Finding Angles in Regular Polygons Worksheet | Cazoom Maths Worksheets - Free Printable

Problem Analysis

The worksheet is about finding angles in regular polygons. Regular polygons have equal side lengths and equal interior angles. The key formulas we will use are:

1. Sum of Interior Angles of a Polygon:
\[
\text{Sum of Interior Angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides.

2. Size of Each Interior Angle:
\[
\text{Each Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
\]

3. Size of Each Exterior Angle:
\[
\text{Each Exterior Angle} = \frac{360^\circ}{n}
\]

Section A: Finding Angles in Regular Polygons

#### 1) Find the size of each interior angle in a regular hexagon.
- A hexagon has \( n = 6 \) sides.
- Using the formula for each interior angle:
\[
\text{Each Interior Angle} = \frac{(6 - 2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ
\]
- Answer: \( 120^\circ \)

#### 2) Find the size of each interior angle in a regular nonagon.
- A nonagon has \( n = 9 \) sides.
- Using the formula for each interior angle:
\[
\text{Each Interior Angle} = \frac{(9 - 2) \times 180^\circ}{9} = \frac{7 \times 180^\circ}{9} = \frac{1260^\circ}{9} = 140^\circ
\]
- Answer: \( 140^\circ \)

#### 3) Find the size of each exterior angle in a regular dodecagon.
- A dodecagon has \( n = 12 \) sides.
- Using the formula for each exterior angle:
\[
\text{Each Exterior Angle} = \frac{360^\circ}{12} = 30^\circ
\]
- Answer: \( 30^\circ \)

#### 4) Write an expression for the sum of the interior angles in a polygon with \( n \) sides.
- The sum of the interior angles of a polygon with \( n \) sides is:
\[
\text{Sum of Interior Angles} = (n - 2) \times 180^\circ
\]
- Answer: \( (n - 2) \times 180^\circ \)

#### 5) Write an expression for the size of each interior angle in a polygon with \( n \) sides.
- The size of each interior angle in a polygon with \( n \) sides is:
\[
\text{Each Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
\]
- Answer: \( \frac{(n - 2) \times 180^\circ}{n} \)

#### 6) Write an expression for the size of each exterior angle in a polygon with \( n \) sides.
- The size of each exterior angle in a polygon with \( n \) sides is:
\[
\text{Each Exterior Angle} = \frac{360^\circ}{n}
\]
- Answer: \( \frac{360^\circ}{n} \)

Section B: Find the Angles Indicated

#### 1)
- The figure is a regular pentagon.
- Each interior angle of a regular pentagon:
\[
\text{Each Interior Angle} = \frac{(5 - 2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ
\]
- \( a \) is an interior angle, so \( a = 108^\circ \).
- \( b \) is an exterior angle, so:
\[
b = 180^\circ - 108^\circ = 72^\circ
\]
- Answers: \( a = 108^\circ \), \( b = 72^\circ \)

#### 2)
- The figure is a regular pentagon.
- Each interior angle of a regular pentagon is \( 108^\circ \).
- \( c \) is an exterior angle, so:
\[
c = 180^\circ - 108^\circ = 72^\circ
\]
- \( d \) is an interior angle, so \( d = 108^\circ \).
- Answers: \( c = 72^\circ \), \( d = 108^\circ \)

#### 3)
- The figure is a regular pentagon.
- Each interior angle of a regular pentagon is \( 108^\circ \).
- \( e \) is an exterior angle, so:
\[
e = 180^\circ - 108^\circ = 72^\circ
\]
- \( f \) is an exterior angle, so:
\[
f = 72^\circ
\]
- Answers: \( e = 72^\circ \), \( f = 72^\circ \)

#### 4)
- The figure is a regular octagon.
- Each interior angle of a regular octagon:
\[
\text{Each Interior Angle} = \frac{(8 - 2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ
\]
- \( g \) is an exterior angle, so:
\[
g = 180^\circ - 135^\circ = 45^\circ
\]
- \( h \) is the angle formed by two diagonals inside the octagon. Since the diagonals intersect at the center, \( h \) is twice the exterior angle:
\[
h = 2 \times 45^\circ = 90^\circ
\]
- Answers: \( g = 45^\circ \), \( h = 90^\circ \)

#### 5)
- The figure is a regular hexagon.
- Each interior angle of a regular hexagon is \( 120^\circ \).
- \( i \) is an exterior angle, so:
\[
i = 180^\circ - 120^\circ = 60^\circ
\]
- \( j \) is the angle formed by two diagonals inside the hexagon. Since the diagonals intersect at the center, \( j \) is twice the exterior angle:
\[
j = 2 \times 60^\circ = 120^\circ
\]
- Answers: \( i = 60^\circ \), \( j = 120^\circ \)

#### 6)
- The figure is a regular octagon.
- Each interior angle of a regular octagon is \( 135^\circ \).
- \( k \) is an exterior angle, so:
\[
k = 180^\circ - 135^\circ = 45^\circ
\]
- \( l \) is the angle formed by two diagonals inside the octagon. Since the diagonals intersect at the center, \( l \) is twice the exterior angle:
\[
l = 2 \times 45^\circ = 90^\circ
\]
- Answers: \( k = 45^\circ \), \( l = 90^\circ \)

Final Answers

\[
\boxed{
\begin{array}{l}
\text{Section A:} \\
1) 120^\circ \\
2) 140^\circ \\
3) 30^\circ \\
4) (n - 2) \times 180^\circ \\
5) \frac{(n - 2) \times 180^\circ}{n} \\
6) \frac{360^\circ}{n} \\
\text{Section B:} \\
1) a = 108^\circ, b = 72^\circ \\
2) c = 72^\circ, d = 108^\circ \\
3) e = 72^\circ, f = 72^\circ \\
4) g = 45^\circ, h = 90^\circ \\
5) i = 60^\circ, j = 120^\circ \\
6) k = 45^\circ, l = 90^\circ \\
\end{array}
}
\]