Geometry and Trigonometry: Area, Polygons, and Special Triangles Study Guide

Apothem Calculation for a Hexagon: The specific apothem mentioned is calculated as $6.25 \times \sqrt{3}$ . This represents the distance from the center of the regular polygon to the midpoint of one of its sides.
Side Length and Uniformity: In the provided example, the side length is identified as $12.5$ . This value is constant for every side of the polygon (uniformity).
Determining the Perimeter:
- For a regular hexagon (six sides), the perimeter is calculated by multiplying the side length by the number of sides.
- Calculation: $6 \times 12.5 = 75$ .
Area Formula for Regular Polygons: The general formula used is $Area = \frac{a \times P}{2}$ , where $a$ is the apothem and $P$ is the perimeter.
Numerical Verification of Area:
  - Calculations were performed to check the final area value.
  - Results initially mentioned were around $406.13$ , corrected to $406.3$ after division by $2$ .
  - The process involves taking the apothem ( $6.25 \times \sqrt{3}$ ), multiplying it by the perimeter ( $75$ ), and then dividing the product by $2$ .

Problem Context: Analyzing a square where a segment (radius-like) is given, necessitating the use of special right triangle rules to find side lengths.
Geometric Construction: A square can be split by diagonals into four right triangles. When a $90^\circ$ corner is bisected, it creates a $45^\circ$ angle.
The 45-45-90 Rule:
- The sides are in the ratio $x : x : x\sqrt{2}$ .
- The hypotenuse is $x\sqrt{2}$ .
Calculating the Side Length from the Hypotenuse:
  - Given a segment length (hypotenuse) of $3$ , the side of the internal triangle is found by dividing by the square root of two.
  - Formula: $\text{Side} = \frac{3}{\sqrt{2}}$ .
  - Decimal approximation: $3 / 1.414 \approx 2.12$ .
Determining the Full Side of the Square:
- The value $2.12$ represents only half of the square's side length.
- Full side calculation: $2.12 + 2.12 = 4.24$ .
Perimeter of the Square: Multiply the calculated side by four: $4.24 \times 4 = 16.96$ , often rounded or approximated based on the specific problem context.

Given Values:
- Shape: Octagon (eight sides).
- Radius ( $r$ ): $7\,mm$ .
Central Angle Calculation:
- To find the internal angles of the embedded triangles, divide the total degrees of a circle ( $360^\circ$ ) by the number of sides ( $n$ ).
- Calculation: $\frac{360^\circ}{8} = 45^\circ$ .
Right Triangle Decomposition:
- To use trigonometry, the central triangle ( $45^\circ$ at the vertex) must be split in half, creating a right triangle.
- New vertex angle: $\frac{45^\circ}{2} = 22.5^\circ$ .
Internal Triangle Angles: Since a triangle totals $180^\circ$ , a right triangle with a $22.5^\circ$ angle will have a third angle of $67.5^\circ$ ( $180^\circ - (90^\circ + 22.5^\circ) = 67.5^\circ$ ).
Using SOHCAHTOA for the Apothem:
  - Reference angle: $22.5^\circ$ .
  - Radius ( $7\,mm$ ) acts as the hypotenuse.
  - Apothem ( $a$ ) is the adjacent side.
  - Formula: $\cos(22.5^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{7}$ .
  - Solving for $a$ : $a = 7 \times \cos(22.5^\circ)$ .
Algebraic Manipulations in Trigonometry:
- If the variable is in the numerator (e.g., $\frac{x}{7}$ ), multiply the whole number by the trig function.
- If the variable is in the denominator (e.g., $\frac{7}{x}$ ), divide the whole number by the trig function.

Question on Area Calculation: A student asked how the value $406.3$ was reached for the hexagon.
- Response: Mrs. Braw clarified that the perimeter ( $75$ ) must be multiplied by the apothem and then divided by $2$ . One part of the calculation involving the side length resulted in $6.25$ when divided correctly.
Question on Side Lengths for Squares: A student asked how the teacher arrived at $4.24$ for the side of a square when the given hypotenuse segment was $3$ .
- Response: Mrs. Braw explained the division by $\sqrt{2}$ to get $2.12$ and then doubling that value to account for the full length of the side.
Logistics regarding Instructor Attendance:
  - Mrs. Braw announced she has jury duty on the following day (Tuesday).
  - Mr. Muhammad will be the regular teacher, but Mrs. Braw is teaching in his absence currenty as the Math Department Chair.
  - There is a scheduled test on Wednesday or Thursday.