160k views
2 votes
Find the area of the equilateral triangle below (remember, this is a regular polygon).

Find the area of the equilateral triangle below (remember, this is a regular polygon-example-1
User Networks
by
8.0k points

1 Answer

4 votes

Equilateral triangles have interior angles of measure 60º. AL bisects angle MAY, so triangle ALY has angles 30º, 60º, and 90º. This means AY and LY occur in a ratio of √3 to 1. AY is half of AN, so AY = 9 and LY = 9/√3 = 3√3.

We can split up triangle MAN into 6 triangles with the same area as ALY, whose area is

1/2 * AY * LY = 1/2 * 9 * 3√3 = (27√3)/2

so that MAN has area

6 * (27√3)/2 = 81√3, or about 140.296.

Alternatively, we can observe that ML has the same length as AL, which by the Pythagorean theorem has length

AL = √(AY^2 + LY^2) = 6√3

Then MAN has area

1/2 * AN * (ML + LY) = 1/2 * 18 * (6√3 + 3√3) = 81√3

User Alfonsina
by
8.2k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories