How to calculate the sum of the interior angles of a polygon
Recently, one of the hot topics on the Internet is the sharing of mathematics learning methods and techniques. As one of the basic concepts in mathematics, the interior angles and formulas of polygons have become the focus of many students and parents. This article will introduce in detail the calculation method of the sum of the interior angles of polygons, and supplement it with structured data to help readers better understand.
Basic concepts of sum of interior angles of polygons
A polygon is a closed figure composed of three or more line segments connected end to end. Depending on the number of sides, polygons can be divided into triangles, quadrilaterals, pentagons, etc. The sum of interior angles is the sum of the measures of all interior angles in a polygon.
Formula for calculating the sum of the interior angles of a polygon
The formula for calculating the sum of interior angles of a polygon is:(n-2) × 180°, among whichnRepresents the number of sides of the polygon. For example, if the number of sides of a triangle is 3, the sum of its interior angles is (3-2)×180°=180°.
| polygon name | Number of sides (n) | Sum of interior angles calculation formula | sum of interior angles result |
|---|---|---|---|
| triangle | 3 | (3-2)×180° | 180° |
| quadrilateral | 4 | (4-2)×180° | 360° |
| pentagon | 5 | (5-2)×180° | 540° |
| hexagon | 6 | (6-2)×180° | 720° |
Calculation of interior angles of regular polygons
A regular polygon is a polygon in which all sides and angles are equal. Since the formula for the sum of interior angles is known, the number of each interior angle of a regular polygon can be found by dividing the sum of the interior angles by the number of sides. The calculation formula is:[(n-2) × 180°] / n.
| regular polygon name | Number of sides (n) | Calculation formula for each interior angle | The result of each interior angle |
|---|---|---|---|
| equilateral triangle | 3 | [(3-2)×180°]/3 | 60° |
| square | 4 | [(4-2)×180°]/4 | 90° |
| regular pentagon | 5 | [(5-2)×180°]/5 | 108° |
| regular hexagon | 6 | [(6-2)×180°]/6 | 120° |
Derivation of the formula for the sum of the interior angles of a polygon
The derivation of the formula for the sum of the interior angles of a polygon is based on the sum of the interior angles of a triangle theorem. By splitting the polygon into triangles, you can intuitively understand where the formula comes from. For example, a quadrilateral can be divided into 2 triangles, so the sum of its interior angles is 2×180°=360°.
Application examples
Assuming that the sum of the interior angles of a heptagon is 900°, we can verify whether the number of sides is correct through the formula:(n-2)×180°=900°, the solution is n=7, and the verification is correct.
Summary
The calculation of the sum of the interior angles of a polygon is a basic knowledge point in mathematics. Mastering its formulas and derivation methods can help solve more complex geometric problems. Whether it is an ordinary polygon or a regular polygon, you can quickly calculate the sum of interior angles or the measure of a single interior angle using the above formula. I hope this article can help readers better understand and apply this knowledge.
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