Area Of Triangle Using Apothem

What is an Apothem?

Field: Geometry
Image Created Past: azavez1

What is an Apothem?

The image to the right shows the shortest distance from the eye to the midpoint of one side in diverse regular polygons.

ane Basic Description
two A More than Mathematical Explanation
- two.i Full general Formula for Finding Surface area
- ii.2 Using the Apothem to Solve the Wire Problem
- 2.3 Eliminating Calculus
3 Teaching Materials
4 References

Bones Description

An apothem extends from the center of a regular polygon to the midpoint of i of its sides. If you know the lengths of the apothem and one side of a regular polygon, y'all can easily find its surface area. If the regular polygon (run into the hexagon in Effigy 1) is divided into triangles, the triangles tin be unrolled to form half of a rectangle.

Let'south start with a simple example. We will use the apothem of a hexagon to observe its area as shown in Figure 2.

Start with ii hexagons.
Assume that each side is 10 units long. The perimeter of each hexagon is 6x units.
Divide each hexagon into vi equilateral triangles (illustrated below). Each equilateral triangle has sides of length x and a pinnacle equal to the apothem. Let's call this a.
"Unwrap" each hexagon to get two rows of coinciding equilateral triangles. These two rows fit together nicely in a rectangle. The base of the rectangle is equal to half-dozenx, then the area of the rectangle is equal to 6xa. Simply what is a? To find out the expanse of the rectangle and therefore the hexagon, nosotros need the rectangle'southward elevation, which is equal to the apothem. Since the apothem cuts each equilateral triangle into two right triangles nosotros can use trigonometry to solve for a.

cos30^\circ = \frac{a}{x}

$a = xcos30^\circ$

Thus the surface area of the rectangle is $6x \cdot xcos30^\circ$
The area of the hexagon is equal to half the expanse of the rectangle so the expanse of the hexagon is equal to
$3x^2cos30^\circ$

A More Mathematical Explanation

Note: understanding of this explanation requires: *Geometry, Basic Algebra

[Click to view A More Mathematical Caption]

General Formula for Finding Area

Consider a regular polygon with northward sides that are each x [...]

[Click to hide A More Mathematical Explanation]

Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

References

National Security Agency'due south K-12 Academia Programme

↑ [ http://en.wikipedia.org/wiki/Centroid "Centroid"], Retrieved on xvi July 2012.

If you are able, please consider adding to or editing this folio! Have questions about the prototype or the explanations on this folio?
Leave a message on the discussion page by clicking the 'word' tab at the acme of this paradigm page.