How do you find the apothem of a regular pentagon? Hexagon?

If I recall, apothem is the distance from the centre to the midpoint of a side

so look at one of the equal 5 isosceles triangles.
the central angles are 360/5 or 72° each, making the base angles 54°.

let the base be length x
then
tan 54° = apoth/(x/2)
apothem = (1/2) base x tan 54°

To find the apothem of a regular polygon, including a pentagon or a hexagon, you need to follow a few steps.

1. Determine the side length of the polygon: For a regular polygon, all sides are equal in length. If you are given the side length, you can proceed to the next step. If not, you may need additional information or measurements to calculate it.

2. Calculate the central angle: To find the central angle of a regular polygon, use the following formula:
Central angle = 360 degrees / number of sides
For a pentagon, the central angle would be 360 degrees / 5 = 72 degrees.
For a hexagon, the central angle would be 360 degrees / 6 = 60 degrees.

3. Find the length of the apothem using trigonometry: The apothem of a regular polygon is the distance from the center of the polygon to the midpoint of any side. To find the apothem, you can use the following formula:
Apothem = (Side length) / (2 * tan(central angle/2))
For example, if the side length of a pentagon is 10 units, you would substitute this value into the formula.
Apothem = (10) / (2 * tan(72/2))
Similarly, for a hexagon with a side length of 8 units:
Apothem = (8) / (2 * tan(60/2))

4. Simplify and calculate: Use a calculator or a trigonometric table to find the tangent value. After substituting the values into the formula and simplifying the expression, calculate the final answer.

These steps should help you find the apothem of a regular pentagon or hexagon.