Question

question
find the area of the regular triangle. the figure has a perimeter of 45.87. round to the nearest hundredth.
(an image of a regular triangle with a dashed line from the center to a side, labeled 18 mm, and a right angle symbol at the side)
answer attempt 1 out of 2
+ additional solution - no solution
a = mm² submit answer

Explanation:

Step1: Find the side length of the regular triangle.

A regular triangle (equilateral triangle) has three equal sides. The perimeter \( P \) is the sum of the three sides, so the length of one side \( s \) is \( s=\frac{P}{3} \). Given \( P = 45.87 \), we have \( s=\frac{45.87}{3}=15.29 \) mm. Wait, but there is a 18 mm segment (the apothem, \( a \))? Wait, maybe I misread. Wait, the formula for the area of a regular polygon is \( A=\frac{1}{2}\times perimeter\times apothem \). Oh, right! A regular triangle is a regular polygon with \( n = 3 \) sides. So the area formula for a regular polygon is \( A=\frac{1}{2}\times P\times a \), where \( P \) is the perimeter and \( a \) is the apothem (the distance from the center to the midpoint of a side, which is the 18 mm here? Wait, no, wait the diagram: the 18 mm is the apothem? Wait, no, maybe the height? Wait, no, in a regular triangle, the apothem is \( \frac{height}{3} \)? Wait, no, let's recall: for a regular polygon with \( n \) sides, side length \( s \), apothem \( a \), the area is \( A=\frac{1}{2}\times n\times s\times a \). Since \( n\times s = perimeter \), so \( A=\frac{1}{2}\times P\times a \). Wait, but in the diagram, the 18 mm: is that the apothem or the height? Wait, the problem says "regular triangle" (equilateral triangle). Wait, maybe there is a mistake, or maybe the 18 mm is the apothem? Wait, no, in an equilateral triangle, the apothem (distance from center to a side) is \( \frac{height}{3} \), and the height \( h=\frac{\sqrt{3}}{2}s \). But the problem gives perimeter 45.87, so side length \( s = 45.87/3 = 15.29 \) mm. Then height \( h=\frac{\sqrt{3}}{2}\times15.29\approx13.23 \) mm. But the diagram shows 18 mm, which is conflicting. Wait, maybe the 18 mm is the apothem? Wait, no, maybe the problem is a regular polygon (triangle) with apothem 18 mm? Wait, that can't be, because the side length would be \( s = 2\times a\times\tan(\frac{\pi}{n}) \), for \( n = 3 \), \( s = 2\times18\times\tan(60^\circ)=36\times\sqrt{3}\approx62.35 \) mm, then perimeter would be \( 3\times62.35\approx187.05 \), which is not 45.87. So there must be a misinterpretation. Wait, maybe the 18 mm is the height? Wait, if the height is 18 mm, then area is \( \frac{1}{2}\times s\times h \). But we need to find \( s \) from perimeter: \( s = 45.87/3 = 15.29 \) mm. Then area would be \( \frac{1}{2}\times15.29\times18 = 137.61 \) mm². But that contradicts the apothem idea. Wait, maybe the diagram's 18 mm is the height? Let's check the problem again: "Find the area of the regular triangle. The figure has a perimeter of 45.87. Round to the nearest hundredth." The diagram shows a triangle with a dashed line (apothem?) of 18 mm. Wait, maybe the problem is using the regular polygon area formula: \( A=\frac{1}{2}\times perimeter\times apothem \). So if perimeter is 45.87, apothem is 18, then \( A=\frac{1}{2}\times45.87\times18 \). Let's calculate that: \( 0.5\times45.87 = 22.935 \), \( 22.935\times18 = 412.83 \) mm². Wait, that makes sense. Maybe the 18 mm is the apothem. So the formula for the area of a regular polygon is \( A=\frac{1}{2}\times P\times a \), where \( P \) is perimeter, \( a \) is apothem. So that's the key. So regardless of the side length, if we use the regular polygon area formula, with \( P = 45.87 \) and \( a = 18 \), then:

Step1: Apply the regular polygon area formula.

The formula for the area \( A \) of a regular polygon is \( A=\frac{1}{2}\times perimeter\times apothem \).

Step2: Substitute the given values.

Given \( perimeter = 45.87 \) and \( apothem = 18 \) mm, we have:
\( A=\frac{1}{2}\times45.87\times18 \)

Step3: Calculate the result.

First, \( \frac{1}{2}\times45.87 = 22.935 \)
Then, \( 22.935\times18 = 412.83 \)

Answer:

\( 412.83 \)