Diagonal: Explanation and Study Guide

Definition:

A diagonal is a straight line that connects two non-adjacent vertices of a polygon or a polyhedron. It is a line segment that does not necessarily intersect with the sides of the polygon or polyhedron.

Properties of Diagonals:

1. In a polygon, the number of diagonals can be calculated using the formula: D = n(n-3)/2, where n is the number of sides of the polygon.
2. The length of a diagonal in a rectangle or square can be found using the Pythagorean theorem: d = √(l² + w²), where d is the length of the diagonal, l is the length, and w is the width of the rectangle or square.
3. In a cube or rectangular prism, the length of a diagonal can be calculated using the formula: d = √(l² + w² + h²), where d is the length of the diagonal, and l, w, and h are the length, width, and height of the shape, respectively.

Examples:

Let's consider a square with sides of length 5 units. The length of the diagonal can be found using the formula:

d = √(5² + 5²)

d = √(25 + 25)

d = √50

d ≈ 7.07 units

Study Guide:

To master the concept of diagonals, it's important to practice the following:

1. Understand the definition of a diagonal and its properties.
2. Practice using the formulas to calculate the number of diagonals in a polygon and the length of diagonals in different shapes.
3. Solve problems involving diagonals in rectangles, squares, cubes, and rectangular prisms.
4. Explore real-life applications of diagonals in various geometric shapes.

By mastering the concept of diagonals, you'll be able to understand the relationships between different vertices of polygons and polyhedra, and apply this knowledge to solve a variety of geometry problems.