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Polygon angle sum via triangulation from a vertex · 7 practice problems

4.G.A.2

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Generated variants — 7

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

Variant 1 easy

In a regular pentagon (a regular polygon with 5 sides), find the measure of angle $a$ .

Your answer

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1 · Understandwhat's really being asked

I have a regular pentagon -- a 5-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 5 sides equal, all 5 interior angles equal).
It has 5 sides (a pentagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 5.

2 · Planchoose the strategy

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

Triangulate the pentagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the pentagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 5 sides breaks into 5 - 2 = 3 triangles.

5 - 2 = 3 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 5 interior angles. So the total of all interior angles is 3 x 180 = 540 degrees.

3 \times 180^\circ = 540^\circ

3 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 5 angles

#7 Identify Subproblems 4.G.A.2

Because the pentagon is regular, all 5 interior angles are equal. Each one, including a, is 540 / 5 = 108 degrees.

540^\circ \div 5 = 108^\circ

Equal angles means just sharing the 540 degrees fairly into 5 equal parts.

Answer: 108 degrees

4 · Reviewdoes it hold up?

108 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 108 for a 5-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 5-gon is 360 / 5 = 72 degrees, so each interior angle is 180 - 72 = 108 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 3 triangles of 180 degrees shared among 5 equal corners gives 108 degrees each!

Variant 2 easy

In a regular hexagon (a regular polygon with 6 sides), find the measure of angle $a$ .

Your answer

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1 · Understandwhat's really being asked

I have a regular hexagon -- a 6-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 6 sides equal, all 6 interior angles equal).
It has 6 sides (a hexagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 6.

2 · Planchoose the strategy

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

Triangulate the hexagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the hexagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 6 sides breaks into 6 - 2 = 4 triangles.

6 - 2 = 4 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 6 interior angles. So the total of all interior angles is 4 x 180 = 720 degrees.

4 \times 180^\circ = 720^\circ

4 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 6 angles

#7 Identify Subproblems 4.G.A.2

Because the hexagon is regular, all 6 interior angles are equal. Each one, including a, is 720 / 6 = 120 degrees.

720^\circ \div 6 = 120^\circ

Equal angles means just sharing the 720 degrees fairly into 6 equal parts.

Answer: 120 degrees

4 · Reviewdoes it hold up?

120 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 120 for a 6-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 6-gon is 360 / 6 = 60 degrees, so each interior angle is 180 - 60 = 120 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 4 triangles of 180 degrees shared among 6 equal corners gives 120 degrees each!

Variant 3 medium

In a regular heptagon (a regular polygon with 7 sides), find the measure of angle $a$ .

Your answer

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1 · Understandwhat's really being asked

I have a regular heptagon -- a 7-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 7 sides equal, all 7 interior angles equal).
It has 7 sides (a heptagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 7.

2 · Planchoose the strategy

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

Triangulate the heptagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the heptagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 7 sides breaks into 7 - 2 = 5 triangles.

7 - 2 = 5 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 7 interior angles. So the total of all interior angles is 5 x 180 = 900 degrees.

5 \times 180^\circ = 900^\circ

5 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 7 angles

#7 Identify Subproblems 4.G.A.2

Because the heptagon is regular, all 7 interior angles are equal. Each one, including a, is 900 / 7 = 128.6 degrees.

900^\circ \div 7 = 128.6^\circ

Equal angles means just sharing the 900 degrees fairly into 7 equal parts.

Answer: 128.6 degrees

4 · Reviewdoes it hold up?

128.6 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 128.6 for a 7-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 7-gon is 360 / 7 = 51.4 degrees, so each interior angle is 180 - 51.4 = 128.6 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 5 triangles of 180 degrees shared among 7 equal corners gives 128.6 degrees each!

Variant 4 medium

In a regular octagon (a regular polygon with 8 sides), find the measure of angle $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

I have a regular octagon -- a 8-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 8 sides equal, all 8 interior angles equal).
It has 8 sides (a octagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 8.

2 · Planchoose the strategy

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

Triangulate the octagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the octagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 8 sides breaks into 8 - 2 = 6 triangles.

8 - 2 = 6 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 8 interior angles. So the total of all interior angles is 6 x 180 = 1080 degrees.

6 \times 180^\circ = 1080^\circ

6 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 8 angles

#7 Identify Subproblems 4.G.A.2

Because the octagon is regular, all 8 interior angles are equal. Each one, including a, is 1080 / 8 = 135 degrees.

1080^\circ \div 8 = 135^\circ

Equal angles means just sharing the 1080 degrees fairly into 8 equal parts.

Answer: 135 degrees

4 · Reviewdoes it hold up?

135 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 135 for a 8-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 8-gon is 360 / 8 = 45 degrees, so each interior angle is 180 - 45 = 135 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 6 triangles of 180 degrees shared among 8 equal corners gives 135 degrees each!

Variant 5 medium

In a regular nonagon (a regular polygon with 9 sides), find the measure of angle $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

I have a regular nonagon -- a 9-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 9 sides equal, all 9 interior angles equal).
It has 9 sides (a nonagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 9.

2 · Planchoose the strategy

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

Triangulate the nonagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the nonagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 9 sides breaks into 9 - 2 = 7 triangles.

9 - 2 = 7 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 9 interior angles. So the total of all interior angles is 7 x 180 = 1260 degrees.

7 \times 180^\circ = 1260^\circ

7 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 9 angles

#7 Identify Subproblems 4.G.A.2

Because the nonagon is regular, all 9 interior angles are equal. Each one, including a, is 1260 / 9 = 140 degrees.

1260^\circ \div 9 = 140^\circ

Equal angles means just sharing the 1260 degrees fairly into 9 equal parts.

Answer: 140 degrees

4 · Reviewdoes it hold up?

140 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 140 for a 9-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 9-gon is 360 / 9 = 40 degrees, so each interior angle is 180 - 40 = 140 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 7 triangles of 180 degrees shared among 9 equal corners gives 140 degrees each!

Variant 6 hard

In a regular decagon (a regular polygon with 10 sides), find the measure of angle $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

I have a regular decagon -- a 10-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 10 sides equal, all 10 interior angles equal).
It has 10 sides (a decagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 10.

2 · Planchoose the strategy

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

Triangulate the decagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the decagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 10 sides breaks into 10 - 2 = 8 triangles.

10 - 2 = 8 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 10 interior angles. So the total of all interior angles is 8 x 180 = 1440 degrees.

8 \times 180^\circ = 1440^\circ

8 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 10 angles

#7 Identify Subproblems 4.G.A.2

Because the decagon is regular, all 10 interior angles are equal. Each one, including a, is 1440 / 10 = 144 degrees.

1440^\circ \div 10 = 144^\circ

Equal angles means just sharing the 1440 degrees fairly into 10 equal parts.

Answer: 144 degrees

4 · Reviewdoes it hold up?

144 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 144 for a 10-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 10-gon is 360 / 10 = 36 degrees, so each interior angle is 180 - 36 = 144 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 8 triangles of 180 degrees shared among 10 equal corners gives 144 degrees each!

Variant 7 hard

In a regular dodecagon (a regular polygon with 12 sides), find the measure of angle $a$ .

Your answer

Show solution

1 · Understandwhat's really being asked

I have a regular dodecagon -- a 12-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.

Givens

The polygon is regular (all 12 sides equal, all 12 interior angles equal).
It has 12 sides (a dodecagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 12.

2 · Planchoose the strategy

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

Triangulate the dodecagon from one vertex to turn an unfamiliar polygon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.

3 · Execute3 carry out the plan

1Cut the dodecagon into triangles

#1 Draw a Diagram 4.G.A.2

From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 12 sides breaks into 12 - 2 = 10 triangles.

12 - 2 = 10 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

2Add up all the interior angles

#9 Solve an Easier Related Problem 4.MD.C.7

Each triangle contributes 180 degrees, and these triangle angles together fill exactly the polygon's 12 interior angles. So the total of all interior angles is 10 x 180 = 1800 degrees.

10 \times 180^\circ = 1800^\circ

10 triangles of 180 degrees combine into the polygon's full interior-angle total.

3Divide equally among the 12 angles

#7 Identify Subproblems 4.G.A.2

Because the dodecagon is regular, all 12 interior angles are equal. Each one, including a, is 1800 / 12 = 150 degrees.

1800^\circ \div 12 = 150^\circ

Equal angles means just sharing the 1800 degrees fairly into 12 equal parts.

Answer: 150 degrees

4 · Reviewdoes it hold up?

150 degrees is between 0 and 180 degrees, which fits a convex regular polygon. As polygons gain sides their interior angles grow toward 180, and 150 for a 12-gon sits sensibly in that range.

Another way: Pattern approach (tool 5): each exterior angle of a regular 12-gon is 360 / 12 = 30 degrees, so each interior angle is 180 - 30 = 150 degrees -- the same answer.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular polygon and using the equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the 180-degree triangles into the total interior-angle measure.

💡Takeaway. Any polygon is just triangles in disguise -- 10 triangles of 180 degrees shared among 12 equal corners gives 150 degrees each!

Where next?

← Back to the problem: Polygon angle sum via triangulation from a vertex