How To Find The Diagonal Of A Triangle

The Diagonal Formula for Squares, Rectangles, Cubes & Polygons

Polygons are the shapes of our earth. Computer and television screens, doors, and sheets of newspaper are all polygons. Diagonals of polygons are also useful. Larn how to instantly know how many diagonals whatever polygon can have past using this formula:

$# o f D i a chiliad o n a fifty south = \frac{northward (n - 3)}{2}$

What's a Simple Polygon?
What's a Diagonal?

Diagonals in Real Life

Diagonal of a Polygon Formula

Diagonal Formula

How To Find The Diagonal of a Rectangle

Diagonal of a Rectangle Formula
Diagonal of a Square Formula
Diagonal of a Cube Formula

What'south a Simple Polygon?

A simple polygon is any two-dimensional (apartment) shape made only with directly sides that shut in a infinite, and with sides that exercise non cross each other (if they practice, information technology is a complex polygon). A triangle is a polygon. A dart, kite, quadrilateral, and star are all polygons. Simple polygons can exist concave or convex. The formula we volition use works for all simple polygons.

What's a Diagonal?

A diagonal of a polygon is a line from a vertex to a non-adjacent vertex. So a triangle, the simplest polygon, has no diagonals. You cannot describe a line from 1 interior angle to whatever other interior angle that is non too a side of the triangle. A quadrilateral, the adjacent-simplest, has 2 diagonals. A pentagon, whether regular or irregular, has five diagonals.

$What is A Diagonal? - Definition$

In convex, simple polygons, diagonals will ever be inside the interior. Consider a rectangular door. You tin can run a line from the top hinge corner to the bottom, opposite corner. You can also run a line from the bottom hinge corner up to the top, contrary corner. Those are the only two diagonals possible.

In concave, elementary polygons, the diagonals may go exterior the polygon, crossing sides and partly lying in the shape'southward outside. They are yet diagonals. Darts and stars are typical examples of concave polygons with diagonals outside their shapes.

$Diagonals In Concave Polygons$

Exercise not attempt to apply these concepts and our diagonal formula to complex polygons (polygons with self-intersecting lines).

Diagonals in Real Life

Diagonals in squares and rectangles add together strength to structure, whether for a firm wall, bridge, or tall building. You may see diagonal wires used to proceed bridges steady. When houses are existence built, look for diagonal braces that concord the walls direct and true.

Bookshelves and scaffolding are braced with diagonals. For a catcher in softball or baseball to throw out a runner at second base, the catcher throws forth a diagonal from home plate to second.

The telephone or calculator screen you lot are viewing this lesson on is measured along its diagonal. A 21" screen never tells yous the width and meridian; it is 21" from one corner to an opposite corner.

Diagonal of a Polygon Formula

To find all possible diagonals of a unproblematic polygon with merely a few sides, you can hands count them. When the polygon gets a bit complicated, counting them can exist very hard.

$Diagonal Formula For Square Rectangle and Cube$

Fortunately, an easy formula exists to tell you lot exactly how many diagonals a polygon has. Remember that any vertex (corner) is continued by sides to two other vertices, and so those connections cannot count equally diagonals. That vertex cannot connect to itself, either. So for $n$ sides, nosotros will immediately reduce the possible number of diagonals past three.

Nosotros too do not desire to count the same diagonal twice. Our door, for example, only has two diagonals; you lot do not count going from the elevation swivel to bottom opposite and back again. You will have to divide any answer by 2.

Diagonal Formula

This leaves united states with an elegant formula, where $due north$ is the number of sides (or vertices):

$# o f D i a g o n a l southward = \frac{north (n - three)}{ii}$

How To Find The Diagonal of a Rectangle

Test this formula with something nosotros know: diagonals of a rectangle. A rectangle has four sides and four vertices.

$# o f D i a thousand o northward a fifty s = \frac{north (n - 3)}{ii}$

$= \frac{4 (4 - three)}{2}$

$= \frac{iv (i)}{2}$

$= \frac{4}{two}$

$= 2$

Be skeptical! Attempt information technology for a pentagon (v sides):

$How To Find The Diagonal$

$= \frac{5 (5 - 3)}{two}$

$= \frac{v (two)}{ii}$

$= \frac{10}{ii}$

$= 5$

A pentagon has merely 5 diagonals; our formula works.

Exist actually skeptical! Try it for a tetracontakaiheptagon, which is a ridiculously long (simply correct) name for a 47-gon:

$= \frac{47 (47 - 3)}{2}$

$= \frac{47 (44)}{2}$

$= \frac{ii,068}{2}$

$= 1,034$

Trust the formula. A 47-gon has 1,034 diagonals. This formula works every time to tell you exactly how many diagonals can exist constructed inside (or outside) of any simple polygon, whether the shape is convex or concave.

Diagonal of a Rectangle Formula

For rectangles, $l$ is the length of the rectangle, and $b$ is the summit of the rectangle.

$D i a g o n a fifty o f a R e c t a n g l eastward = \sqrt{l^{2} + b^{2}}$

Diagonal of a Square Formula

Now permit's look at a few different diagonal formulas to find the length of a diagonal.

$D i a g o due north a l o f a S q u a r due east = a \sqrt{2}$

Where $a$ is the side of the foursquare.

Diagonal of a Cube Formula

For a cube, we find the diagonal by using a iii-dimensional version of the Pythagorean Theorem/distance formula:

$D i a g o north a l o f a C u b east = \sqrt{{southward}^{ii} + {due south}^{ii} + s^{2}}$

Lesson Summary

You have learned a lot near specially important parts of polygons, their diagonals. You lot at present know how to place the diagonals of whatsoever polygon, what some existent-life examples of diagonals are, and how to utilize the formula, $# o f D i a g o n a l s = \frac{n (due north - three)}{2}$ , where north is the number of sides (or vertices) of the polygon. Too, we briefly covered diagonal forumals to find the length of a diagonal in cubes squares, and rectangles.