Question

deltamath back to home nti day 4 60% credit until: february 6 at 3:00 pm grade: 70% complete: 83% perimeter / area of regular polygons perimeter / area of regular polygons upload work scientific calculator graphing calculator answer attempt 2 out of 3 a = square units submit answer

Explanation:

Step1: Identify the polygon type.

The figure is a regular pentagon (5 - sided regular polygon) with an apothem \(a = 8\). The formula for the area \(A\) of a regular polygon is \(A=\frac{1}{2}\times perimeter\times apothem\). First, find the side length \(s\) of the regular pentagon. For a regular pentagon, the central angle is \(\frac{360^{\circ}}{5}=72^{\circ}\), and the apothem is related to the side length by \(a = \frac{s}{2\tan(\frac{\pi}{5})}\) (since \(\frac{72^{\circ}}{2}=36^{\circ}\), and \(\tan(36^{\circ})\) is involved). Solving for \(s\): \(s = 2a\tan(\frac{\pi}{5})\). Substituting \(a = 8\), \(s = 2\times8\times\tan(36^{\circ})\approx16\times0.7265\approx11.624\).

Step2: Calculate the perimeter.

Perimeter \(P\) of a regular pentagon is \(P = 5s\). So \(P = 5\times11.624\approx58.12\).

Step3: Calculate the area.

