Question

question
find the area of the regular triangle. the figure has a perimeter of 45.87. round to the nearest hundredth.
(there is a triangle figure with a side labeled 18 mm and a dashed line with a right angle symbol from the center to the side)

Explanation:

Step1: Find the side length of the regular triangle

Since the perimeter \(P\) of a regular triangle (equilateral triangle) is \(P = 3s\) (where \(s\) is the side - length), and \(P=45.87\). Then \(s=\frac{P}{3}=\frac{45.87}{3}=15.29\)

Step2: Use the formula for the area of a regular polygon \(A=\frac{1}{2}aP\)

For a regular triangle (a regular polygon with \(n = 3\) sides), the formula for the area \(A\) of a regular polygon \(A=\frac{1}{2}aP\) (where \(a\) is the apothem and \(P\) is the perimeter). Here, \(a = 18\) (given as the length from the center to the side) and \(P=45.87\)

Substitute the values into the formula: \(A=\frac{1}{2}\times18\times45.87\)

First, calculate \(\frac{1}{2}\times18=9\)

Then, \(A = 9\times45.87=412.83\)

Answer:

\(412.83\)