Regular Polygons — Hard Practice Quiz

A Geometry cheat sheet for Regular Polygons — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. A regular polygon has an interior angle of $150^\circ$. If its side length is $s=6$, what is its area?

   • $108(2+\sqrt{3})$
   • $54(2+\sqrt{3})$
   • $108\sqrt{3}$
   • $216$

   Answer: $108(2+\sqrt{3})$

2. Consider two regular polygons, Polygon A with $n_A$ sides and Polygon B with $n_B$ sides. If $n_B = 2n_A$ and both polygons have the same side length $s$, what is the ratio of their areas, $\frac{A_B}{A_A}$?

   • $\frac{2 \cot\left(\frac{\pi}{2n_A}\right)}{\cot\left(\frac{\pi}{n_A}\right)}$
   • $\frac{\cot\left(\frac{\pi}{2n_A}\right)}{2 \cot\left(\frac{\pi}{n_A}\right)}$
   • $2$
   • $\frac{1}{2}$

   Answer: $\frac{2 \cot\left(\frac{\pi}{2n_A}\right)}{\cot\left(\frac{\pi}{n_A}\right)}$

3. For a regular polygon with side length $s$, express its apothem $r$ in terms of $s$ and its exterior angle $\theta_{exterior}$. Assume $\theta_{exterior}$ is in degrees.

   • $r = \frac{1}{2}s\cot\left(\frac{\pi \theta_{exterior}}{360^\circ}\right)$
   • $r = s\cot\left(\frac{\pi \theta_{exterior}}{180^\circ}\right)$
   • $r = \frac{1}{2}s\tan\left(\frac{\pi \theta_{exterior}}{360^\circ}\right)$
   • $r = s\tan\left(\frac{\pi \theta_{exterior}}{180^\circ}\right)$

   Answer: $r = \frac{1}{2}s\cot\left(\frac{\pi \theta_{exterior}}{360^\circ}\right)$

4. As the number of sides $n$ of a regular polygon with a fixed perimeter $P$ approaches infinity, what shape does the polygon approximate, and how does its area relate to this shape?

   • A circle; its area approaches $\frac{P^2}{4\pi}$.
   • A circle; its area approaches $\frac{P^2}{2\pi}$.
   • A square; its area approaches $\frac{P^2}{16}$.
   • An ellipse; its area approaches $\frac{P^2}{4}$.

   Answer: A circle; its area approaches $\frac{P^2}{4\pi}$.

5. A regular polygon has an area of $A = 150\sqrt{3}$ and an apothem of $r=5\sqrt{3}$. What is its interior angle?

   • $120^\circ$
   • $108^\circ$
   • $135^\circ$
   • $144^\circ$

   Answer: $120^\circ$

6. Express the side length $s$ of a regular polygon in terms of its area $A$ and its interior angle $\theta_{interior}$. Assume $\theta_{interior}$ is in degrees.

   • $s = \sqrt{\frac{A (180^\circ - \theta_{interior})}{90^\circ \cot\left(\frac{\pi (180^\circ - \theta_{interior})}{360^\circ}\right)}}$
   • $s = \sqrt{\frac{A (180^\circ - \theta_{interior})}{180^\circ \cot\left(\frac{\pi (180^\circ - \theta_{interior})}{360^\circ}\right)}}$
   • $s = \sqrt{\frac{A \theta_{interior}}{90^\circ \cot\left(\frac{\pi \theta_{interior}}{360^\circ}\right)}}$
   • $s = \sqrt{\frac{A (180^\circ - \theta_{interior})}{90^\circ \tan\left(\frac{\pi (180^\circ - \theta_{interior})}{360^\circ}\right)}}$

   Answer: $s = \sqrt{\frac{A (180^\circ - \theta_{interior})}{90^\circ \cot\left(\frac{\pi (180^\circ - \theta_{interior})}{360^\circ}\right)}}$

7. If the ratio of the interior angle to the exterior angle of a regular polygon is $5:1$, and its side length is $s=4$, what is its area?

   • $48(2+\sqrt{3})$
   • $24(2+\sqrt{3})$
   • $48\sqrt{3}$
   • $96$

   Answer: $48(2+\sqrt{3})$

8. A regular polygon has a perimeter of $P=60$ and an exterior angle of $\theta_{exterior} = 60^\circ$. What is its area?

   • $150\sqrt{3}$
   • $100\sqrt{3}$
   • $75\sqrt{3}$
   • $300\sqrt{3}$

   Answer: $150\sqrt{3}$

9. For a regular polygon with a fixed side length $s$, how does its apothem $r$ change as the number of sides $n$ increases?

   • The apothem $r$ increases.
   • The apothem $r$ decreases.
   • The apothem $r$ remains constant.
   • The apothem $r$ first increases then decreases.

   Answer: The apothem $r$ increases.

10. Express the number of sides $n$ of a regular polygon in terms of its area $A$, side length $s$, and exterior angle $\theta_{exterior}$. Assume $\theta_{exterior}$ is in degrees.

    • $n = \frac{4A}{s^2\cot\left(\frac{\pi \theta_{exterior}}{360^\circ}\right)}$
    • $n = \frac{4A}{s^2\tan\left(\frac{\pi \theta_{exterior}}{360^\circ}\right)}$
    • $n = \frac{360^\circ}{\theta_{exterior}}$
    • $n = \frac{A \theta_{exterior}}{90^\circ s^2}$

    Answer: $n = \frac{4A}{s^2\cot\left(\frac{\pi \theta_{exterior}}{360^\circ}\right)}$