Answer:
1) 779.4 square units (nearest tenth)
2) 584.6 square units (nearest tenth)
Step-by-step explanation:
To find the areas of the given regular polygons, first determine their side lengths and apothems, then use the area formula:
[tex]\boxed{A=\dfrac{n\cdot s\cdot a}{2}}[/tex]
Question 1
The given diagram shows a six-sided regular polygon with an apothem measuring 15 units. Therefore:
- Number of sides: n = 6
- Apothem: a = 15
The formula for the apothem of a regular polygon is:
[tex]\boxed{\begin{minipage}{5.5cm}\underline{Apothem of a regular polygon}\\\\$a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\where:\\\phantom{ww}$\bullet$ $s$ is the side length.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]
Therefore, to find the side length, s, of the given regular polygon, substitute the values of a and n into the apothem formula and solve for s:
[tex]\implies 15=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{6}\right)}[/tex]
[tex]\implies 15=\dfrac{s}{2 \tan\left(30^{\circ}\right)}[/tex]
[tex]\implies s=30\tan\left(30^{\circ}\right)[/tex]
[tex]\implies s=30\cdot \dfrac{\sqrt{3}}{3}[/tex]
[tex]\implies s=10\sqrt{3}[/tex]
Therefore, the side length of the polygon is 10√3 units.
The formula for the area of a regular polygon is:
[tex]\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{n\cdot s\cdot a}{2}$\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the length of one side.\\ \phantom{ww}$\bullet$ $a$ is the apothem.\\\end{minipage}}[/tex]
Therefore, to find the area of the given regular polygon, substitute the values of n, s and a into the area formula and solve for A:
[tex]\implies A=\dfrac{6 \cdot 10\sqrt{3} \cdot 15}{2}[/tex]
[tex]\implies A=\dfrac{900\sqrt{3}}{2}[/tex]
[tex]\implies A=450\sqrt{3}[/tex]
[tex]\implies A=779.4\; \sf square \; units\;(nearest\;tenth)[/tex]
Therefore, the area of the given regular polygon is 779.4 square units (nearest tenth).
[tex]\hrulefill[/tex]
Question 2
The given diagram shows a six-sided regular polygon with a side length measuring 15 units. Therefore:
- Number of sides: n = 6
- Side length: s = 15
The formula for the apothem of a regular polygon is:
[tex]\boxed{\begin{minipage}{5.5cm}\underline{Apothem of a regular polygon}\\\\$a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\where:\\\phantom{ww}$\bullet$ $s$ is the side length.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]
Therefore, to find the apothem, a, of the given regular polygon, substitute the values of s and n into the apothem formula and solve for a:
[tex]\implies a=\dfrac{15}{2 \tan\left(\dfrac{180^{\circ}}{6}\right)}[/tex]
[tex]\implies a=\dfrac{15}{2 \tan\left(30^{\circ}\right)}[/tex]
[tex]\implies a=\dfrac{15}{2 \cdot \dfrac{\sqrt{3}}{3}}[/tex]
[tex]\implies a=\dfrac{15\sqrt{3}}{2}[/tex]
Therefore, the apothem of the polygon is (15√3)/2 units.
The formula for the area of a regular polygon is:
[tex]\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{n\cdot s\cdot a}{2}$\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the length of one side.\\ \phantom{ww}$\bullet$ $a$ is the apothem.\\\end{minipage}}[/tex]
Therefore, to find the area of the given regular polygon, substitute the values of n, s and a into the area formula and solve for A:
[tex]\implies A=\dfrac{6 \cdot 15 \cdot \dfrac{15\sqrt{3}}{2}}{2}[/tex]
[tex]\implies A=\dfrac{675\sqrt{3}}{2}[/tex]
[tex]\implies A=584.6\; \sf square \; units\;(nearest\;tenth)[/tex]
Therefore, the area of the given regular polygon is 584.6 square units (nearest tenth).
