Find the area of a regular decagon with radius 12 cm.A = _
The Correct Answer and Explanation is:
Answer:
The area AAA of a regular decagon (10-sided polygon) with a radius (distance from center to a vertex) of 12 cm is approximately: A=10×12×r2×sin(360∘10)=10×12×122×sin(36∘)=5×144×sin(36∘)≈720×0.5878≈423.22 cm2A = 10 \times \frac{1}{2} \times r^2 \times \sin\left(\frac{360^\circ}{10}\right) = 10 \times \frac{1}{2} \times 12^2 \times \sin(36^\circ) = 5 \times 144 \times \sin(36^\circ) \approx 720 \times 0.5878 \approx \boxed{423.22\ \text{cm}^2}A=10×21×r2×sin(10360∘)=10×21×122×sin(36∘)=5×144×sin(36∘)≈720×0.5878≈423.22 cm2
Explanation (Textbook Style):
A regular decagon is a ten-sided polygon where all sides and angles are equal. To find the area of a regular polygon given the radius (distance from the center to any vertex), the formula derived from trigonometry is: A=12⋅n⋅r2⋅sin(360∘n)A = \frac{1}{2} \cdot n \cdot r^2 \cdot \sin\left(\frac{360^\circ}{n}\right)A=21⋅n⋅r2⋅sin(n360∘)
Here:
- nnn is the number of sides (10 for a decagon),
- rrr is the radius (12 cm),
- sin(360∘n)\sin\left(\frac{360^\circ}{n}\right)sin(n360∘) comes from dividing the polygon into nnn congruent isosceles triangles from the center.
Plugging the values into the formula: A=12⋅10⋅122⋅sin(36∘)A = \frac{1}{2} \cdot 10 \cdot 12^2 \cdot \sin\left(36^\circ\right)A=21⋅10⋅122⋅sin(36∘)
Calculate 122=14412^2 = 144122=144. Then: A=5⋅144⋅sin(36∘)A = 5 \cdot 144 \cdot \sin(36^\circ)A=5⋅144⋅sin(36∘)
Using a calculator, sin(36∘)≈0.5878\sin(36^\circ) \approx 0.5878sin(36∘)≈0.5878. So: A≈720⋅0.5878≈423.22 cm2A \approx 720 \cdot 0.5878 \approx 423.22\ \text{cm}^2A≈720⋅0.5878≈423.22 cm2
Thus, the area of a regular decagon with a radius of 12 cm is approximately 423.22 square centimeters. This method uses trigonometry and is applicable for any regular polygon if the radius and number of sides are known.