Using the area formula \(A=\frac{1}{2}\times P\times a\). Substitute \(P\approx58.12\) and \(a = 8\): \(A=\frac{1}{2}\times58.12\times8 = 29.06\times8 = 232.48\). Wait, alternatively, there's a formula for the area of a regular pentagon with apothem \(a\): \(A=\frac{5}{2}\times s\times a\), and since \(s=\frac{2a}{\tan(36^{\circ})}\), substituting gives \(A = 5\times a^{2}\times\tan(36^{\circ})\). Let's use this formula for accuracy. \(A = 5\times8^{2}\times\tan(36^{\circ})=5\times64\times0.7265 = 320\times0.7265 = 232.48\). Wait, another way: The formula for the area of a regular pentagon is also \(A=\frac{5}{2}sa\), and for a regular pentagon, the ratio of side to apothem is \(s=\frac{2a}{\tan(\pi/5)}\), so \(A=\frac{5}{2}\times\frac{2a}{\tan(\pi/5)}\times a=\frac{5a^{2}}{\tan(\pi/5)}\). \(\tan(\pi/5)=\tan(36^{\circ})\approx0.7265\), so \(A=\frac{5\times64}{0.7265}\approx\frac{320}{0.7265}\approx440.47\)? Wait, I made a mistake earlier. Wait, the apothem is the distance from the center to the mid - side, and the formula \(A=\frac{1}{2}Pa\) is correct. Let's re - derive: For a regular polygon with \(n\) sides, central angle \(\theta=\frac{360}{n}\), the triangle formed by the center, a vertex, and the mid - point of a side is a right triangle with angle \(\frac{\theta}{2}\), apothem \(a\) (adjacent side), and half - side \(\frac{s}{2}\) (opposite side). So \(\tan(\frac{\theta}{2})=\frac{\frac{s}{2}}{a}\), so \(s = 2a\tan(\frac{\theta}{2})\). For \(n = 5\), \(\theta = 72^{\circ}\), \(\frac{\theta}{2}=36^{\circ}\), so \(s = 2\times8\times\tan(36^{\circ})\approx16\times0.7265\approx11.624\). Perimeter \(P = 5\times11.624 = 58.12\). Then \(A=\frac{1}{2}\times58.12\times8=232.48\). Wait, but actually, the correct formula for the area of a regular pentagon with apothem \(a\) is \(A = \frac{5}{2}a^{2}\tan(72^{\circ})\)? No, wait, let's use the formula \(A=\frac{1}{2}Pa\) correctly. Let's check with a known value: For a regular pentagon with apothem \(a = 8\), the area can also be calculated as follows. The formula for the area of a regular polygon is \(A = n\times\frac{1}{2}\times s\times a\), where \(n\) is the number of sides. So \(n = 5\), \(s\) is side length, \(a\) is apothem. We can also use the formula \(A=\frac{5a^{2}}{\tan(\pi/5)}\). Let's compute \(\tan(\pi/5)=\tan(36^{\circ})\approx0.72654\). Then \(A=\frac{5\times8^{2}}{0.72654}=\frac{320}{0.72654}\approx440.43\). Wait, I see my mistake earlier: when I calculated \(s = 2a\tan(36^{\circ})\), that's wrong. The correct relation is \(\tan(36^{\circ})=\frac{\frac{s}{2}}{a}\), so \(\frac{s}{2}=a\tan(36^{\circ})\), so \(s = 2a\tan(36^{\circ})\) is correct, but when calculating the perimeter, \(P = 5s=10a\tan(36^{\circ})\). Then \(A=\frac{1}{2}\times P\times a=\frac{1}{2}\times10a\tan(36^{\circ})\times a = 5a^{2}\tan(36^{\circ})\). Ah! There we go. I had a mistake in the perimeter calculation earlier. So \(A = 5\times8^{2}\times\tan(36^{\circ})=5\times64\times0.72654 = 320\times0.72654\approx232.49\). Wait, now I'm confused. Let's use the formula for the area of a regular pentagon: The area of a regular pentagon with side length \(s\) is \(A=\frac{5s^{2}}{4\tan(\pi/5)}\), and with apothem \(a\), \(a=\frac{s}{2\tan(\pi/5)}\), so \(s = 2a\tan(\pi/5)\). Substituting into the area formula: \(A=\frac{5(2a\tan(\pi/5))^{2}}{4\tan(\pi/5)}=\frac{5\times4a^{2}\tan^{2}(\pi/5)}{4\tan(\pi/5)} = 5a^{2}\tan(\pi/5)\). So that's correct. So \(a = 8\), so \(A = 5\times64\times\tan(36^{\circ})\approx320\times0.7265 = 232.48\). Wait, but let's check with a different approach. The formula for the area of a regular polygon is \(A = \frac{n}{2}a^{2}\tan(\frac{\pi}{n})\). For \(n = 5\), \(A=\frac{5}{2}\times8^{2}\times\tan(\frac{\pi}{5})=\frac{5}{2}\times64\times0.7265=\frac{320}{2}\times0.7265 = 160\times0.7265 = 116.24\)? No, that's not right. Wait, no, the formula \(A=\frac{n}{2}a^{2}\tan(\frac{\pi}{n})\) is incorrect. The correct formula from the right triangle: in the right triangle with angle \(\frac{\pi}{n}\) (half - central angle), adjacent side \(a\) (apothem), opposite side \(\frac{s}{2}\), so \(\tan(\frac{\pi}{n})=\frac{\frac{s}{2}}{a}\), so \(\frac{s}{2}=a\tan(\frac{\pi}{n})\), \(s = 2a\tan(\frac{\pi}{n})\). The area of one of the \(n\) isosceles triangles (formed by the center and a side) is \(\frac{1}{2}\times s\times a\). So total area \(A=n\times\frac{1}{2}\times s\times a=n\times\frac{1}{2}\times2a\tan(\frac{\pi}{n})\times a=n\times a^{2}\tan(\frac{\pi}{n})\). Ah! So for \(n = 5\), \(A = 5\times a^{2}\times\tan(\frac{\pi}{5})\). So that's correct. So \(a = 8\), \(\frac{\pi}{5}=36^{\circ}\), \(\tan(36^{\circ})\approx0.7265\), so \(A = 5\times64\times0.7265 = 320\times0.7265 = 232.48\). Wait, but I think I made a mistake in the initial assumption. Wait, the figure is a regular pentagon? Wait, the apothem is given as 8, and the formula for the area of a regular pentagon is also known to be approximately \(1.720a^{2}\) (where \(a\) is the apothem). Let's check: \(1.720\times64 = 110.08\), no, that's not matching. Wait, no, the correct coefficient for the area of a regular pentagon in terms of apothem \(a\) is \(\frac{5}{2}\times\frac{a}{\cos(36^{\circ})}\times a\)? No, I'm getting confused. Let's use the formula for the area of a regular polygon: \(A=\frac{1}{2}Pa\), where \(P\) is perimeter, \(a\) is apothem. For a regular pentagon, the side length \(s\) and apothem \(a\) are related by \(a=\frac{s}{2\tan(36^{\circ})}\), so \(s = 2a\tan(36^{\circ})\). Perimeter \(P = 5s=10a\tan(36^{\circ})\). Then \(A=\frac{1}{2}\times10a\tan(36^{\circ})\times a = 5a^{2}\tan(36^{\circ})\). So with \(a = 8\), \(A = 5\times64\times\tan(36^{\circ})\). \(\tan(36^{\circ})\approx0.7265\), so \(A = 5\times64\times0.7265 = 320\times0.7265 = 232.48\). So the area is approximately \(232.48\) square units.

Answer:

\(232.48\) (or more accurately, using \(\tan(36^{\circ})=\frac{\sqrt{5 - 2\sqrt{5}}}{2}\approx0.7265\), the exact value calculation gives approximately \(232.47\))