Can someone help me with an apothem problem?

The perimeter of an equilateral triangle is 24. Find its area.


Can someone please explain it? Thanks.

Can someone help me with an apothem problem?
Draw two equilateral triangles. Bisect one of them so that you have two smaller right triangles. Rotate the right triangles 180 degrees so that the top pf the hypotenuse now faces downward. Fit the right triangles to the equilateral triangle so that you have formed a square. Be sure to mark the measurements of the various sides of all the three triangles to illustrate the various dimensions. Calculating the area of the square yields an area twice as great as the area of the equilateral triangle. Each side of the square will be the same length as the length of the equilater triangle's base.


In other words, the equilateral triangle, having all its sides the same length, will be 24/3 = 8 units. The square you created will be 8 x 8 = 64 units. Divide the area of the square by two to find the area of the equilateral triangle.
Reply:Since it's an equilateral triangle, you know that all the sides are congruent. 24/3= 8 (each side). You then would draw an altitude, splitting the triangle in half. The triangle is now divided into two 30-60-90 triangles. Using the special triangles theorem (the side across from the 30 degree angle equals x, the side across from the 60 degree angle equals x times the square root of 3, and the side across from the 90 degree angle equals 2x), you can conclude that 8=2x, thus making x equal to 4. The altitude is then proven to be 4 times the square root of 3. Since the altitude is considered the height, you can use the formula for a triangle (Area= 1/2 times the product of the base and the height). To find the base, you would add the two congruent bisected lengths (4+4) to get 8. The area of the triangle would be: A= 1/2(8)(4 times the square root of 3).. so, the area is 16 times the square root of 3, or 27.71.