← 4-1 · Split a polygon into triangles to sum angles · Angle Facts in a Figure

Split a polygon into triangles to sum angles · 8 practice problems

4.MD.C.74.G.A.1

Generated variants — 8

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

Variant 1 answer: 180 degrees

Find the sum of the measures of the three angles of the figure.

Show solution

Understand

A triangle has 3 sides and 3 corners. We need the total of all three inside angles.

Givens

The figure is a triangle: 3 sides, 3 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the three interior angles of the triangle.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the triangle into 1 triangle that exactly cover it.

3 \text{ sides} \rightarrow 3 - 2 = 1 \text{ triangle}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 1 triangle' angles together make up exactly the triangle's three interior angles with nothing left over. So multiply.

1 \times 180^\circ = 180^\circ

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 180 degrees

Review

A regular triangle corner is 60 degrees, and 3 x 60 = 180 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 2 answer: 1260 degrees

Find the sum of the measures of the nine angles of the figure.

Show solution

Understand

A nonagon has 9 sides and 9 corners. We need the total of all nine inside angles.

Givens

The figure is a nonagon: 9 sides, 9 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the nine interior angles of the nonagon.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the nonagon into 7 triangles that exactly cover it.

9 \text{ sides} \rightarrow 9 - 2 = 7 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 7 triangles' angles together make up exactly the nonagon's nine interior angles with nothing left over. So multiply.

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

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 1260 degrees

Review

A regular nonagon corner is 140 degrees, and 9 x 140 = 1260 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 3 answer: 720 degrees

Find the sum of the measures of the six angles of the figure.

Show solution

Understand

A hexagon has 6 sides and 6 corners. We need the total of all six inside angles.

Givens

The figure is a hexagon: 6 sides, 6 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the six interior angles of the hexagon.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the hexagon into 4 triangles that exactly cover it.

6 \text{ sides} \rightarrow 6 - 2 = 4 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 4 triangles' angles together make up exactly the hexagon's six interior angles with nothing left over. So multiply.

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

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 720 degrees

Review

A regular hexagon corner is 120 degrees, and 6 x 120 = 720 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 4 answer: 1080 degrees

Find the sum of the measures of the eight angles of the figure.

Show solution

Understand

A octagon has 8 sides and 8 corners. We need the total of all eight inside angles.

Givens

The figure is a octagon: 8 sides, 8 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the eight interior angles of the octagon.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the octagon into 6 triangles that exactly cover it.

8 \text{ sides} \rightarrow 8 - 2 = 6 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 6 triangles' angles together make up exactly the octagon's eight interior angles with nothing left over. So multiply.

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

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 1080 degrees

Review

A regular octagon corner is 135 degrees, and 8 x 135 = 1080 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 5 answer: 1440 degrees

Find the sum of the measures of the ten angles of the figure.

Show solution

Understand

A decagon has 10 sides and 10 corners. We need the total of all ten inside angles.

Givens

The figure is a decagon: 10 sides, 10 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the ten interior angles of the decagon.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the decagon into 8 triangles that exactly cover it.

10 \text{ sides} \rightarrow 10 - 2 = 8 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 8 triangles' angles together make up exactly the decagon's ten interior angles with nothing left over. So multiply.

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

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 1440 degrees

Review

A regular decagon corner is 144 degrees, and 10 x 144 = 1440 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 6 answer: 360 degrees

Find the sum of the measures of the four angles of the figure.

Show solution

Understand

A quadrilateral has 4 sides and 4 corners. We need the total of all four inside angles.

Givens

The figure is a quadrilateral: 4 sides, 4 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the four interior angles of the quadrilateral.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the quadrilateral into 2 triangles that exactly cover it.

4 \text{ sides} \rightarrow 4 - 2 = 2 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 2 triangles' angles together make up exactly the quadrilateral's four interior angles with nothing left over. So multiply.

2 \times 180^\circ = 360^\circ

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 360 degrees

Review

A regular quadrilateral corner is 90 degrees, and 4 x 90 = 360 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 7 answer: 900 degrees

Find the sum of the measures of the seven angles of the figure.

Show solution

Understand

A heptagon has 7 sides and 7 corners. We need the total of all seven inside angles.

Givens

The figure is a heptagon: 7 sides, 7 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the seven interior angles of the heptagon.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the heptagon into 5 triangles that exactly cover it.

7 \text{ sides} \rightarrow 7 - 2 = 5 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 5 triangles' angles together make up exactly the heptagon's seven interior angles with nothing left over. So multiply.

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

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 900 degrees

Review

A regular heptagon corner is 128 degrees, and 7 x 128 = 900 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!

Variant 8 answer: 540 degrees

Find the sum of the measures of the five angles of the figure.

Show solution

Understand

A pentagon has 5 sides and 5 corners. We need the total of all five inside angles.

Givens

The figure is a pentagon: 5 sides, 5 vertices.
We already know the three angles of any triangle add to 180 degrees.

Unknowns

The sum of the five interior angles of the pentagon.

Constraints

Use triangles to build up the answer (triangulation).

Plan

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

Cut the figure into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the total is the number of triangles times 180.

Execute

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

Pick one vertex and draw straight lines (diagonals) to the non-neighboring vertices. This divides the pentagon into 3 triangles that exactly cover it.

5 \text{ sides} \rightarrow 5 - 2 = 3 \text{ triangles}

Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

#7 Identify Subproblems 4.MD.C.7

Each triangle's three angles add to 180 degrees, and the 3 triangles' angles together make up exactly the pentagon's five interior angles with nothing left over. So multiply.

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

All the little triangle corners glue back together into the figure's corners, so their measures add up.

Answer: 540 degrees

Review

A regular pentagon corner is 108 degrees, and 5 x 108 = 540 degrees, matching our triangulation answer. Each extra side adds another 180 degrees.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the triangles' 180-degree sums into the polygon total.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the polygon into triangles.

💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a polygon's angle total!