Problem 76
Question
In Exercises 73 - 76, find the number of diagonals of the polygon. (A line segment connecting any two non adjacent vertices is called a diagonal of the polygon.) Decagon (\( 10 \) sides)
Step-by-Step Solution
Verified Answer
A decagon has 35 diagonals.
1Step 1: Understanding the formula
Firstly, recall the formula used to find the number of diagonals in any polygon. The formula is \( n(n-3)/2 \), where \( n \) is the number of sides of the polygon.
2Step 2: Substitute the number of sides into the formula
Now, since a decagon has 10 sides, substitute \( n = 10 \) into the formula. This becomes \( 10(10-3)/2 \).
3Step 3: Calculating the number of diagonals
Perform the calculation in the brackets first due to the order of operations (BIDMAS/BODMAS), yielding \( 10(7)/2 \), which simplifies to \( 35 \). Therefore, a decagon has 35 diagonals.
Key Concepts
DecagonPolygon GeometryNumber of Sides Formula
Decagon
A decagon is a type of polygon characterized by having exactly ten sides and ten angles. This shape is common in geometry and can be either regular or irregular.
In a regular decagon, all sides are of equal length, and all interior angles are equal, which simplifies many calculations, such as finding the number of diagonals.
Each angle in a regular decagon measures 144 degrees because the sum of the interior angles of a decagon is \( (10-2) \times 180 = 1440 \) degrees.
The understanding of a decagon's attributes enhances our ability to compute geometric properties, such as area and perimeter, as well as apply formulas to find specific details like its diagonals.
In a regular decagon, all sides are of equal length, and all interior angles are equal, which simplifies many calculations, such as finding the number of diagonals.
Each angle in a regular decagon measures 144 degrees because the sum of the interior angles of a decagon is \( (10-2) \times 180 = 1440 \) degrees.
The understanding of a decagon's attributes enhances our ability to compute geometric properties, such as area and perimeter, as well as apply formulas to find specific details like its diagonals.
Polygon Geometry
Polygon geometry is the study of shapes with multiple sides. A polygon is essentially a flat shape consisting of straight lines that are connected to form a closed figure. These lines are called sides, and the points where sides meet are vertices.
Common polygons include triangles, quadrilaterals, and pentagons, with more complex ones like hexagons and decagons. The number of sides determines the complexity and the characteristics of a polygon, such as its angles and diagonals.
In polygon geometry, understanding how to compute various properties like perimeter, area, and diagonals is crucial.
Diagonals of a polygon are line segments that connect non-adjacent vertices within the shape. The process to calculate these, especially for complex polygons like a decagon, often involves specific formulas that consider the number of sides of the polygon.
Common polygons include triangles, quadrilaterals, and pentagons, with more complex ones like hexagons and decagons. The number of sides determines the complexity and the characteristics of a polygon, such as its angles and diagonals.
In polygon geometry, understanding how to compute various properties like perimeter, area, and diagonals is crucial.
Diagonals of a polygon are line segments that connect non-adjacent vertices within the shape. The process to calculate these, especially for complex polygons like a decagon, often involves specific formulas that consider the number of sides of the polygon.
Number of Sides Formula
When calculating specific characteristics of polygons, such as the number of diagonals, it's essential to use accurate formulas.
One common formula used is the number of sides formula for diagonals: \( \frac{n(n-3)}{2} \), where \( n \) is the number of sides. This formula helps determine how many diagonals can be drawn in a polygon.
To use this formula for a decagon (), simply substitute the number of sides, which is 10, into \( n \) to compute the number of diagonals: \( \frac{10(10-3)}{2} = 35 \).
Understanding and applying this formula is a fundamental aspect of solving problems related to polygon geometry, making it easier to handle exercises involving shapes with various numbers of sides.
One common formula used is the number of sides formula for diagonals: \( \frac{n(n-3)}{2} \), where \( n \) is the number of sides. This formula helps determine how many diagonals can be drawn in a polygon.
To use this formula for a decagon (), simply substitute the number of sides, which is 10, into \( n \) to compute the number of diagonals: \( \frac{10(10-3)}{2} = 35 \).
Understanding and applying this formula is a fundamental aspect of solving problems related to polygon geometry, making it easier to handle exercises involving shapes with various numbers of sides.
Other exercises in this chapter
Problem 75
In Exercises 71-76, write the first five terms of the sequence. (Assume that \( n \) begins with 0.) \( a_n = \dfrac{(-1)^{2n}}{(2n)!} \)
View solution Problem 76
Toss two coins 100 times and write down the number of heads that occur on each toss (0, 1, or 2). How many times did two heads occur? How many times would you e
View solution Problem 76
In Exercises 73 - 78, use the Binomial Theorem to expand the complex number. Simplify your result. \( \left(5 + \sqrt{-9}\right)^3 \)
View solution Problem 76
In Exercises 73 - 78, find a quadratic model for the sequence with the indicated terms. \( a_0 = 3, a_2 = 0, a_6 = 36 \)
View solution