← Exterior angle sum of a regular polygon is 360 · Angle Facts in a Figure

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

4.G.A.24.MD.C.7

Generated variants — 8

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 medium answer: 360 degrees

Each side of a regular triangle is extended, as shown. Find the sum of the measures of angles aa, bb, cc.

a b c
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.
(3−2)×180∘3=180∘3=60∘\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∘−60∘=120∘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.
3×120∘=360∘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 answer: 360 degrees

Each side of a regular nonagon is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd, ee, ff, gg, hh, ii.

a b c d e f g h i
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.
(9−2)×180∘9=1260∘9=140∘\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∘−140∘=40∘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.
9×40∘=360∘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 answer: 360 degrees

Each side of a regular hexagon is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd, ee, ff.

a b c d e f
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.
(6−2)×180∘6=720∘6=120∘\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∘−120∘=60∘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.
6×60∘=360∘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 answer: 360 degrees

Each side of a regular octagon is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd, ee, ff, gg, hh.

a b c d e f g h
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.
(8−2)×180∘8=1080∘8=135∘\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∘−135∘=45∘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.
8×45∘=360∘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 answer: 360 degrees

Each side of a regular decagon is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd, ee, ff, gg, hh, ii, jj.

a b c d e f g h i j
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.
(10−2)×180∘10=1440∘10=144∘\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∘−144∘=36∘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.
10×36∘=360∘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 answer: 360 degrees

Each side of a regular quadrilateral (square) is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd.

a b c d
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.
(4−2)×180∘4=360∘4=90∘\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∘−90∘=90∘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.
4×90∘=360∘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 answer: 360 degrees

Each side of a regular heptagon is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd, ee, ff, gg.

a b c d e f g
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.
(7−2)×180∘7=900∘7=128.6∘\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∘−128.6∘=51.4∘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.
7×51.4∘=360∘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 answer: 360 degrees

Each side of a regular pentagon is extended, as shown. Find the sum of the measures of angles aa, bb, cc, dd, ee.

a b c d e
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.
(5−2)×180∘5=540∘5=108∘\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∘−108∘=72∘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.
5×72∘=360∘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!