Problem
Split a polygon into triangles to sum angles
A pentagon has 5 sides and 5 corners. We need the total of all five inside angles.
Measurement & data
Your answer
degrees
Watch out It's tempting to multiply the number of sides directly by 180, getting 5 × 180 = 900 degrees, but only 3 triangles actually fit inside the pentagon, not 5.
How to solve
Strategy Identify Subproblems — 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.
1STEP 1
Split the pentagon into triangles
Draw diagonals from one vertex to the non-neighboring vertices, splitting the pentagon into 3 triangles.
5 sides → 5 - 2 = 3 triangles
Drawing diagonals from one corner always makes (number of sides minus 2) triangles.
4.G.A.1Draw A Diagram2STEP 2
Add up the triangle angle sums
Each triangle's angles add to 180 degrees, and the 3 triangles make up the whole pentagon. So multiply.
3 × 180° = 540°
All the little triangle corners glue back together into the pentagon's corners, so their measures add up.
4.MD.C.7Identify SubproblemsThe pentagon's five interior angles together add up to three triangles' worth of angle, that is 3 \times 180^\circ.
Why?
The three triangles fit inside the pentagon exactly, so all their corner angles together are the same as the pentagon's whole angle total.
Why?
The diagonals from one corner split the pentagon into three triangles with no gaps between them and no overlaps, and every triangle corner lands on a pentagon corner, so nothing is added or lost.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each one of the three triangles supplies 180 degrees of angle.
Why?
The three corners of any flat triangle always add up to one straight angle, 180 degrees.
🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.
Why?
Counting 180 once for each of the three triangles is three equal groups of 180, and that is exactly what 3 \times 180 means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
540 degrees
3 × 180° = 540°
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.
Takeaway
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!
- Split the pentagon into triangles
- Add up the triangle angle sums
Where next?
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