Answer:
101.8 square units
Step-by-step explanation:
The given diagram shows a regular octagon with a radius of 6 units.
The radius of a regular polygon is the distance from the center of the polygon to one of its vertices.
Therefore:
- Number of sides: n = 8
- Radius: r = 6
To find the area of a regular polygon given its radius and number of sides, we can use the following formula:
[tex]\boxed{\begin{minipage}{5.5cm}\underline{Area of a regular polygon}\\\\$A=nr^2\sin \left(\dfrac{180^{\circ}}{n}\right)\cos\left(\dfrac{180^{\circ}}{n}\right)$\\\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
Substitute n = 8 and r = 6 into the formula and solve for A:
[tex]A=8\cdot 6^2\sin \left(\dfrac{180^{\circ}}{8}\right)\cos\left(\dfrac{180^{\circ}}{8}\right)[/tex]
[tex]A=288\sin \left(22.5^{\circ}\right)\cos\left(22.5^{\circ}\right)[/tex]
[tex]A=101.823376...[/tex]
[tex]A=101.8\; \sf square\;units\;(nearest\;tenth)[/tex]
Therefore, the area of a regular octagon with a radius of 6 units is 101.8 square units, to the nearest tenth.
