Math Problem Statement

Complete the table by filling in the details for polygons based on the number of sides provided.

Solution

The table you've provided appears to be a math problem related to polygons, specifically asking to complete information about different polygons based on their number of sides. Let's fill in the missing information.

Columns of the table:

  1. Number of Sides: Given in the table.
  2. Kind of Polygon: Name based on the number of sides.
  3. Number of Triangles: This is given by (Number of Sides2)(\text{Number of Sides} - 2).
  4. Sum of Interior Angles: This is given by (Number of Sides2)×180(\text{Number of Sides} - 2) \times 180^\circ.
  5. Measure of Each Angle in a Regular Polygon: This is given by Sum of Interior AnglesNumber of Sides\frac{\text{Sum of Interior Angles}}{\text{Number of Sides}}.

Let's fill in the table:

Number of SidesKind of PolygonNumber of TrianglesSum of Interior Angles (in degrees)Measure of Each Angle in a Regular Polygon (in degrees)
3Triangle11×180=1801 \times 180 = 180^\circ1803=60\frac{180}{3} = 60^\circ
4Quadrilateral22×180=3602 \times 180 = 360^\circ3604=90\frac{360}{4} = 90^\circ
5Pentagon33×180=5403 \times 180 = 540^\circ5405=108\frac{540}{5} = 108^\circ
6Hexagon44×180=7204 \times 180 = 720^\circ7206=120\frac{720}{6} = 120^\circ
8Octagon66×180=10806 \times 180 = 1080^\circ10808=135\frac{1080}{8} = 135^\circ
10Decagon88×180=14408 \times 180 = 1440^\circ144010=144\frac{1440}{10} = 144^\circ
12Dodecagon1010×180=180010 \times 180 = 1800^\circ180012=150\frac{1800}{12} = 150^\circ

Would you like a detailed explanation on how each of these calculations is performed, or do you have any specific questions about this table?

Here are 5 related questions you might consider:

  1. How do you calculate the sum of interior angles for any polygon?
  2. Why does the number of triangles matter in calculating the sum of interior angles?
  3. How does the measure of each angle change as the number of sides increases in a regular polygon?
  4. What is the relationship between the number of sides and the name of the polygon?
  5. Can you derive the formula for the measure of each angle in a regular polygon?

Tip:

Remember, the exterior angle of a regular polygon is always 360Number of Sides\frac{360^\circ}{\text{Number of Sides}}. This can be useful for solving various problems involving polygons.

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Math Problem Analysis

Mathematical Concepts

Geometry
Polygons
Interior Angles
Regular Polygons

Formulas

Number of Triangles = (Number of Sides - 2)
Sum of Interior Angles = (Number of Sides - 2) × 180°
Measure of Each Angle in a Regular Polygon = Sum of Interior Angles / Number of Sides

Theorems

Polygon Interior Angle Sum Theorem

Suitable Grade Level

Grades 6-8