Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 2: Angles and Triangles

Lesson 3: Angles of Polygons

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, Chapter 2, students learn how to find the sum of interior angle measures of polygons using the relationship between the number of sides and the number of triangles formed, expressed by the formula S = (n - 2) × 180°. Students also discover that the sum of the exterior angle measures of any convex polygon is always 360°. The lesson develops these concepts through hands-on exploration with quadrilaterals, pentagons, hexagons, and octagons before applying the patterns to find angle measures in polygons with more sides.

Section 1

Polygon Interior Angle Sum Formula

Property

The sum of interior angles of any polygon with nn sides is given by:

S=(n−2)×180°S = (n-2) \times 180°

This formula is derived by dividing any polygon into (n−2)(n-2) triangles from one vertex.

Section 2

Formula for a Single Interior Angle in a Regular Polygon

Property

The measure of an interior angle of a regular polygon with nn sides is given by the formula:

Interior Angle=(n−2)⋅180∘n \text{Interior Angle} = \frac{(n-2) \cdot 180^\circ}{n}

Examples

Section 3

Polygon Exterior Angle Sum Property

Property

An exterior angle of a polygon is formed by extending one side of the polygon at a vertex. The sum of all exterior angles of any convex polygon is always 360°360°.

Sum of exterior angles=360°\text{Sum of exterior angles} = 360°

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Chapter 2: Angles and Triangles

  1. Lesson 1

    Lesson 1: Parallel Lines and Transversals

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  2. Lesson 2

    Lesson 2: Angles of Triangles

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  3. Lesson 3Current

    Lesson 3: Angles of Polygons

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  4. Lesson 4

    Lesson 4: Using Similar Triangles

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Section 1

Polygon Interior Angle Sum Formula

Property

The sum of interior angles of any polygon with nn sides is given by:

S=(n−2)×180°S = (n-2) \times 180°

This formula is derived by dividing any polygon into (n−2)(n-2) triangles from one vertex.

Section 2

Formula for a Single Interior Angle in a Regular Polygon

Property

The measure of an interior angle of a regular polygon with nn sides is given by the formula:

Interior Angle=(n−2)⋅180∘n \text{Interior Angle} = \frac{(n-2) \cdot 180^\circ}{n}

Examples

Section 3

Polygon Exterior Angle Sum Property

Property

An exterior angle of a polygon is formed by extending one side of the polygon at a vertex. The sum of all exterior angles of any convex polygon is always 360°360°.

Sum of exterior angles=360°\text{Sum of exterior angles} = 360°

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 2: Angles and Triangles

  1. Lesson 1

    Lesson 1: Parallel Lines and Transversals

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  2. Lesson 2

    Lesson 2: Angles of Triangles

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  3. Lesson 3Current

    Lesson 3: Angles of Polygons

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  4. Lesson 4

    Lesson 4: Using Similar Triangles

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