Practice · A regular polygon has as many axes as sides · Sensim Math

Variant 1 easy answer: 5 lines of symmetry

Find the total number of lines of symmetry of the regular pentagon below.

The figure is a regular pentagon: a regular polygon with $5$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular pentagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 5 equal sides and 5 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 5-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 5 sides

#5 Look for a Pattern 4.G.A.3

The pentagon has 5 sides, so it has 5 lines of symmetry. Because 5 is odd there are no opposite vertices at all: every line runs from one vertex straight to the middle of the side across from it, and there are 5 vertices.

5 \text{ vertices} \to 5

5 lines of symmetry.

Answer: 5 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 5 = 36 degrees. Going a full half turn covers 5 of them and then repeats, which agrees with the count.

Another way: An odd-sided polygon has no opposite corner to any corner, so every axis is the vertex-to-side-midpoint kind. Counting the vertices counts the axes: 5.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 2 easy answer: 6 lines of symmetry

Find the total number of lines of symmetry of the regular hexagon below.

The figure is a regular hexagon: a regular polygon with $6$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular hexagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 6 equal sides and 6 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 6-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 6 sides

#5 Look for a Pattern 4.G.A.3

The hexagon has 6 sides, so it has 6 lines of symmetry. Because 6 is even they come in two kinds: 3 lines through pairs of opposite vertices and 3 through the midpoints of pairs of opposite sides.

3 + 3 = 6

6 lines of symmetry.

Answer: 6 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 6 = 30 degrees. Going a full half turn covers 6 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 3 axes join opposite corners and 3 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 3 easy answer: 7 lines of symmetry

Find the total number of lines of symmetry of the regular heptagon below.

The figure is a regular heptagon: a regular polygon with $7$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular heptagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 7 equal sides and 7 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 7-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 7 sides

#5 Look for a Pattern 4.G.A.3

The heptagon has 7 sides, so it has 7 lines of symmetry. Because 7 is odd there are no opposite vertices at all: every line runs from one vertex straight to the middle of the side across from it, and there are 7 vertices.

7 \text{ vertices} \to 7

7 lines of symmetry.

Answer: 7 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 7 = 25.71 degrees. Going a full half turn covers 7 of them and then repeats, which agrees with the count.

Another way: An odd-sided polygon has no opposite corner to any corner, so every axis is the vertex-to-side-midpoint kind. Counting the vertices counts the axes: 7.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 4 easy answer: 8 lines of symmetry

Find the total number of lines of symmetry of the regular octagon below.

The figure is a regular octagon: a regular polygon with $8$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular octagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 8 equal sides and 8 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 8-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 8 sides

#5 Look for a Pattern 4.G.A.3

The octagon has 8 sides, so it has 8 lines of symmetry. Because 8 is even they come in two kinds: 4 lines through pairs of opposite vertices and 4 through the midpoints of pairs of opposite sides.

4 + 4 = 8

8 lines of symmetry.

Answer: 8 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 8 = 22.5 degrees. Going a full half turn covers 8 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 4 axes join opposite corners and 4 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 5 medium answer: 9 lines of symmetry

Find the total number of lines of symmetry of the regular nonagon below.

The figure is a regular nonagon: a regular polygon with $9$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular nonagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 9 equal sides and 9 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 9-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 9 sides

#5 Look for a Pattern 4.G.A.3

The nonagon has 9 sides, so it has 9 lines of symmetry. Because 9 is odd there are no opposite vertices at all: every line runs from one vertex straight to the middle of the side across from it, and there are 9 vertices.

9 \text{ vertices} \to 9

9 lines of symmetry.

Answer: 9 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 9 = 20 degrees. Going a full half turn covers 9 of them and then repeats, which agrees with the count.

Another way: An odd-sided polygon has no opposite corner to any corner, so every axis is the vertex-to-side-midpoint kind. Counting the vertices counts the axes: 9.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 6 medium answer: 10 lines of symmetry

Find the total number of lines of symmetry of the regular decagon below.

The figure is a regular decagon: a regular polygon with $10$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular decagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 10 equal sides and 10 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 10-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 10 sides

#5 Look for a Pattern 4.G.A.3

The decagon has 10 sides, so it has 10 lines of symmetry. Because 10 is even they come in two kinds: 5 lines through pairs of opposite vertices and 5 through the midpoints of pairs of opposite sides.

5 + 5 = 10

10 lines of symmetry.

Answer: 10 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 10 = 18 degrees. Going a full half turn covers 10 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 5 axes join opposite corners and 5 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 7 medium answer: 11 lines of symmetry

Find the total number of lines of symmetry of the regular hendecagon below.

The figure is a regular hendecagon: a regular polygon with $11$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular hendecagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 11 equal sides and 11 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 11-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 11 sides

#5 Look for a Pattern 4.G.A.3

The hendecagon has 11 sides, so it has 11 lines of symmetry. Because 11 is odd there are no opposite vertices at all: every line runs from one vertex straight to the middle of the side across from it, and there are 11 vertices.

11 \text{ vertices} \to 11

11 lines of symmetry.

Answer: 11 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 11 = 16.36 degrees. Going a full half turn covers 11 of them and then repeats, which agrees with the count.

Another way: An odd-sided polygon has no opposite corner to any corner, so every axis is the vertex-to-side-midpoint kind. Counting the vertices counts the axes: 11.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 8 medium answer: 12 lines of symmetry

Find the total number of lines of symmetry of the regular dodecagon below.

