Practice · Exterior angle sum of a regular polygon is 360 · Sensim Math

Variant 1 medium

Each side of a regular triangle is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular triangle has each of its 3 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 3 exterior angles a, b, c.

Givens

The polygon is a regular triangle (3 equal sides, 3 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 3 vertices.
The 3 exterior angles are labeled a, b, c around the polygon.

Unknowns

The sum a + b + c of the 3 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular triangle

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 1 triangles by drawing diagonals from one vertex; the 3 interior angles add to 1 x 180 = 180 degrees. Because the polygon is regular, each interior angle is 180 / 3 = 60 degrees.

\frac{(3-2)\times 180^\circ}{3} = \frac{180^\circ}{3} = 60^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 60 = 120 degrees.

180^\circ - 60^\circ = 120^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 3 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 3 exterior angles are equal to 120 degrees, so the sum is 3 x 120 = 360 degrees.

3 \times 120^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

3 angles of 120 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (3 x 120 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 2 medium

Each side of a regular nonagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ , $h$ , $i$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular nonagon has each of its 9 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 9 exterior angles a, b, c, d, e, f, g, h, i.

Givens

The polygon is a regular nonagon (9 equal sides, 9 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 9 vertices.
The 9 exterior angles are labeled a, b, c, d, e, f, g, h, i around the polygon.

Unknowns

The sum a + b + c + d + e + f + g + h + i of the 9 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular nonagon

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 7 triangles by drawing diagonals from one vertex; the 9 interior angles add to 7 x 180 = 1260 degrees. Because the polygon is regular, each interior angle is 1260 / 9 = 140 degrees.

\frac{(9-2)\times 180^\circ}{9} = \frac{1260^\circ}{9} = 140^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 140 = 40 degrees.

180^\circ - 140^\circ = 40^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 9 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 9 exterior angles are equal to 40 degrees, so the sum is 9 x 40 = 360 degrees.

9 \times 40^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

9 angles of 40 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (9 x 40 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 3 medium

Each side of a regular hexagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ , $f$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular hexagon has each of its 6 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 6 exterior angles a, b, c, d, e, f.

Givens

The polygon is a regular hexagon (6 equal sides, 6 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 6 vertices.
The 6 exterior angles are labeled a, b, c, d, e, f around the polygon.

Unknowns

The sum a + b + c + d + e + f of the 6 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular hexagon

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 4 triangles by drawing diagonals from one vertex; the 6 interior angles add to 4 x 180 = 720 degrees. Because the polygon is regular, each interior angle is 720 / 6 = 120 degrees.

\frac{(6-2)\times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 120 = 60 degrees.

180^\circ - 120^\circ = 60^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 6 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 6 exterior angles are equal to 60 degrees, so the sum is 6 x 60 = 360 degrees.

6 \times 60^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

6 angles of 60 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (6 x 60 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 4 medium

Each side of a regular octagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ , $h$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular octagon has each of its 8 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 8 exterior angles a, b, c, d, e, f, g, h.

Givens

The polygon is a regular octagon (8 equal sides, 8 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 8 vertices.
The 8 exterior angles are labeled a, b, c, d, e, f, g, h around the polygon.

Unknowns

The sum a + b + c + d + e + f + g + h of the 8 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular octagon

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 6 triangles by drawing diagonals from one vertex; the 8 interior angles add to 6 x 180 = 1080 degrees. Because the polygon is regular, each interior angle is 1080 / 8 = 135 degrees.

\frac{(8-2)\times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 135 = 45 degrees.

180^\circ - 135^\circ = 45^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 8 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 8 exterior angles are equal to 45 degrees, so the sum is 8 x 45 = 360 degrees.

8 \times 45^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

8 angles of 45 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (8 x 45 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 5 medium

Each side of a regular decagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ , $h$ , $i$ , $j$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular decagon has each of its 10 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 10 exterior angles a, b, c, d, e, f, g, h, i, j.

Givens

The polygon is a regular decagon (10 equal sides, 10 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 10 vertices.
The 10 exterior angles are labeled a, b, c, d, e, f, g, h, i, j around the polygon.

