Hi there. This is a short math post covering the topic of the area of an equilateral triangle. Knowing the Pythagorean Theorem formula is helpful here.
I use a little bit of Quicklatex.com for math images. Witeboard.com with screenshot images are also used here.
Equilateral Triangle Has All Three Side Lengths Being The Same
The equilateral triangle is a special triangle where all three sides have the same side length. Each of the three angles in the equilateral triangle is sixty degrees. Picture below from https://www.mathsisfun.com/definitions/equilateral-triangle.html.
Area Of Triangle Formula
Area refers to the amount of space in a two dimensional shape.
For the area of the triangle, the formula is the base multiplied by the height of the triangle then divided by two.
Area Of Equilateral Triangle Example With The Pythagorean Theorem
With the equilateral triangle the side length can be used as the base length. What needs to determined is the height of the triangle. From one corner of the triangle draw a line from that corner to the opposite side of the triangle. The line that meets the opposite side of the triangle meets at a right angle.
The screenshot below features an equilateral triangle with side lengths of 2 units each.
This equilateral triangle can be split into two right angled triangles. Look into the right angled triangle on the right side. The diagonal side length is 2 units, the bottom side length is 1 unit as it was split in half from 2 units. The height is unknown which needs to be solved. See the next screenshot for a visual.
This height of the right angled triangle can be solved with the use of the Pythagorean Theorem formula.
As the height of the equilateral triangle is known the area of this equilateral triangle can be determined.
Extension - Area of Pentagon, Hexagon, Etc.
The area of a equilateral triangle is extremely helpful for obtaining the area of a pentagon, the area of a hexagon, octagon, etc.
A pentagon can be split into five equilateral triangles. The area of a pentagon would be the area of the five equilateral triangles inside the pentagon. More details can be found from https://www.wikihow.com/Find-the-Area-of-a-Regular-Pentagon. Note that they use the tangent ratio to obtain the height of the triangle. I use the Pythagorean Theorem. Picture from Wikihow.
A further extension would be determining the surface area or volume of pentagonal prisms, hexagonal prisms, pentagonal based pyramids, octagonal based pyramids and so on. The calculations for those heavily involve the area of pentagons, hexagons which are related to the area of equilateral triangles.