For each problem, use the formula for area of a polygon A = 12 ap , where a is the length of the
apothem and p is the perimeter of the polygon, to find the area of the polygon.
Show the exact values of a and p on your figure and give your answers as exact values - no rounding – no decimals. Show all your work clearly. All polygons represented are regular polygons.

Part 1) Triangle:  and  

Part 2) Square:  and  

Part 3) Hexagon 1:  and  

Part 4) Hexagon 2:  and  

Step-by-step explanation:

Part 1) we have an equilateral triangle

we know that

where

a is the apothem

P is the perimeter

step 1

Find the length side of the equilateral triangle

Let

b ----> the length side of triangle

we know that

The measure of each interior angle is 60 degrees

so

---> by TOA (opposite side divided by the adjacent side)

equate

step 2

Find the perimeter

step 3

Find the area

Part 2) we have a square

we know that

where

a is the apothem

P is the perimeter

step 1

Find the length side of the square

Let

b ----> the length side of the square

we know that

The diagonal is half the radius of the square

Applying the Pythagorean Theorem

we have

simplify

step 2

Find the perimeter

step 3

Find the area

The apothem is half the length side of the square

Part 3) we have a regular hexagon

we know that

where

a is the apothem

P is the perimeter

step 1

Find the length side of the hexagon

we know that

The length side of a regular hexagon is equal to the radius

Let

b ----> the length side of the hexagon

we have

----> the radius is half the diameter

step 2

Find the perimeter

step 3

Find the area

The apothem is the height of an equilateral triangle

Applying the Pythagorean Theorem

simplify

substitute in the formula of area

Part 4) we have a regular hexagon

we know that

where

a is the apothem

P is the perimeter

step 1

Let

b ---> the length side of the hexagon

we have

Find the perimeter

step 2

Find the area

The apothem is the height of an equilateral triangle

Applying the Pythagorean Theorem

simplify

substitute in the formula of area