decagon

What is a Decagon in Math? Definition

In mathematics, a decagon is a polygon with ten sides and ten angles. The word "decagon" is derived from the Latin words "decem" meaning ten and "gonia" meaning angle. It is a two-dimensional shape that lies entirely on a plane.

Knowledge Points of a Decagon and Detailed Explanation

A decagon contains several important knowledge points:

1. Sides: A decagon has ten sides, which are all equal in length.
2. Angles: A decagon has ten angles, each measuring 144 degrees.
3. Diagonals: A decagon has a total of 35 diagonals, which are line segments connecting non-adjacent vertices.
4. Symmetry: A decagon possesses rotational symmetry of order 10, meaning it can be rotated by multiples of 36 degrees to coincide with its original position.

Formula or Equation for a Decagon

The formula to calculate the sum of the interior angles of a decagon is given by:

Sum of Interior Angles = (n - 2) * 180 degrees

For a decagon, substituting n = 10 into the formula:

Sum of Interior Angles = (10 - 2) * 180 degrees = 8 * 180 degrees = 1440 degrees

Application of the Decagon Formula

The formula for the sum of interior angles of a decagon is useful in various mathematical applications. For example, it can be used to determine the measure of each interior angle of a regular decagon by dividing the sum by the number of angles (10 in this case).

Symbol for a Decagon

There is no specific symbol for a decagon in mathematics. It is usually represented by drawing a polygon with ten sides and labeling the vertices accordingly.

Methods for a Decagon

There are several methods to construct a decagon:

1. Compass and Straightedge: Using a compass to draw circles and a straightedge to connect the points of intersection, a decagon can be constructed.
2. Trigonometry: By utilizing trigonometric functions and properties, a decagon can be constructed based on the angles and side lengths.

Solved Examples on Decagon

Example 1: Find the measure of each interior angle of a regular decagon. Solution: Using the formula for the sum of interior angles, we know that the sum is 1440 degrees. Dividing this by the number of angles (10), we get 144 degrees. Therefore, each interior angle of a regular decagon measures 144 degrees.

Example 2: Determine the number of diagonals in a decagon. Solution: The formula to calculate the number of diagonals in a polygon is given by n * (n - 3) / 2, where n is the number of sides. Substituting n = 10, we have 10 * (10 - 3) / 2 = 35 diagonals in a decagon.

Practice Problems on Decagon

1. Calculate the measure of each exterior angle of a regular decagon.
2. Construct a regular decagon using a compass and straightedge.
3. Find the length of a diagonal in a decagon with side length 5 cm.

FAQ on Decagon

Question: What is a decagon? Answer: A decagon is a polygon with ten sides and ten angles.

Question: How many diagonals does a decagon have? Answer: A decagon has a total of 35 diagonals.

Question: What is the sum of the interior angles of a decagon? Answer: The sum of the interior angles of a decagon is 1440 degrees.

Question: How can I construct a decagon? Answer: A decagon can be constructed using a compass and straightedge or by utilizing trigonometric functions and properties.

Question: What is the measure of each interior angle of a regular decagon? Answer: Each interior angle of a regular decagon measures 144 degrees.