Table of Contents
Diagonals in Geometry
In geometry, diagonals refer to line segments that connect non-adjacent vertices of a polygon. The term is commonly used in the context of polygons, and the number of diagonals depends on the number of sides the polygon has. H
Diagonals of Different Polygons
- Triangle:
- A triangle has no diagonals because all of its vertices are adjacent, and there are no non-adjacent vertices to connect.
- Quadrilateral (Square, Rectangle, Parallelogram):
- A quadrilateral has two diagonals connecting non-adjacent vertices.
- Pentagon:
- A pentagon has five vertices, and each vertex can be connected to any non-adjacent vertex by a diagonal. Therefore, a pentagon has five diagonals.
- Hexagon:
- A hexagon has six vertices, and the number of diagonals is given by the formula D = \( \frac{n \times (n – 3)}{2}\), where n is the number of vertices. For a hexagon, this results in D = \(\frac{6 \times (6 – 3)}{2} = 9\) diagonals.
- Octagon:
- An octagon has eight vertices, and using the formula mentioned above, the number of diagonals is D = \(\frac{8 \times (8 – 3)}{2} = 20\) diagonals.
| Polygon Sides (\(n\)) | Number of Vertices | Number of Diagonals (\(D\)) |
|---|---|---|
| 3 (Triangle) | 3 | 0 |
| 4 (Quadrilateral) | 4 | 2 |
| 5 (Pentagon) | 5 | 5 |
| 6 (Hexagon) | 6 | 9 |
| 7 (Heptagon) | 7 | 14 |
| 8 (Octagon) | 8 | 20 |
Diagonal Formula
The formula to find the number of diagonals (D) in a polygon with nn sides is given by:
D = \(\frac{n \times (n – 3)}{2}\)
Diagonals of Solid Shapes
In three-dimensional geometry, the concept of diagonals, as commonly understood in two-dimensional shapes, is extended to “space diagonals” or “interior diagonals” in solid shapes. Unlike traditional diagonals that connect vertices in flat polygons, space diagonals connect non-adjacent vertices through the interior of three-dimensional solids.
For instance, in a cube, there are space diagonals that traverse through the center, connecting opposite vertices within the solid. A rectangular prism exhibits a similar pattern with interior diagonals connecting non-adjacent vertices. Pyramids, whether square-based or triangular, have space diagonals that link the apex with vertices on the base.
Cylinders and cones, while not having traditional diagonals, are characterized by features like height and radius that convey a sense of dimensionality. Spheres, being perfectly symmetrical, lack traditional diagonals but have a diameter representing the distance across the center.
Solved Examples on Diagonals
Example 1: Consider a rectangle with sides measuring 20 units and 15 units. Find the length of the diagonal (dd).
Solution:
Use the Pythagorean Theorem:
d = \(\sqrt{a^2 + b^2}\)
d = \(\sqrt{20^2 + 15^2}\)
= \(\sqrt{400 + 225}\)
diagonal d = \( \sqrt{625} \)
d= \(25 \text{ units}\)
Example 2: In a square with each side measuring a = 3 units, find the length of the diagonal (d).
Solution:
Use the formula for the diagonal of a square:
d = \( a \times \sqrt{2}\)
d = \(3 \times \sqrt{2}\)
diagonal d = \( 3 \times 1.414\)
d = 4.41 \text{ units (rounded)}\)
