Regular Polygons
Inscribed in a Circle
Master the construction and properties of regular polygons inscribed in circles using compass and straightedge techniques.
Geometry
What Is an Inscribed Polygon?
A polygon is inscribed in a circle when every one of its vertices lies exactly on the circle. The circle is called the circumscribed circle (or circumcircle), and its center is the circumcenter of the polygon.
Why Inscribed Polygons Matter
Construction
Compass and straightedge methods have been central to geometry since Euclid, forming the basis of formal geometric proof.
Architecture
Rose windows, domes, and tiled floors rely on regular polygons inscribed in circles for their symmetric beauty.
Engineering
Gears, bolt patterns, and satellite dish arrays use inscribed polygon geometry for even spacing and structural balance.
Key Vocabulary
1
Circumscribed circle: The circle passing through all vertices of the inscribed polygon.
2
Circumradius (\( r \)): The radius of the circumscribed circle: the distance from center to any vertex.
3
Central angle: The angle at the center subtended by one side of the polygon: \( \frac{360^\circ}{n} \) for an \( n \)-sided regular polygon.
4
Apothem (\( a \)): The perpendicular distance from the center to the midpoint of a side: \( a = r\cos\!\left(\frac{\pi}{n}\right) \).
5
Side length (\( s \)): Each side of the regular polygon: \( s = 2r\sin\!\left(\frac{\pi}{n}\right) \).
The Central Angle Theorem
The central angle determines where to place each vertex around the circle. Equal central angles guarantee equal spacing, which produces a regular polygon.
Side Length and Apothem Formulas
Side Length
\( s = 2r\sin\!\left(\dfrac{\pi}{n}\right) \)
Apothem
\( a = r\cos\!\left(\dfrac{\pi}{n}\right) \)
Area
\( A = \dfrac{1}{2} \cdot n \cdot s \cdot a \)
Inscribed vs. Circumscribed
| Inscribed | Circumscribed | |
|---|---|---|
| Vertices | On the circle | Outside the circle |
| Sides | Inside the circle | Tangent to the circle |
| Circle role | Circumscribed circle | Inscribed circle (incircle) |
| Radius used | Circumradius \( R \) | Inradius \( r \) |
How to Construct an Inscribed Polygon
1
Draw the circle. Set your compass to the desired circumradius \( r \) and draw the circle with center \( O \).
2
Calculate the central angle. Divide \( 360^\circ \) by \( n \) (the number of sides) to find \( \theta = \frac{360^\circ}{n} \).
3
Mark the first vertex. Choose any point on the circle as vertex \( V_1 \).
4
Mark successive vertices. Using a protractor (or compass-based angle construction), mark each vertex at intervals of \( \theta \) around the circle.
5
Connect consecutive vertices. Draw straight segments between adjacent vertices to complete the polygon.
Example 1: Inscribed Equilateral Triangle
Example 2: Inscribed Regular Hexagon
Example 3: Regular Pentagon, \( r = 8 \) cm
1
Central angle: \( \theta = \frac{360^\circ}{5} = 72^\circ \)
2
Side length: \( s = 2(8)\sin\!\left(\frac{\pi}{5}\right) = 16\sin(36^\circ) \approx \) \( 9.41 \) cm
3
Apothem: \( a = 8\cos\!\left(\frac{\pi}{5}\right) = 8\cos(36^\circ) \approx \) \( 6.47 \) cm
4
Area: \( A = \frac{1}{2}(5)(9.41)(6.47) \approx \) \( 152.2 \text{ cm}^2 \)
Compass and Straightedge Construction
Not all regular polygons can be constructed with compass and straightedge alone. Gauss proved that a regular \( n \)-gon is constructible if and only if \( n \) is a product of a power of 2 and distinct Fermat primes.
Constructible
\( n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, \ldots \)
Not Constructible
\( n = 7, 9, 11, 13, 14, 18, 19, \ldots \)
Constructing a Regular Hexagon
As \( n \) Increases: Approaching a Circle
Common Mistakes
✕ Confusing \( r \) and \( a \)
Using the apothem where the circumradius belongs (or vice versa). Remember: \( r \) goes to vertices, \( a \) goes to side midpoints.
✕ Wrong angle mode
Using degrees in a function expecting radians (or vice-versa). Use your calculator's DEG mode for angles like \( 180^\circ/n \) and RAD mode for angles like \( \pi/n \).
✕ Unequal compass setting
Changing the compass width during construction. The compass must stay locked at the circumradius for consistent vertex spacing.
Quick Reference Table
| Polygon | Central Angle | Side \( s \) (if \( r = 1 \)) | Apothem \( a \) (if \( r=1 \)) |
|---|---|---|---|
| Triangle (\( n = 3 \)) | \( 120^\circ \) | \( \sqrt{3} \approx 1.732 \) | \( 0.500 \) |
| Square (\( n = 4 \)) | \( 90^\circ \) | \( \sqrt{2} \approx 1.414 \) | \( 0.707 \) |
| Pentagon (\( n = 5 \)) | \( 72^\circ \) | \( 1.176 \) | \( 0.809 \) |
| Hexagon (\( n = 6 \)) | \( 60^\circ \) | \( 1.000 \) | \( 0.866 \) |
| Octagon (\( n = 8 \)) | \( 45^\circ \) | \( 0.765 \) | \( 0.924 \) |
| Dodecagon (\( n = 12 \)) | \( 30^\circ \) | \( 0.518 \) | \( 0.966 \) |
Question 1 of 5
What is the central angle of a regular octagon inscribed in a circle?
The central angle is \( \frac{360^\circ}{n} = \frac{360^\circ}{8} = 45^\circ \). An octagon has 8 sides, so each central angle is \( 45^\circ \).
Question 2 of 5
A regular hexagon is inscribed in a circle of radius 12 cm. What is the side length of the hexagon?
For a regular hexagon, \( s = 2r\sin\!\left(\frac{\pi}{6}\right) = 2(12)\left(\frac{1}{2}\right) = 12 \) cm. The side length of an inscribed regular hexagon always equals the circumradius.
Question 3 of 5
The apothem of a regular polygon inscribed in a circle is the:
The apothem is the perpendicular distance from the center to the midpoint of a side. The distance from center to vertex is the circumradius, not the apothem.
Question 4 of 5
A regular pentagon is inscribed in a circle of radius 10 cm. Which expression gives the side length?
The side length formula is \( s = 2r\sin\!\left(\frac{\pi}{n}\right) \). With \( r = 10 \) and \( n = 5 \): \( s = 2(10)\sin\!\left(\frac{\pi}{5}\right) = 20\sin\!\left(\frac{\pi}{5}\right) \). Option (d) gives the apothem, not the side length.
Question 5 of 5
Which of the following regular polygons cannot be constructed with compass and straightedge alone?
A regular heptagon (\( n = 7 \)) cannot be constructed with compass and straightedge. By Gauss's theorem, \( n \) must be a product of a power of 2 and distinct Fermat primes. Since 7 is not a Fermat prime (\( 7 \neq 2^{2^k} + 1 \) for any \( k \)), the regular 7-gon is not constructible.