here's the question

Dear Dr. Math,

I wanted tto find the area of this hexagon. I few the apothem as I've shown. Then I measured the apothem and perimeter. I used the formula a=1/2ap and got 3.2cm^2 for the area but that seems way too small for me. I think my teacher got lucky when she used that formula because it only works in a while. Do you agree?

K. L. Ooless

I need help with this and i need to know how to show the work.

The area of a hexagon is 1/2 a*P

The formula works all the time for a regular polygon.

However, the way you wrote the formula makes no sense. You cannot use the same variable "a" for area and for apothem.
Furthermore, we cannot draw diagrams in this forum, and lastly you did not give the dimensions of your hexagon, so I cannot check your answer.

Here is why the formula works.
A hexagon has 6 equal triangles, so lets find the area of one of them and multiply by 6
Let the base of the triangle be n, and its height h
so each triangle is (1/2)nh
But isn't the height h of the triangle the same as the apothem a ?
so each triangle is (1/2)an
so lets multiply by 6 to get the total area
which is (1/2)a(6n)
BUT, isn't 6n the perimeter ????

So the area is (1/2)a(perimeter)

60

To find the area of a regular hexagon, you can use the formula you mentioned: A = (1/2)ap, where "a" is the apothem (the distance from the center of the hexagon to a side), and "p" is the perimeter of the hexagon.

However, if the result seems too small, it's important to double-check your calculations and measurements. Let's go through the process step by step:

1. Measure the apothem: Make sure you measure the apothem accurately. It should be the distance from the center of the hexagon to a side. If you're unsure about the measurement, you can try remeasuring or seeking assistance.

2. Measure the perimeter: Carefully measure the perimeter of the hexagon, which is the total length of all six sides combined.

3. Calculate the area: Substitute the measured values into the formula A = (1/2)ap. Multiply the apothem by the perimeter, then divide the result by 2. This will give you the area in square units.

If you believe your teacher "got lucky" with the formula, it's worth noting that the formula is indeed valid for regular hexagons. However, if the hexagon you're working with is irregular (meaning the sides and angles are not all equal), then the formula may not give an accurate result.

In any case, if you're still uncertain about the accuracy, you can consider alternative methods for finding the area, such as dividing the hexagon into triangles or using trigonometric functions. These methods may require additional measurements and calculations, but they can provide a more accurate result.