Calculating the Area of a Regular Decagon Inscribed in a Circle

In this article, we will explore how to calculate the area of a regular decagon inscribed in a circle with a diameter of 15 cm. We will use the formula for the area of a regular polygon and break down the process step by step.

Formula for the Area of a Regular Polygon

The area of a regular polygon inscribed in a circle can be calculated using the formula:

Area (frac{1}{2} n r^2 sinleft(frac{2pi}{n}right))

Understanding the Variables

n The number of sides of the decagon. For a decagon, n 10. r The radius of the circle in which the decagon is inscribed. (sin(cdot)) The sine function, which is used to calculate the length of the side of the decagon.

Given Data

The diameter of the circle is 15 cm. Therefore, the radius r is:

r (frac{15}{2}) 7.5 cm

Step-by-Step Calculation

We will now substitute n 10 and r 7.5 into the area formula.

Calculate the square of the radius:

7.5^2 56.25

Calculate the sine of the central angle:

(sinleft(frac{2pi}{10}right) sinleft(frac{pi}{5}right))

The value of (sinleft(frac{pi}{5}right)) is approximately 0.5878.

Substitute the values into the area formula:

Area (frac{1}{2} times 10 times 56.25 times 0.5878)

Simplify the expression:

(frac{1}{2} times 10 5)

5 times 56.25 281.25)

Therefore, the area is approximately:

281.25 times 0.5878 approx 165.1 text{ cm}^2)

Conclusion

The area of a regular decagon inscribed in a circle with a diameter of 15 cm is approximately 165.1 square centimeters.

Additional Information

A regular decagon is a polygon with 10 equal sides. When inscribed in a circle, the radius of the circle is also the circumradius of the decagon. Knowing the radius and the number of sides, we can calculate the area of the decagon.

The Role of the Apothem

In the construction of the octagon, a perpendicular is drawn from the center of the circle to one of the sides, which is the apothem. The apothem is crucial in finding the length of the side of the decagon.

Circular Geometry

The central angle of a regular decagon is 36°. This means that each exterior angle is also 36°, and the decagon fits perfectly within the circle, with its vertices touching the circumference.

Understanding and applying these concepts can help in solving similar problems involving inscribed polygons and circles.