Question

calculate the area of the regular polygon. round the final answer to two decimal places if necessary. 3.53 3

Explanation:

Step1: Identify the polygon and formula

The polygon is a regular pentagon. The formula for the area of a regular polygon is \( A=\frac{1}{2} \times \text{perimeter} \times \text{apothem} \), but here we can also use the formula for a regular pentagon with radius \( r \) (distance from center to a vertex) and side length \( s \). First, find the perimeter: a pentagon has 5 sides, so perimeter \( P = 5\times s \), where \( s = 3.53 \). So \( P=5\times3.53 = 17.65 \).

Step2: Use the formula for area of regular pentagon

Another formula for the area of a regular pentagon with side length \( s \) and radius \( r \) (or we can use the formula \( A=\frac{5}{2} \times s \times a \), where \( a \) is the apothem. Wait, alternatively, for a regular pentagon, the area can be calculated as \( A = \frac{5}{2} \times r \times s \times \sin(\frac{2\pi}{5}) \)? Wait, no, actually, the regular pentagon can be divided into 5 isosceles triangles, each with two sides equal to the radius \( r = 3 \) and base \( s = 3.53 \). The area of one triangle is \( \frac{1}{2} \times r \times r \times \sin(\theta) \), where \( \theta=\frac{360^\circ}{5}=72^\circ \). So area of one triangle: \( \frac{1}{2} \times 3 \times 3 \times \sin(72^\circ) \). Then total area is 5 times that.

Wait, let's recast: the formula for the area of a regular polygon is \( A=\frac{1}{2} \times n \times s \times r \times \sin(\frac{2\pi}{n}) \), where \( n \) is the number of sides, \( s \) is side length, \( r \) is the radius (distance from center to vertex). Here \( n = 5 \), \( s = 3.53 \), \( r = 3 \).

So \( A=\frac{1}{2} \times 5 \times 3.53 \times 3 \times \sin(72^\circ) \).

First, calculate \( \sin(72^\circ)\approx0.9511 \).

Then, \( \frac{1}{2} \times 5 \times 3.53 \times 3 = \frac{5\times3.53\times3}{2}=\frac{52.95}{2}=26.475 \).

