Diagonal

A diagonal is a line segment that joins two non-adjacent vertices of a polygon. In other words, it is a line connecting two corners of a shape that is not next to each other. Diagonals are commonly found in polygons such as squares, rectangles, and triangles.

Types of Polygons and Diagonals

Square: A square has two diagonals, and each of them is of equal length. The length of the diagonal can be found using the formula: diagonal = side length * √2.
Rectangle: A rectangle also has two diagonals, but they are not of equal length. The length of the diagonals can be found using the formula: diagonal = √(length² + width²).
Triangle: A triangle has three sides but no diagonals. However, some geometric shapes have diagonals, such as a quadrilateral.

Properties of Diagonals

Some important properties of diagonals in polygons include:

Diagonals of a square are equal in length and bisect each other at 90 degrees.
Diagonals of a rectangle are not equal in length, and they bisect each other at 90 degrees.
The sum of the interior angles formed by the diagonals of a polygon can be calculated using the formula: sum of interior angles = (n - 2) * 180°, where n is the number of sides of the polygon.

Examples and Practice Problems

Let's solve some practice problems to understand the concept better:

Find the length of the diagonal of a square with a side length of 5 cm.

Answer: The length of the diagonal can be found using the formula: diagonal = side length * √2. Substituting the given value, we get: diagonal = 5 cm * √2 ≈ 7.07 cm.

Calculate the sum of the interior angles formed by the diagonals of a hexagon.

Answer: Using the formula: sum of interior angles = (n - 2) * 180°, where n = 6 (number of sides), we get: sum of interior angles = (6 - 2) * 180° = 4 * 180° = 720°.

By understanding the concept of diagonals and practicing related problems, you can gain a strong grasp of this topic in geometry.

◂Math Worksheets and Study Guides Fifth Grade. Place Value

The resources above cover the following skills:

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.

Algebra (NCTM)

Use mathematical models to represent and understand quantitative relationships.

Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.

Connections to the Grade 5 Focal Points (NCTM)

Number and Operations: Building on their work in grade 4, students extend their understanding of place value to numbers through millions and millionths in various contexts. They apply what they know about multiplication of whole numbers to larger numbers. Students also explore contexts that they can describe with negative numbers (e.g., situations of owing money or measuring elevations above and below sea level).