Sum of interior angles of a polygon | Angles and intersecting lines | Geometry | Khan Academy

TL;DR

The sum of the interior angles of a polygon can be found by multiplying the number of triangles that can fit inside the polygon by 180 degrees.

Transcript

We already know that the sum of the interior angles of a triangle add up to 180 degrees. So if the measure of this angle is a, the measure of this angle over here is b, and the measure of this angle is c, we know that a plus b plus c is equal to 180 degrees. But what happens when we have polygons with more than three sides? So let's try the case wh... Read More

Key Insights

• 🔺 The sum of the interior angles of a triangle is always 180 degrees.
• 🔺 Dividing a polygon into triangles helps in determining the sum of its interior angles.
• #️⃣ The number of triangles that can fit inside a polygon can be calculated as the number of sides minus 2.
• 🔺 Each triangle within the polygon has an interior angle sum of 180 degrees.
• 🔺 The sum of the interior angles of a polygon can be found by multiplying the number of triangles by 180 degrees.
• 💠 This method applies to polygons of any shape.
• 🔺 Regular polygons have equal interior angles, while irregular polygons have varying interior angles.

Questions & Answers

Q: How can we find the sum of the interior angles of a polygon with more than three sides?

To find the sum of the interior angles of a polygon with more than three sides, you can divide the polygon into triangles and calculate the sum of their angles. Each triangle has 180 degrees, so by multiplying the number of triangles by 180 degrees, you can find the sum of the interior angles.

Q: Why do we subtract 2 from the number of sides to find the number of triangles that can fit inside a polygon?

When dividing a polygon into triangles, the first four sides are used to create two triangles. The remaining sides (s - 4) can each form one triangle. Therefore, the number of triangles is equal to s - 2, where s is the number of sides.

Q: How does dividing a polygon into triangles help in calculating the sum of the interior angles?

Dividing a polygon into triangles allows us to utilize the fact that the sum of the interior angles of a triangle is 180 degrees. By counting the number of triangles, we can multiply this count by 180 degrees to find the sum of the interior angles of the polygon.

Q: Can this method be applied to any polygon, irrespective of its shape?

Yes, this method can be applied to any polygon, regardless of its shape. It works for both regular and irregular polygons since it is based on the principle of dividing the polygon into triangles.

Summary & Key Takeaways

• The sum of the interior angles of a triangle is 180 degrees.

• For polygons with more than three sides, the polygon can be divided into triangles to determine the sum of the interior angles.

• The number of triangles that can fit inside a polygon can be calculated as the number of sides minus 2.