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Grade 4 (age 9-10)

grade 4 Polygons

A polygon is a shape closed in only by straight sides. Polygons are named by their number of sides, like triangle, quadrilateral and pentagon, and a polygon whose sides are all the same length and whose angles are all the same size is called a regular polygon. You can find regular polygons around you, such as the hexagonal cells of a honeycomb or the pentagons and hexagons on a soccer ball. Change the number of sides to redraw the regular polygon and its diagonals, and see how diagonals drawn from one vertex split it into triangles that give the sum of its angles.

What Is a Polygon?

🔷 Shapes Closed In by Straight Sides
①A shape closed in only by straight sides is a polygon; a shape with a curved edge, like a circle, is not
②Polygons are named by their number of sides: 5 sides make a pentagon and 6 sides make a hexagon
③A polygon with all sides equal and all angles equal is a regular polygon
④A diagonal is a segment joining two vertices that are not next to each other

Exploring Regular Polygons

5

Angle Sums from Triangles

💡 Finding the Angle Sum with Triangles
①Drawing diagonals from one vertex splits a polygon into several triangles
②The number of triangles is 2 fewer than the number of sides
③Each triangle's angles add up to 180°, so the angle sum is (number of sides − 2) × 180°
④In a regular polygon all the angles are equal, so dividing the angle sum by the number of sides gives each angle
⑤As the number of sides goes up to 12, the regular polygon looks more and more like a circle

Summary

Angle Sum of a Polygon
(n - 2) × 180°
n is the number of sides; it is the total of the angles of (n − 2) triangles
Each Angle of a Regular Polygon
(n - 2) × 180°n
Divide the angle sum by the number of sides
Number of Diagonals
n(n - 3)2
Each vertex has n − 3 diagonals, and each diagonal is counted twice, so divide by 2
🎯 Summary
①A polygon is closed in only by straight sides
②A regular polygon has all sides equal and all angles equal
③The angle sum of a polygon is (number of sides − 2) × 180°
④Each angle of a regular polygon is the angle sum divided by the number of sides
⑤The number of diagonals is number of sides × (number of sides − 3) ÷ 2

Examples and Unit Test

Example 1
What is the angle sum of a pentagon?
1
Diagonals from one vertex split a pentagon into 5 − 2 = 3 triangles.
2
3 × 180° = 540°.
▸ 540°
The number of triangles is the number of sides minus 2.
Example 2
How large is each angle of an equilateral triangle?
1
The three angles of a triangle add up to 180°.
2
All three angles of an equilateral triangle are equal, so each is 180° ÷ 3 = 60°.
▸ 60°
Each angle of a regular polygon is the angle sum divided by the number of sides.
Unit Test
What is the sum of the four angles of a quadrilateral?
①180°
②360°
③540°
④720°
▸ ② 360°
1
One diagonal splits a quadrilateral into 2 triangles.
2
2 × 180° = 360°.
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Types of Quadrilaterals
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Plane Figure Transformations

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