Unknowns

The sum a + b + c + d + e + f + g + h + i + j of the 10 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular decagon

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 8 triangles by drawing diagonals from one vertex; the 10 interior angles add to 8 x 180 = 1440 degrees. Because the polygon is regular, each interior angle is 1440 / 10 = 144 degrees.

\frac{(10-2)\times 180^\circ}{10} = \frac{1440^\circ}{10} = 144^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 144 = 36 degrees.

180^\circ - 144^\circ = 36^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 10 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 10 exterior angles are equal to 36 degrees, so the sum is 10 x 36 = 360 degrees.

10 \times 36^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

10 angles of 36 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (10 x 36 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 6 medium

Each side of a regular quadrilateral (square) is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular quadrilateral (square) has each of its 4 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 4 exterior angles a, b, c, d.

Givens

The polygon is a regular quadrilateral (square) (4 equal sides, 4 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 4 vertices.
The 4 exterior angles are labeled a, b, c, d around the polygon.

Unknowns

The sum a + b + c + d of the 4 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular quadrilateral (square)

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 2 triangles by drawing diagonals from one vertex; the 4 interior angles add to 2 x 180 = 360 degrees. Because the polygon is regular, each interior angle is 360 / 4 = 90 degrees.

\frac{(4-2)\times 180^\circ}{4} = \frac{360^\circ}{4} = 90^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 90 = 90 degrees.

180^\circ - 90^\circ = 90^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 4 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 4 exterior angles are equal to 90 degrees, so the sum is 4 x 90 = 360 degrees.

4 \times 90^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

4 angles of 90 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (4 x 90 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 7 medium

Each side of a regular heptagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular heptagon has each of its 7 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 7 exterior angles a, b, c, d, e, f, g.

Givens

The polygon is a regular heptagon (7 equal sides, 7 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 7 vertices.
The 7 exterior angles are labeled a, b, c, d, e, f, g around the polygon.

Unknowns

The sum a + b + c + d + e + f + g of the 7 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular heptagon

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 5 triangles by drawing diagonals from one vertex; the 7 interior angles add to 5 x 180 = 900 degrees. Because the polygon is regular, each interior angle is 900 / 7 = 128.6 degrees.

\frac{(7-2)\times 180^\circ}{7} = \frac{900^\circ}{7} = 128.6^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 128.6 = 51.4 degrees.

180^\circ - 128.6^\circ = 51.4^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 7 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 7 exterior angles are equal to 51.4 degrees, so the sum is 7 x 51.4 = 360 degrees.

7 \times 51.4^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

7 angles of 51.4 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (7 x 51.4 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Variant 8 medium

Each side of a regular pentagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A regular pentagon has each of its 5 sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all 5 exterior angles a, b, c, d, e.

Givens

The polygon is a regular pentagon (5 equal sides, 5 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 5 vertices.
The 5 exterior angles are labeled a, b, c, d, e around the polygon.

Unknowns

The sum a + b + c + d + e of the 5 exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the polygon is regular, all interior angles are equal and all exterior angles are equal.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

Break the total into two easy facts: the interior angles of the polygon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

3 · Execute3 carry out the plan

1Find each interior angle of the regular pentagon

#9 Solve an Easier Related Problem 4.G.A.2

Split the polygon into 3 triangles by drawing diagonals from one vertex; the 5 interior angles add to 3 x 180 = 540 degrees. Because the polygon is regular, each interior angle is 540 / 5 = 108 degrees.

\frac{(5-2)\times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

2Get one exterior angle using the straight line

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 108 = 72 degrees.

180^\circ - 108^\circ = 72^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

3Add all 5 exterior angles

#7 Identify Subproblems 4.MD.C.7

All 5 exterior angles are equal to 72 degrees, so the sum is 5 x 72 = 360 degrees.

5 \times 72^\circ = 360^\circ

Adding equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

4 · Reviewdoes it hold up?

5 angles of 72 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the polygon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Another way: Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with this polygon (5 x 72 = 360) confirms the rule.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular polygon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the equal exterior angles.

💡Takeaway. Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!

Exterior angle sum of a regular polygon is 360 · 8 practice problems

Generated variants — 8

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?