothem, which is the distance from the center of the triangle to any side.
We can use this information to find the length of one side by using the formula for the apothem of an equilateral triangle:
apothem = (side length) * sqrt(3) / 2
In this case, the apothem is given as 5 cm. Plugging it into the formula:
5 = (side length) * sqrt(3) / 2
To solve for the side length, we can rearrange the equation:
(side length) * sqrt(3) / 2 = 5
(side length) * sqrt(3) = 10
(side length) = 10 / sqrt(3)
Now, we can find the perimeter of the triangle by multiplying the side length by 3, since all sides of an equilateral triangle are equal:
Perimeter = (side length) * 3
Plugging in the side length:
Perimeter ≈ (10 / sqrt(3)) * 3
Calculating this value, we get:
Perimeter ≈ 17.3 cm
Therefore, the perimeter of the equilateral triangle is approximately 17.3 cm (rounded to the nearest centimeter).