Unit notes and practice questions

Grade 9 Mathematics unit 6: Regular Polygons

Everything here is for unit 6 of the Grade 9 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

3
Cards: 10
Practice questions on unit 6: 181–208
The textbook pages for unit 6

What unit 6 covers

1.Sum of Interior Angles of a Convex Polygon
2.Sum of Exterior Angles of a Convex Polygon
3.Measures of Each Interior Angle and Exterior Angle of a Regular Polygon
4.Properties of Regular Polygons: pentagon, hexagon, octagon and decagon

Short notes for unit 6

The cards filed under this chapter on the Grade 9 Mathematics page. Open one at a time.

Area of a Regular PolygonFormula

A = \frac{1}{2} a P

A = \frac{1}{2} a P

$A$: area of the regular polygon $\text{m}^2$
$a$: apothem of the regular polygon $\text{m}$
$P$: perimeter of the regular polygon $\text{m}$

When you use it

Use this formula to calculate the area of any regular polygon when its apothem and perimeter are known.

Watch out

The apothem is the perpendicular distance from the center to a side, not the radius to a vertex.

Drafted from Grade 9 Mathematics, pages 181-208, then checked twice before it went up

Interior Angle of a Regular PolygonFormula

\theta = \frac{(n - 2) \times 180^\circ}{n}

\theta = \frac{(n - 2) \times 180^\circ}{n}

$\theta$: measure of each interior angle $^\circ$
$n$: number of sides of the regular polygon $1$

When you use it

Use this formula to find the measure of a single interior angle in a regular polygon where all angles are equal.

Watch out

This formula applies only to regular polygons. Irregular polygons do not have equal interior angles.

Drafted from Grade 9 Mathematics, pages 181-208, then checked twice before it went up

Side Length of Inscribed Regular PolygonFormula

s = 2r \sin\left(\frac{180^\circ}{n}\right)

s = 2r \sin\left(\frac{180^\circ}{n}\right)

$s$: side length of the regular polygon $\text{m}$
$r$: radius of the circumscribed circle $\text{m}$
$n$: number of sides of the regular polygon $1$

When you use it

Use this formula to calculate the side length of an n-sided regular polygon inscribed in a circle of radius r.

Watch out

Remember to multiply by 2 because the perpendicular bisector splits the full side into two equal halves.

Drafted from Grade 9 Mathematics, pages 181-208, then checked twice before it went up

Practice questions on unit 6

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

According to the definition, what is a regular polygon?

A.A polygon that is equiangular (equal angles) and equilateral (equal sides)
B.A polygon with at least one interior angle greater than 180 degrees
C.Any closed plane figure with four unequal sides
D.A polygon whose vertices point inwards

Show the answer and why

Answer: A A polygon that is equiangular (equal angles) and equilateral (equal sides)

Why

A polygon is regular if it is both equiangular (all interior angles are equal) and equilateral (all sides are equal in length).

Question 2

How many lines of symmetry does an n-sided regular polygon have?

A.n - 2
B.n
C.2n
D.n / 2

Show the answer and why

Answer: B n

Why

An n-sided regular polygon always has exactly n lines of symmetry. For example, a regular hexagon (6 sides) has 6 lines of symmetry.

Question 3

What is the sum of the measures of the exterior angles of any convex polygon?

A.180°
B.360°
C.540°
D.720°

Show the answer and why

Answer: B 360°

Why

For any n-sided convex polygon, the sum of the measures of the exterior angles is always 360°.

Question 4

What is the measure of each exterior angle of a regular pentagon (5-sided polygon)?

A.60°
B.72°
C.108°
D.120°

Show the answer and why

Answer: B 72°

Why

The measure of each exterior angle of an n-sided regular polygon is 360° / n. For a pentagon with n = 5, the measure is 360° / 5 = 72°.

Question 5

What is the sum of the interior angles of a convex heptagon (a 7-sided polygon)?

A.720°
B.900°
C.1,080°
D.1,260°

Show the answer and why

Answer: B 900°

Why

The sum of the interior angles of an n-sided polygon is (n - 2) × 180°. For a heptagon, n = 7: (7 - 2) × 180° = 5 × 180° = 900°.

Question 6

A student in Hawassa finds that the sum of the interior angles of a polygon is 1,440°. How many sides does the polygon have?

A.8
B.9
C.10
D.12

Show the answer and why

Answer: C 10

Why

Using the formula (n - 2) × 180° = 1,440°, we divide both sides by 180° to get n - 2 = 8, which gives n = 10 (a decagon).

Question 7

What is the measure of each interior angle of a regular octagon (8-sided polygon)?

A.108°
B.120°
C.135°
D.144°

Show the answer and why

Answer: C 135°

Why

Each interior angle of an n-sided regular polygon is ((n - 2) × 180°) / n. For an octagon (n = 8), this is ((8 - 2) × 180°) / 8 = 1,080° / 8 = 135°.

Question 8

What is the measure of each central angle of a regular decagon (10-sided polygon)?

A.36°
B.45°
C.60°
D.72°

Show the answer and why

Answer: A 36°

Why

The measure of each central angle of an n-sided regular polygon is 360° / n. For a decagon, n = 10, so 360° / 10 = 36°.

Question 9

Four interior angles of a pentagon measure 95°, 100°, 120°, and 70°. What is the measure of the fifth interior angle?

A.135°
B.145°
C.155°
D.165°

Show the answer and why

Answer: C 155°

Why

The sum of the interior angles of a pentagon is (5 - 2) × 180° = 540°. The sum of the four given angles is 95° + 100° + 120° + 70° = 385°. The fifth angle is 540° - 385° = 155°.

Question 10

A square is inscribed in a circle of radius r = 5 cm. Using the formula s = 2r sin(180° / n), what is the side length s of the square?