Regular Polygons — Practice Quiz

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

Formulas & key concepts

Area of Regular Polygon

$$A = \frac{1}{2}nsr = \frac{1}{4}ns^2\cot\left(\frac{\pi}{n}\right)$$

Where: \(n\) = sides, \(s\) = side length, \(r\) = apothem

Interior Angle

$$\theta_{interior} = \frac{(n-2) \cdot 180^\circ}{n}$$

Exterior Angle

$$\theta_{exterior} = \frac{360^\circ}{n}$$

Practice quiz

  1. What is the measure of an exterior angle of a regular octagon?

    • $45^\circ$
    • $135^\circ$
    • $90^\circ$
    • $60^\circ$

    Answer: $45^\circ$

  2. Calculate the measure of an interior angle of a regular hexagon.

    • $60^\circ$
    • $90^\circ$
    • $120^\circ$
    • $150^\circ$

    Answer: $120^\circ$

  3. A regular polygon has an exterior angle of $30^\circ$. How many sides does it have?

    • $10$
    • $12$
    • $15$
    • $18$

    Answer: $12$

  4. If the interior angle of a regular polygon is $144^\circ$, how many sides does the polygon have?

    • $8$
    • $9$
    • $10$
    • $12$

    Answer: $10$

  5. For any regular polygon, what is the sum of its interior and exterior angles at a single vertex?

    • $90^\circ$
    • $180^\circ$
    • $270^\circ$
    • $360^\circ$

    Answer: $180^\circ$

  6. A regular pentagon has a side length of $8 \text{ cm}$. Calculate its area.

    • $110.1 \text{ cm}^2$
    • $120.5 \text{ cm}^2$
    • $100.0 \text{ cm}^2$
    • $95.8 \text{ cm}^2$

    Answer: $110.1 \text{ cm}^2$

  7. A regular hexagon has a side length of $6 \text{ m}$ and an apothem of $3\sqrt{3} \text{ m}$. What is its area?

    • $36\sqrt{3} \text{ m}^2$
    • $54\sqrt{3} \text{ m}^2$
    • $72\sqrt{3} \text{ m}^2$
    • $108\sqrt{3} \text{ m}^2$

    Answer: $54\sqrt{3} \text{ m}^2$

  8. A regular decagon has an area of $150 \text{ cm}^2$ and a side length of $3 \text{ cm}$. What is its apothem?

    • $5 \text{ cm}$
    • $10 \text{ cm}$
    • $15 \text{ cm}$
    • $20 \text{ cm}$

    Answer: $10 \text{ cm}$

  9. A regular polygon has an interior angle of $108^\circ$. If its side length is $10 \text{ units}$, what is its area?

    • $172.0 \text{ units}^2$
    • $150.5 \text{ units}^2$
    • $180.2 \text{ units}^2$
    • $165.8 \text{ units}^2$

    Answer: $172.0 \text{ units}^2$

  10. A regular polygon has $12$ sides. What is the ratio of its exterior angle to its interior angle?

    • $\frac{1}{4}$
    • $\frac{1}{5}$
    • $\frac{1}{6}$
    • $\frac{1}{3}$

    Answer: $\frac{1}{5}$

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