Quiz: Regular Polygons — 20 domande

1. A polygon has every interior angle measuring less than 180° and all of its vertices point outward. What type of polygon is it?

A. An open polygon

B. A concave polygon

C. A convex polygon

D. A self-intersecting polygon

Answer: (C)

Spiegazione

A convex polygon has every interior angle less than 180° and has vertices that point outward.

2. What does it mean for a line to be a line of symmetry of a figure?

A. It passes through the center of the figure

B. It divides the figure into two regions with equal area

C. It divides the figure into two exact mirror-image halves

D. It connects two vertices of the figure

Answer: (C)

Spiegazione

A line of symmetry divides a figure so that its two halves match exactly as mirror images. Equal area or passing through the center alone is not sufficient.

3. How many lines of symmetry does a regular octagon have?

A. 4

B. 8

C. 6

D. 16

Answer: (B)

Spiegazione

An n-sided regular polygon has exactly n lines of symmetry. Therefore, a regular octagon has 8 lines of symmetry.

4. What is the measure of each interior angle of a regular 9-sided polygon?

A. 120°

B. 160°

C. 135°

D. 140°

Answer: (D)

Spiegazione

Using ((n − 2) × 180°) / n with n = 9 gives (7 × 180°) / 9 = 140°. This is one interior angle, not the total interior-angle sum.

5. Which condition must a polygon satisfy to be regular?

A. It must have both equal angles and equal sides

B. It must have equal angles but may have unequal sides

C. It must have equal diagonals and equal angles

D. It must have equal sides but may have unequal angles

Answer: (A)

Spiegazione

A regular polygon is both equiangular, meaning all angles are equal, and equilateral, meaning all sides are equal. Equal angles alone do not guarantee regularity.

6. What is the sum of one exterior angle at each vertex of any convex polygon?

A. 180°

B. 360°

C. It depends on the polygon's interior angles

D. The number of sides multiplied by 180°

Answer: (B)

Spiegazione

For every convex polygon, one exterior angle at each vertex sums to 360°, regardless of the number of sides.

7. What is the measure of each central angle in a regular 12-sided polygon?

A. 30°

B. 36°

C. 45°

D. 15°

Answer: (A)

Spiegazione

The central angle of a regular n-gon is 360°/n. For n = 12, the angle is 360°/12 = 30°.

8. When diagonals are drawn from one vertex of a convex pentagon to its nonadjacent vertices, into how many non-overlapping triangles is the pentagon divided?

A. Two triangles

B. Four triangles

C. Three triangles

D. Five triangles

Answer: (C)

Spiegazione

Joining one vertex to its nonadjacent vertices divides an n-sided convex polygon into n−2 triangles. For a pentagon, 5−2=3.

9. Which description correctly defines a polygon?

A. A collection of line segments that may remain unjoined in the plane

B. A simple closed plane figure made from three or more line segments joined end to end

C. A single line segment whose endpoints are connected by a curve

D. A curved plane figure that encloses a region without straight sides

Answer: (B)

Spiegazione

A polygon is a simple closed plane figure formed by at least three line segments joined end to end. Unjoined segments do not form a closed polygon.

10. If an interior angle and its adjacent exterior angle at a polygon vertex measure 125° and x°, what is x?

A. 65°

B. 55°

C. 125°

D. 235°

Answer: (B)

Spiegazione

Adjacent interior and exterior angles form a straight angle, so they add to 180°. Therefore, x = 180° − 125° = 55°.

11. Which statement correctly distinguishes an inscribed polygon from a polygon circumscribed about a circle?

A. An inscribed polygon has vertices on the circle, while a circumscribed polygon has sides touching the circle

B. An inscribed polygon surrounds the circle, while a circumscribed polygon lies entirely inside the circle

C. An inscribed polygon has sides touching the circle, while a circumscribed polygon has vertices on the circle

D. An inscribed polygon has its center on the circle, while a circumscribed polygon has its center inside the circle

Answer: (A)

Spiegazione

An inscribed polygon has all its vertices on the circle. A circumscribed polygon has each side tangent to, or touching, the circle.

12. What is the difference between a vertex and a side in a polygon?

A. A vertex is a forming line segment, while a side is a common endpoint of two vertices

B. A vertex is an interior angle, while a side is the region enclosed by the polygon

C. A vertex is a common endpoint of two sides, while a side is a forming line segment

D. A vertex is the entire polygon, while a side is one of its interior angles

Answer: (C)

Spiegazione

Vertices are the common endpoints where two sides meet, while sides are the line segments that form the polygon.

13. A regular pentagon has circumradius r. Which expression gives its apothem?

A. 2r cos(36°)

B. 2r sin(36°)

C. r sin(36°)

D. r cos(36°)

Answer: (D)

Spiegazione

The apothem of a regular n-gon is a = r cos(180°/n). For a pentagon, 180°/5 = 36°, so a = r cos(36°).

14. What is the sum of the interior angles of a polygon with 8 sides?

A. 1260°

B. 1080°

C. 900°

D. 720°

Answer: (B)

Spiegazione

Using S=(n−2)×180°, an octagon has S=(8−2)×180°=1080°.

15. A regular polygon has apothem 6 units and perimeter 40 units. What is its area?

A. 46 square units

B. 120 square units

C. 20 square units

D. 240 square units

Answer: (B)

Spiegazione

The area of a regular polygon is A = 1/2 aP. Substituting a = 6 and P = 40 gives A = 1/2(6)(40) = 120 square units.

16. Which description correctly identifies an exterior angle of a convex polygon?

A. An angle inside the polygon formed by two adjacent sides

B. An angle between two opposite sides outside the polygon

C. An angle formed by two nonadjacent diagonals inside the polygon

D. An angle outside the polygon formed by a side and the extension of an adjacent side

Answer: (D)

Spiegazione

An exterior angle is outside the polygon and is formed by one side and the extension of an adjacent side. An interior angle lies inside the polygon.

17. What is the perimeter of a regular 10-gon inscribed in a circle of radius r?

A. 20r sin(18°)

B. 10r sin(18°)

C. 10r cos(18°)

D. 20r cos(18°)

Answer: (A)

Spiegazione

The perimeter formula is P = 2nr sin(180°/n). With n = 10, this becomes P = 20r sin(18°).

18. What geometric segment represents the apothem of a regular polygon?

A. A segment from the center to any vertex of the polygon

B. A segment from one vertex to the opposite vertex through the center

C. A perpendicular segment from the center to the midpoint of a side

D. A tangent segment from a vertex to the circumcircle

Answer: (C)

Spiegazione

The apothem is perpendicular to a side and extends from the polygon’s center to that side’s midpoint. It is also the radius of the incircle.

19. A regular octagon is inscribed in a circle of radius r. Which expression gives the length of one side?

A. r cos(45°)

B. 2r sin(22.5°)

C. 2r cos(22.5°)

D. r sin(45°)

Answer: (B)

Spiegazione

For an inscribed regular n-gon, the side length is s = 2r sin(180°/n). Substituting n = 8 gives s = 2r sin(22.5°).

20. A polygon has one interior angle measuring 225°. How should the polygon be classified?

A. As a regular polygon

B. As a convex polygon

C. As a concave polygon

D. As an open polygon

Answer: (C)

Spiegazione

An interior angle greater than 180° indicates a concave polygon. Convex polygons have no interior angle above 180°.