Answer:
3) 377.0 cm² (nearest tenth)
4) 14.5 yd² (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 3
The given diagram shows a ten-sided regular polygon with a side length measuring 7 cm. Therefore:
- Number of sides: n = 10
- Side length: s = 7
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 an expression for the apothem, a, of the given regular polygon, substitute the values of s and n into the apothem formula:
[tex]\implies a=\dfrac{7}{2 \tan\left(\dfrac{180^{\circ}}{10}\right)}[/tex]
[tex]\implies a=\dfrac{7}{2 \tan\left(18^{\circ}\right)}[/tex]
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{10 \cdot 7 \cdot \dfrac{7}{2 \tan\left(18^{\circ}\right)}}{2}[/tex]
[tex]\implies A=\dfrac{245}{2\tan\left(18^{\circ}\right)}}[/tex]
[tex]\implies A=377.0\; \sf cm^2\;(nearest\;tenth)[/tex]
Therefore, the area of the given regular polygon is 377.0 cm² (nearest tenth).
[tex]\hrulefill[/tex]
Question 4
The given diagram shows a seven-sided regular polygon with a side length measuring 2 yds. Therefore:
- Number of sides: n = 7
- Side length: s = 2
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 an expression for 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{2}{2 \tan\left(\dfrac{180^{\circ}}{7}\right)}[/tex]
[tex]\implies a=\dfrac{1}{ \tan\left(\dfrac{180^{\circ}}{7}\right)}[/tex]
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{7\cdot 2\cdot \dfrac{1}{ \tan\left(\dfrac{180^{\circ}}{7}\right)}}{2}[/tex]
[tex]\implies A=\dfrac{7}{ \tan\left(\dfrac{180^{\circ}}{7}\right)}[/tex]
[tex]\implies A=14.5\; \sf yd^2\;(nearest\;tenth)[/tex]
Therefore, the area of the given regular polygon is 14.5 yd² (nearest tenth).
