Grade 4 (age 9-10)

grade 4 Polygons

Polygons

A polygon is a shape enclosed only by straight segments. Its name follows the number of sides, like pentagon or hexagon, and if every side and angle is equal it is regular. Soccer balls, road signs, and honeycombs are full of these shapes. Change the number of sides and the polygon splits into triangles for the angle sum, and the diagonals are counted too.

What is a polygon?

🔷 A Shape Enclosed by Line Segments

①A polygon is a closed shape made only of straight lines

②By number of sides: triangle, quadrilateral, pentagon, ...

③If all sides and angles are equal, it's a regular polygon!

Explore Regular Polygons

Number of sides5

Interior Angle Sum from Triangles!

💡 Finding the Interior Angle Sum

①Drawing diagonals from one vertex splits it into triangles

②Number of triangles = sides − 2. n in the formula is the number of sides

③Sum of angles = (n − 2) × 180°

④Diagonals = n(n − 3)/2. From one vertex you can draw n − 3 after skipping itself and its neighbors

⑤Slide sides up to 12 to see it get closer to a circle

Summary

Sum of Interior Angles

(n - 2) × 180°

Sum for an n-gon (triangle-splitting principle)

Each Angle (Regular)

(n - 2) × 180°n

Divide the sum by number of sides

Number of Diagonals

n(n - 3)2

Diagonals of an n-gon (n-3 from each vertex)

🎯 Key Points

①Polygon = closed shape of straight lines

②Regular polygon = all sides and angles equal

③Sum = (sides - 2) × 180°

④More sides → closer to a circle

⑤Diagonal count is computed from sides

Examples and Unit Test

Example 1

What is the sum of interior angles of a pentagon (5 sides)?

The interior angle sum formula is (sides − 2) × 180°.

(5 − 2) × 180° = 3 × 180° = 540°.

▸ 540°

Drawing diagonals from one vertex splits it into (sides − 2) triangles.

Example 2

What is each interior angle of an equilateral triangle?

A triangle's angle sum is (3 − 2) × 180° = 180°.

An equilateral triangle has three equal angles: 180 ÷ 3 = 60°.

▸ 60°

Each angle of a regular polygon = angle sum ÷ number of sides.

Unit Test

What is the sum of interior angles of a quadrilateral (4 sides)?

①180°

②360°

③540°

④720°

▸ ② 360°

Put 4 sides into (sides − 2) × 180°.

(4 − 2) × 180° = 2 × 180° = 360°.

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Types of Quadrilaterals

Plane Figure Transformations