Regular Polygons: Angles, Diagonals & Formulas

9th Grade

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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.

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| Attempts: 12 | Questions: 20 | Updated: Jan 16, 2026

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1) The sum of interior angles of a regular hexagon is:

360°

540°

720°

900°

Sum of interior angles = ( n − 2 ) × 180 = ( 6 − 2 ) × 180 = 720°.

Explanation

Sum of interior angles = ( n − 2 ) × 180 = ( 6 − 2 ) × 180 = 720°.

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About This Quiz

Regular Polygons: Angles, Diagonals & Formulas - Quiz

Get ready to discover the hidden math inside regular polygons! In this quiz, you’ll calculate interior and exterior angles, find the sum of angles, and work out how many diagonals a polygon has. You’ll also see how special formulas help with hexagons, octagons, and more. With each problem, you’ll grow... see moremore confident in measuring and describing these shapes.
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2) The measure of each exterior angle of a regular octagon is:

36°

40°

45°

60°

The exterior angle of a regular polygon can be calculated using the formula 360° divided by the number of sides. For an octagon, which has 8 sides, each exterior angle is 360°/8 = 45°.

Explanation

The exterior angle of a regular polygon can be calculated using the formula 360° divided by the number of sides. For an octagon, which has 8 sides, each exterior angle is 360°/8 = 45°.

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3) The formula for the sum of interior angles of an n-sided polygon is:

180n

180(n–1)

90(n–2)

180(n–2)

The sum of the interior angles of an n-sided polygon is calculated using the formula 180(n–2) degrees. This formula derives from the fact that a polygon can be divided into (n–2) triangles, and since each triangle has interior angles that sum to 180 degrees, multiplying by the number of triangles gives the total sum of the interior angles.

Explanation

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4) Each interior angle of a regular decagon measures:

144°

150°

156°

160°

To find the measure of each interior angle of a regular decagon, you can use the formula: (n-2) × 180° / n, where n is the number of sides. For a decagon (n=10), the calculation is (10-2) × 180° / 10 = 144°.

Explanation

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5) The number of diagonals in a polygon with n sides is:

[n(n–3)]/2

[n(n–2)]/2

N²–n

N–3

The formula for calculating the number of diagonals in a polygon is derived from the fact that each vertex can connect to (n-3) other vertices to form diagonals, and since each diagonal connects two vertices, the total is divided by 2.

Explanation

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6) How many diagonals does a regular hexagon have?

A regular hexagon has 9 diagonals, which can be calculated using the formula n(n-3)/2, where n is the number of sides (6 in this case). Thus, 6(6-3)/2 = 9.

Explanation

A regular hexagon has 9 diagonals, which can be calculated using the formula n(n-3)/2, where n is the number of sides (6 in this case). Thus, 6(6-3)/2 = 9.

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7) The central angle of a regular polygon with n sides is:

180/n

360/n

90/n

(n−2)×180/n

The apothem is a line segment from the center of a regular polygon that is perpendicular to one of its sides, specifically connecting to the midpoint of that side.

Explanation

The apothem is a line segment from the center of a regular polygon that is perpendicular to one of its sides, specifically connecting to the midpoint of that side.

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8) Each central angle of a regular nonagon measures:

30°

36°

40°

45°

A regular nonagon has 9 sides, and the formula for calculating the central angle is (360°/number of sides). Thus, 360°/9 = 40°.

Explanation

A regular nonagon has 9 sides, and the formula for calculating the central angle is (360°/number of sides). Thus, 360°/9 = 40°.

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9) In a regular polygon, the apothem connects the center to:

A vertex

The midpoint of a side

The midpoint of a diagonal

The circumcircle

The apothem is drawn from the center and meets the side at a right angle.

Explanation

The apothem is drawn from the center and meets the side at a right angle.

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10) The apothem of a regular polygon is always:

Perpendicular to a side

Parallel to a side

Equal to the radius

A diagonal

The apothem of a regular polygon is the line segment from the center to the midpoint of one of its sides, and it is always perpendicular to that side.

Explanation

The apothem of a regular polygon is the line segment from the center to the midpoint of one of its sides, and it is always perpendicular to that side.

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11) Which of the following regular polygons can tessellate the plane?

Equilateral triangle, square, hexagon

Pentagon, hexagon, octagon

Triangle, pentagon, decagon

Octagon, decagon, dodecagon

These three have interior angles that fit exactly into 360°.

Explanation

These three have interior angles that fit exactly into 360°.

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12) A regular polygon with each exterior angle 24° has how many sides?

The sum of the exterior angles of a polygon is always 360 degrees. To find the number of sides, you can divide 360 by the measure of each exterior angle. In this case, 360 / 24 = 15, so the polygon has 15 sides.

Explanation

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13) Which statement is always true about a regular polygon?

Its diagonals are always equal

Its sides are congruent and angles are congruent

Its sides are perpendicular to each other

Its diagonals are always perpendicular

A regular polygon is defined as a polygon that is both equilateral (all sides are of equal length) and equiangular (all interior angles are equal). Therefore, Its sides are congruent and angles are congruent is the only statement that is universally true for all regular polygons.

Explanation

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14) The sum of exterior angles of any polygon is:

180°

270°

360°

540°

A regular polygon is defined as a polygon with all sides and angles equal. The sum of the interior angles of a polygon can be calculated using the formula (n-2) × 180°, where n is the number of sides. For regular polygons with 3 or more sides, the sum of angles will always be greater than 180°, and 360° is the angle sum for a quadrilateral, making it applicable to simple regular shapes like squares.

Explanation

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15) The area of a regular polygon can be found using:

The area of a regular polygon is calculated using the formula that relates the number of sides and the length of a side, typically involving the number 6 when using the apothem or side length in standard formulas.

Explanation

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16) The area of a regular polygon can be found using:

½ × perimeter × apothem

½ × base × height

πr²

S²

The formula for the area of a regular polygon involves the perimeter and the apothem, which is the distance from the center to the midpoint of a side. This formula is essential for calculating the area when the shape has equal sides and angles.

Explanation

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17) A regular polygon has each central angle measuring 40°. how many sides does it have?

The sum of all central angles in a polygon is 360 degrees. To find the number of sides (n), you can use the formula: n = 360 / central angle. Here, n = 360 / 40 = 9.

Explanation

The sum of all central angles in a polygon is 360 degrees. To find the number of sides (n), you can use the formula: n = 360 / central angle. Here, n = 360 / 40 = 9.

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18) Which regular polygon has each exterior angle equal to 72°?

Pentagon

Hexagon

Octagon

Decagon

A regular pentagon has 5 sides, and the formula for calculating the measure of each exterior angle of a regular polygon is 360° divided by the number of sides. Thus, for a pentagon, 360°/5 = 72°.

Explanation

A regular pentagon has 5 sides, and the formula for calculating the measure of each exterior angle of a regular polygon is 360° divided by the number of sides. Thus, for a pentagon, 360°/5 = 72°.

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19) Which regular polygon has the fewest sides?

Triangle

Quadrilateral

Pentagon

Hexagon

A triangle is a polygon with three sides, which is the minimum number of sides required to form a polygon.

Explanation

A triangle is a polygon with three sides, which is the minimum number of sides required to form a polygon.

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20) The measure of each interior angle of a regular pentagon is:

90°

108°

120°

135°

A regular pentagon has five sides. The formula to calculate the measure of each interior angle of a regular polygon is (n-2) × 180° / n, where n is the number of sides. For a pentagon, this becomes (5-2) × 180° / 5 = 108°.

Explanation

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