The figure is a regular dodecagon: a regular polygon with $12$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular dodecagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 12 equal sides and 12 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 12-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 12 sides

#5 Look for a Pattern 4.G.A.3

The dodecagon has 12 sides, so it has 12 lines of symmetry. Because 12 is even they come in two kinds: 6 lines through pairs of opposite vertices and 6 through the midpoints of pairs of opposite sides.

6 + 6 = 12

12 lines of symmetry.

Answer: 12 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 12 = 15 degrees. Going a full half turn covers 12 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 6 axes join opposite corners and 6 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 9 hard answer: 15 lines of symmetry

Find the total number of lines of symmetry of the regular pentadecagon below.

The figure is a regular pentadecagon: a regular polygon with $15$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular pentadecagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 15 equal sides and 15 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 15-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 15 sides

#5 Look for a Pattern 4.G.A.3

The pentadecagon has 15 sides, so it has 15 lines of symmetry. Because 15 is odd there are no opposite vertices at all: every line runs from one vertex straight to the middle of the side across from it, and there are 15 vertices.

15 \text{ vertices} \to 15

15 lines of symmetry.

Answer: 15 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 15 = 12 degrees. Going a full half turn covers 15 of them and then repeats, which agrees with the count.

Another way: An odd-sided polygon has no opposite corner to any corner, so every axis is the vertex-to-side-midpoint kind. Counting the vertices counts the axes: 15.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 10 hard answer: 16 lines of symmetry

Find the total number of lines of symmetry of the regular hexadecagon below.

The figure is a regular hexadecagon: a regular polygon with $16$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular hexadecagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 16 equal sides and 16 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 16-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 16 sides

#5 Look for a Pattern 4.G.A.3

The hexadecagon has 16 sides, so it has 16 lines of symmetry. Because 16 is even they come in two kinds: 8 lines through pairs of opposite vertices and 8 through the midpoints of pairs of opposite sides.

8 + 8 = 16

16 lines of symmetry.

Answer: 16 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 16 = 11.25 degrees. Going a full half turn covers 16 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 8 axes join opposite corners and 8 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 11 hard answer: 18 lines of symmetry

Find the total number of lines of symmetry of the regular octadecagon below.

The figure is a regular octadecagon: a regular polygon with $18$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular octadecagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 18 equal sides and 18 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 18-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 18 sides

#5 Look for a Pattern 4.G.A.3

The octadecagon has 18 sides, so it has 18 lines of symmetry. Because 18 is even they come in two kinds: 9 lines through pairs of opposite vertices and 9 through the midpoints of pairs of opposite sides.

9 + 9 = 18

18 lines of symmetry.

Answer: 18 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 18 = 10 degrees. Going a full half turn covers 18 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 9 axes join opposite corners and 9 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

Variant 12 hard answer: 20 lines of symmetry

Find the total number of lines of symmetry of the regular icosagon below.

The figure is a regular icosagon: a regular polygon with $20$ sides, all of equal length and with all angles equal.

Show solution

1 · Understandwhat's really being asked

We need how many fold lines a regular icosagon has -- lines that lay one half exactly onto the other.

Givens

The polygon is regular: 20 equal sides and 20 equal angles.

Unknowns

The number of lines of symmetry.

Constraints

A line of symmetry must lay the whole figure onto itself, not just look plausible.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #9 Solve an Easier Related Problem#5 Look for a Pattern

Counting creases on a 20-sided figure directly is fiddly. Count them on the small regular shapes first, spot the rule, and then apply it.

3 · Execute4 carry out the plan

1Picture the fold lines

#1 Draw a Diagram 4.G.A.3

Draw the regular polygon and imagine folding it so one half lands exactly on the other. Each crease that works is a line of symmetry, and for a regular shape they all pass through the centre.

Every crease goes through the middle.

2Try easier regular shapes first

#9 Solve an Easier Related Problem 4.G.A.3

Count lines of symmetry for smaller regular polygons by sketching their folds: an equilateral triangle has 3, a square 4, a regular pentagon 5, a regular hexagon 6.

\text{triangle} \to 3,\ \text{square} \to 4,\ \text{pentagon} \to 5,\ \text{hexagon} \to 6

Each count matches the number of sides.

3Spot the pattern

#5 Look for a Pattern 4.G.A.3

In every case the number of lines of symmetry equals the number of sides. A regular polygon with n sides has exactly n lines of symmetry.

\text{lines of symmetry} = n \text{ (number of sides)}

One rule covers every regular polygon.

4Apply the pattern to 20 sides

#5 Look for a Pattern 4.G.A.3

The icosagon has 20 sides, so it has 20 lines of symmetry. Because 20 is even they come in two kinds: 10 lines through pairs of opposite vertices and 10 through the midpoints of pairs of opposite sides.

10 + 10 = 20

20 lines of symmetry.

Answer: 20 lines of symmetry

4 · Reviewdoes it hold up?

The axes are evenly spaced around the centre, one every 180 / 20 = 9 degrees. Going a full half turn covers 20 of them and then repeats, which agrees with the count.

Another way: For an even-sided polygon the two families are easy to keep apart: 10 axes join opposite corners and 10 join opposite side midpoints. An odd-sided one has only the one family, which is why the pattern is easier to state as 'the same as the number of sides'.

Standardsmin grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Identifying and counting the lines of symmetry of a figure.

💡Takeaway. A regular shape has one fold line for every side it has.

A regular polygon has as many axes as sides · 12 practice problems

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