Question

Pentagons have 5 diagonals, as illustrated below. pentagon can't copy How many diagonals does the octagon below have? octagon can't copy F. 8 G. 16 H. 20 J. 30 K. 40

Short answer

Answer: An octagon has 20 diagonals.

Step-by-step solution

Understand the concept of diagonals

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. We can find the number of diagonals by calculating the combinations of vertices, as every pair of vertices of the shape forms either a side or a diagonal.

Use the formula to find the number of diagonals.

The formula for calculating the number of diagonals in a polygon is given by: \(n(n-3)/2\), where 'n' is the number of vertices in the polygon. For an octagon, there are 8 vertices. So, we can plug 'n=8' into the formula.

Calculate the number of diagonals in the given octagon.

Using the formula, let's find the number of diagonals in the octagon: Diagonals = \(8(8-3)/2\) Diagonals = \(8(5)/2\) Diagonals = \(40/2\) Diagonals = 20 So, the octagon has a total of 20 diagonals. The correct answer is H. 20.

Key concepts

Understanding Geometry through Polygons and Diagonals

Geometry, the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids, offers engaging problems such as counting the diagonals in polygons. A polygon is a two-dimensional shape with straight sides, and a diagonal in this context is a segment connecting any two non-consecutive vertices.

For students exploring geometric shapes, it is important to understand that a polygon's diagonals are not only fascinating in terms of geometric aesthetics but also serve as a practical tool for understanding the shape's structure. In a polygon, no diagonal ever connects two adjacent vertices, as this would result in a side, rather than a diagonal. The problem of finding diagonals in an octagon, akin to our exercise, introduces students to the visual understanding of geometric figures and their internal segments.

Math Problem Solving: Formulas and Applications

Effective math problem solving often involves knowing the right formulas and understanding how to apply them. In our example, the diagonal-counting problem is transformed into a straightforward task once we have the formula: \[(n(n-3)/2)\].

This formula embodies the conceptual step that transitions you from simply understanding what a diagonal is to being able to calculate its total quantity in any polygon. The intuitive reasoning behind this formula is that each vertex can be connected to all other vertices except itself and the two adjacent vertices, leading to the \((n-3)\) factor. Multiplying this by the total number of vertices \((n)\) accounts for all possible connections, and finally, dividing by two corrects for the double counting of each diagonal. Applying this formula requires little more than basic multiplication and division, making it accessible to all students. The key to successful math problem solving lies in analytically breaking down the problem, recognizing patterns, and making use of algebraic expressions.

Combinatorics in Diagonal Calculations

The study of counting, arrangement, and combination, known as combinatorics, plays a vital role in calculating the number of diagonals in polygons. In essence, when we calculate diagonals using the formula \[(n(n-3)/2)\], we are employing the fundamental principles of combinatorics. Each diagonal represents a specific combination of vertices.

Since each polygon's vertex can be paired with others to form a diagonal, and since these pairings are unique (that is, the diagonal from vertex A to vertex C is the same as the diagonal from vertex C to vertex A), the problem simplifies into the combination of vertices taken two at a time, minus the sides of the polygon. The division by two accounts for the fact that the order of vertices in a diagonal does not matter—another key concept in combinatorics. This practical application of combinatorial principles fascinates students and allows for a deeper appreciation of the interconnectedness of mathematical disciplines.