← 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 medium answer: 180 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the triangle into triangles

#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 sides→3−2=1 triangle3 \text{ sides} \rightarrow 3 - 2 = 1 \text{ triangle}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
1×180∘=180∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 1260 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the nonagon into triangles

#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 sides→9−2=7 triangles9 \text{ sides} \rightarrow 9 - 2 = 7 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
7×180∘=1260∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 720 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the hexagon into triangles

#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 sides→6−2=4 triangles6 \text{ sides} \rightarrow 6 - 2 = 4 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
4×180∘=720∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 1080 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the octagon into triangles

#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 sides→8−2=6 triangles8 \text{ sides} \rightarrow 8 - 2 = 6 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
6×180∘=1080∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 1440 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the decagon into triangles

#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 sides→10−2=8 triangles10 \text{ sides} \rightarrow 10 - 2 = 8 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
8×180∘=1440∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 360 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the quadrilateral into triangles

#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 sides→4−2=2 triangles4 \text{ sides} \rightarrow 4 - 2 = 2 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
2×180∘=360∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 900 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the heptagon into triangles

#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 sides→7−2=5 triangles7 \text{ sides} \rightarrow 7 - 2 = 5 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
5×180∘=900∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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 medium answer: 540 degrees

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

Show solution
1 · Understandwhat's really being asked

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).
2 · Planchoose the strategy

#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.

3 · Execute2 carry out the plan

1Split the pentagon into triangles

#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 sides→5−2=3 triangles5 \text{ sides} \rightarrow 5 - 2 = 3 \text{ triangles}
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.

2Add up the triangle angle sums

#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.
3×180∘=540∘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
4 · Reviewdoes it hold up?

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

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

Standardsmin 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.
💡Takeaway. 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!