Then multiply by \( \sin(72^\circ) \): \( 26.475\times0.9511\approx25.18 \). Wait, but maybe a simpler way: the regular pentagon with side length \( s \) and apothem \( a \). Wait, the apothem \( a \) is the distance from center to the midpoint of a side. Alternatively, if we use the formula \( A = \frac{5}{2} \times s \times a \), where \( a \) is apothem. But we can also use the formula for a regular pentagon: \( A = \frac{5s^2}{4\tan(\frac{\pi}{5})} \). Let's check with \( s = 3.53 \). \( \tan(\frac{\pi}{5})=\tan(36^\circ)\approx0.7265 \). So \( \frac{5\times(3.53)^2}{4\times0.7265} \). Calculate \( (3.53)^2 = 12.4609 \), then \( 5\times12.4609 = 62.3045 \), \( 4\times0.7265 = 2.906 \), then \( 62.3045\div2.906\approx21.44 \). Wait, maybe my first approach was wrong. Wait, the radius (distance from center to vertex) is 3, so \( r = 3 \), side length \( s = 3.53 \). Let's use the formula for area of a regular polygon: \( A=\frac{1}{2} \times n \times r^2 \times \sin(\frac{2\pi}{n}) \). For \( n = 5 \), \( \frac{2\pi}{5}=72^\circ \), \( \sin(72^\circ)\approx0.9511 \). So \( A=\frac{1}{2} \times 5 \times 3^2 \times 0.9511 \). Calculate \( 3^2 = 9 \), \( \frac{1}{2} \times 5 \times 9 = \frac{45}{2}=22.5 \), then \( 22.5\times0.9511\approx21.40 \). Wait, but let's check the side length. The formula for the side length of a regular pentagon with radius \( r \) is \( s = 2r\sin(\frac{\pi}{5}) \). \( \sin(\frac{\pi}{5})=\sin(36^\circ)\approx0.5878 \), so \( s = 2\times3\times0.5878 = 3.5268\approx3.53 \), which matches the given side length. So that's correct. So then the area of one triangle (isosceles with two sides \( r = 3 \), included angle \( 72^\circ \)) is \( \frac{1}{2} \times 3 \times 3 \times \sin(72^\circ) \), and there are 5 such triangles. So total area \( A = 5\times\frac{1}{2} \times 3 \times 3 \times \sin(72^\circ) \). Calculate that: \( 5\times\frac{9}{2}\times0.9511 = \frac{45}{2}\times0.9511 = 22.5\times0.9511\approx21.40 \). Wait, but let's use the perimeter and apothem. The apothem \( a = r\cos(\frac{\pi}{5}) \), since in the isosceles triangle, the apothem is the height, so \( a = 3\cos(36^\circ)\approx3\times0.8090 = 2.427 \). Then perimeter \( P = 5\times3.53 = 17.65 \). Then area \( A=\frac{1}{2} \times P \times a = \frac{1}{2} \times 17.65 \times 2.427 \approx \frac{1}{2} \times 42.84\approx21.42 \). Rounding to two decimal places, approximately 21.42. Wait, but let's check with the side length formula. Since \( s = 2r\sin(\frac{\pi}{5}) \), \( r = 3 \), so \( s = 2\times3\times\sin(36^\circ)\approx6\times0.5878 = 3.5268\approx3.53 \), which is correct. So the area calculation: using \( A=\frac{5}{2} \times s \times a \), where \( a = r\cos(\frac{\pi}{5}) = 3\times\cos(36^\circ)\approx2.427 \), \( s = 3.53 \). So \( \frac{5}{2} \times 3.53 \times 2.427 \approx \frac{5}{2} \times 8.56\approx21.40 \). So the area is approximately 21.40 (or 21.42, depending on rounding). Wait, let's do the calculation more accurately. \( \sin(72^\circ)\approx0.9510565163 \), \( \cos(36^\circ)\approx0.809016994 \). So area using the triangle method: \( 5\times\frac{1}{2}\times3\times3\times\sin(72^\circ)= \frac{45}{2}\times0.9510565163 = 22.5\times0.9510565163 = 21.40 \) (rounded to two decimal places). Alternatively, using perimeter and apothem: perimeter \( P = 5\times3.53 = 17.65 \), apothem \( a = 3\times\cos(36^\circ)\approx2.42705 \). Then \( \frac{1}{2}\times17.65\times2.42705 = 8.825\times2.42705\approx21.42 \). The slight difference is due to rounding during intermediate steps. But since the side length is given as 3.53 (which is approximately \( 2\times3\times\sin(36^\circ)\approx3.5268 \)), so using the more accurate side length: \( s = 3.5268 \), then perimeter \( P = 5\times3.5268 = 17.634 \), apothem \( a = 3\times\cos(36^\circ)\approx2.42705 \), then area \( \frac{1}{2}\times17.634\times2.42705\approx\frac{1}{2}\times42.80\approx21.40 \). So the correct area, when calculated precisely, is approximately 21.40 (or 21.42, depending on rounding the side length). But let's use the formula \( A = \frac{5s^2}{4\tan(\frac{\pi}{5})} \). \( s = 3.53 \), \( \tan(36^\circ)\approx0.72654 \). So \( \frac{5\times(3.53)^2}{4\times0.72654} = \frac{5\times12.4609}{2.90616} = \frac{62.3045}{2.90616}\approx21.44 \). Hmm, different results. Wait, maybe the problem is that the radius is 3, and the side length is 3.53, so we can use the formula for area of a regular polygon with \( n = 5 \), \( s = 3.53 \), \( r = 3 \). The correct formula for the area of a regular polygon is \( A = \frac{1}{2}n r^2 \sin(\frac{2\pi}{n}) \). So \( n = 5 \), \( r = 3 \), so \( A=\frac{1}{2}\times5\times9\times\sin(72^\circ)=\frac{45}{2}\times0.9510565\approx22.5\times0.9510565\approx21.40 \). So I think the correct answer is approximately 21.40 (or 21.42, depending on rounding). Let's check with the given side length. Since \( s = 3.53 \), and \( s = 2r\sin(\frac{\pi}{5}) \), \( r = 3 \), so \( \sin(\frac{\pi}{5}) = \frac{s}{2r}=\frac{3.53}{6}\approx0.5883 \), which is close to \( \sin(36^\circ)\approx0.5878 \), so the approximation is good. So the area is approximately 21.42 (or 21.40) when rounded to two decimal places.

Answer:

21.42 (or 21.40, depending on calculation method, but likely 21.42 when using perimeter and apothem with given side length)