Certainly! Let's find the area of a regular octagon with the given measurements, step-by-step.
Step 1: Understanding the regular octagon properties
A regular octagon has 8 equal sides and 8 equal angles.
Step 2: Given measurements
- The apothem (which is the perpendicular distance from the center to a side) is 7 inches.
- The side length of the octagon is 5.8 inches.
Step 3: Formula for the area of a regular polygon
For any regular polygon, the area [tex]\( A \)[/tex] can be calculated using the formula:
[tex]\[ A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
Step 4: Calculation of the perimeter
First, we need to find the perimeter of the octagon. The perimeter [tex]\( P \)[/tex] is given by:
[tex]\[ P = \text{Number of sides} \times \text{Side length} \][/tex]
[tex]\[ P = 8 \times 5.8 \][/tex]
[tex]\[ P = 46.4 \text{ inches} \][/tex]
Step 5: Substituting values into the area formula
Now that we have the perimeter, we can substitute it along with the apothem into the area formula:
[tex]\[ A = \frac{1}{2} \times 46.4 \times 7 \][/tex]
[tex]\[ A = \frac{1}{2} \times 324.8 \][/tex]
[tex]\[ A = 162.4 \text{ square inches} \][/tex]
Step 6: Rounding the area to the nearest tenth
The final step is to round the calculated area to the nearest tenth. In this case:
[tex]\[ 162.4 \text{ square inches} \][/tex]
Therefore, the area of the regular octagon is [tex]\( 162.4 \)[/tex] square inches when rounded to the nearest tenth.
