What is the diameter of a circle inscribed in a regular polygon of 15 sides with side length unity?
Calculating Inscribed Circle Diameter in Regular Polygons
The problem asks for the diameter of a circle inscribed in a regular polygon with 15 sides and a side length of unity (which means side length is 1). Let's break down how to solve this geometry problem.
Understanding the Inscribed Circle and Apothem
A circle inscribed in a regular polygon is tangent to all the sides of the polygon. The center of the inscribed circle is the same as the center of the polygon. The radius of the inscribed circle is the perpendicular distance from the center to the midpoint of any side. This radius is also known as the apothem of the regular polygon.
Let:
- \( n \) be the number of sides of the regular polygon.
- \( s \) be the side length of the polygon.
- \( r \) be the radius (apothem) of the inscribed circle.
- \( D \) be the diameter of the inscribed circle, so \( D = 2r \).
Deriving the Formula for the Apothem
Consider a regular polygon with \( n \) sides. We can divide the polygon into \( n \) congruent isosceles triangles by connecting the vertices to the center. The angle at the center of each triangle is \( \frac{360^\circ}{n} \). The apothem \( r \) is the altitude from the center to the base (which is a side of the polygon) of one of these triangles. This altitude bisects the central angle and the base side.
So, we form a right-angled triangle with the following properties:
- Hypotenuse: Line from the center to a vertex.
- One leg: The apothem \( r \).
- Other leg: Half of the side length, \( \frac{s}{2} \).
- Angle opposite the leg \( \frac{s}{2} \): Half of the central angle, which is \( \frac{1}{2} \times \frac{360^\circ}{n} = \frac{180^\circ}{n} \).
In this right-angled triangle, using the tangent trigonometric ratio:
\( \tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{s/2}{r} \)
Now, we can solve for the radius \( r \):
\( r = \frac{s/2}{\tan\left(\frac{180^\circ}{n}\right)} = \frac{s}{2} \times \frac{1}{\tan\left(\frac{180^\circ}{n}\right)} \)
Since \( \frac{1}{\tan(\theta)} = \cot(\theta) \), the formula for the apothem (radius) is:
\( r = \frac{s}{2} \cot\left(\frac{180^\circ}{n}\right) \)
Calculating the Diameter
The diameter \( D \) is twice the radius:
\( D = 2r = 2 \times \left( \frac{s}{2} \cot\left(\frac{180^\circ}{n}\right) \right) \)
\( D = s \cot\left(\frac{180^\circ}{n}\right) \)
Applying Values for the 15-sided Polygon
We are given:
- Number of sides, \( n = 15 \).
- Side length, \( s = 1 \) (unity).
Substitute these values into the diameter formula:
\( D = 1 \times \cot\left(\frac{180^\circ}{15}\right) \)
First, calculate the angle \( \frac{180^\circ}{15} \):
\( \frac{180}{15} = \frac{180 \div 3}{15 \div 3} = \frac{60}{5} = 12^\circ \)
So, the diameter is:
\( D = 1 \times \cot(12^\circ) = \cot(12^\circ) \)
Therefore, the diameter of the circle inscribed in a regular polygon of 15 sides with side length unity is \( \cot 12^\circ \).
Revision Table: Inscribed Circle in Regular Polygon
Let \( n \) be the number of sides and \( s \) be the side length.
| Property | Formula |
|---|---|
| Angle at center per side | \( \frac{360^\circ}{n} \) |
| Angle used in right triangle | \( \frac{180^\circ}{n} \) |
| Radius (Apothem) \( r \) | \( \frac{s}{2} \cot\left(\frac{180^\circ}{n}\right) \) |
| Diameter \( D \) | \( s \cot\left(\frac{180^\circ}{n}\right) \) |
Additional Information: Circumscribed Circle
Besides the inscribed circle, a regular polygon also has a circumscribed circle that passes through all its vertices. The radius of the circumscribed circle (let's call it \( R \)) is the distance from the center to any vertex.
In the same right-angled triangle we used before, the hypotenuse is the radius of the circumscribed circle \( R \). Using the sine or cosine ratios:
- Using sine: \( \sin\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{s/2}{R} \)
- So, \( R = \frac{s/2}{\sin\left(\frac{180^\circ}{n}\right)} = \frac{s}{2 \sin\left(\frac{180^\circ}{n}\right)} = \frac{s}{2} \csc\left(\frac{180^\circ}{n}\right) \)
Understanding both inscribed and circumscribed circles is key to solving various geometry problems involving regular polygons.
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