Regular icosagon and a regular tetracontagon are inscribed in a circle with a radius of 15 units Which of the two polygons has bigger area and how much? Tetracontagon with A = 695.28 sq.units Icosagon with A = 703.96 sq.units Icosagon with A = 695.28 sq.units
The Correct Answer and Explanation is:
To determine which of the two polygons—a regular icosagon (20 sides) and a regular tetracontagon (40 sides)—has a larger area when inscribed in a circle of radius 15 units, we apply the standard formula for the area AAA of a regular nnn-gon inscribed in a circle of radius rrr:A=n2r2sin(2πn)A = \frac{n}{2} r^2 \sin\left(\frac{2\pi}{n}\right)A=2nr2sin(n2π)
Step 1: Calculate the area of the icosagon (n = 20)
A20=202⋅152⋅sin(2π20)=10⋅225⋅sin(π5)=2250⋅sin(36∘)A_{20} = \frac{20}{2} \cdot 15^2 \cdot \sin\left(\frac{2\pi}{20}\right) = 10 \cdot 225 \cdot \sin\left(\frac{\pi}{5}\right) = 2250 \cdot \sin\left(36^\circ\right)A20=220⋅152⋅sin(202π)=10⋅225⋅sin(5π)=2250⋅sin(36∘)sin(36∘)≈0.5878⇒A20≈2250⋅0.5878≈1322.55 square units\sin(36^\circ) \approx 0.5878 \Rightarrow A_{20} \approx 2250 \cdot 0.5878 \approx 1322.55 \text{ square units}sin(36∘)≈0.5878⇒A20≈2250⋅0.5878≈1322.55 square units
Step 2: Calculate the area of the tetracontagon (n = 40)
A40=402⋅152⋅sin(2π40)=20⋅225⋅sin(π10)=4500⋅sin(18∘)A_{40} = \frac{40}{2} \cdot 15^2 \cdot \sin\left(\frac{2\pi}{40}\right) = 20 \cdot 225 \cdot \sin\left(\frac{\pi}{10}\right) = 4500 \cdot \sin(18^\circ)A40=240⋅152⋅sin(402π)=20⋅225⋅sin(10π)=4500⋅sin(18∘)sin(18∘)≈0.3090⇒A40≈4500⋅0.3090≈1390.5 square units\sin(18^\circ) \approx 0.3090 \Rightarrow A_{40} \approx 4500 \cdot 0.3090 \approx 1390.5 \text{ square units}sin(18∘)≈0.3090⇒A40≈4500⋅0.3090≈1390.5 square units
Comparison and Conclusion
Comparing the two:
- Area of icosagon ≈ 1322.55 sq. units
- Area of tetracontagon ≈ 1390.5 sq. units
Thus, the regular tetracontagon has the larger area.
Correct Answer:
Tetracontagon with A = 1390.5 sq. units
This result follows a geometric principle: as the number of sides of a regular polygon inscribed in a circle increases, the polygon more closely approximates the area of the circle itself. Therefore, the more sides, the larger the area, given the same radius.
