Sensim Math · Depth 한국어

4-1 · Angles

Split a polygon into triangles to sum angles

4.MD.C.7 · take · grade 4

Archetype: Angle Facts in a Figure · step in a 13-type progression

▶ Practice — 8 problems

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 pentagon into triangles by drawing diagonals from one corner. We already know each triangle's angles total 180 degrees, so the pentagon's 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 two 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.
#7 Identify Subproblems 4.MD.C.7
Each triangle's three angles add to 180 degrees, and the three 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 pentagon'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. It is larger than a quadrilateral's 360 degrees, as expected for one more side.

Look for a pattern (tool 5): triangle 180, quadrilateral 360, each extra side adds 180 degrees, so pentagon = 360 + 180 = 540 degrees.

Standards · min grade 4

  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the three triangles' 180-degree sums into the pentagon total.
  • 4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing diagonals to split the pentagon into triangles.
💡 Cut any shape into triangles you already understand, then add 180 for each one - that is all you need to find a pentagon's angle total!