find (a) the sum of interior angles, (b) the measures of each interior angle and (c) the measures of each exterior angle of a regular 27gon, 30-gon and 35-gon
Answer (1)
Answer:27-gon:Sum of interior angles: 4500°Each interior angle: 166.67°Each exterior angle: 13.33°30-gon:Sum of interior angles: 5040°Each interior angle: 168°Each exterior angle: 12°35-gon:Sum of interior angles: 5940°Each interior angle: 169.71°Each exterior angle: 10.29°Step-by-step explanation:To solve for the sum of interior angles, the measure of each interior angle, and the measure of each exterior angle for a regular polygon, we use the following formulas:1. Sum of interior angles for an -sided polygon:[tex]\text{Sum of interior angles} = (n - 2) \times 180^\circ[/tex]2. Measure of each interior angle for a regular -sided polygon:[tex]\text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n}[/tex]3. Measure of each exterior angle for a regular -sided polygon:[tex]\text{Each exterior angle} = \frac{360^\circ}{n}[/tex]Let’s apply these formulas to a 27-gon, 30-gon, and 35-gon.1. For a 27-gon:(a) Sum of interior angles:[tex]\text{Sum} = (27 - 2) \times 180^\circ = 25 \times 180^\circ = 4500^\circ[/tex](b) Measure of each interior angle:[tex]\text{Each interior angle} = \frac{4500^\circ}{27} = 166.67^\circ[/tex](c) Measure of each exterior angle:[tex]\text{Each exterior angle} = \frac{360^\circ}{27} = 13.33^\circ[/tex]2. For a 30-gon:(a) Sum of interior angles:[tex]\text{Sum} = (30 - 2) \times 180^\circ = 28 \times 180^\circ = 5040^\circ[/tex](b) Measure of each interior angle:[tex]\text{Each interior angle} = \frac{5040^\circ}{30} = 168^\circ[/tex](c) Measure of each exterior angle:[tex]\text{Each exterior angle} = \frac{360^\circ}{30} = 12^\circ[/tex]3. For a 35-gon:(a) Sum of interior angles:[tex]\text{Sum} = (35 - 2) \times 180^\circ = 33 \times 180^\circ = 5940^\circ[/tex](b) Measure of each interior angle:[tex]\text{Each interior angle} = \frac{5940^\circ}{35} = 169.71^\circ[/tex](c) Measure of each exterior angle:[tex]\text{Each exterior angle} = \frac{360^\circ}{35} = 10.29^\circ[/tex]Summary of Results:27-gon:Sum of interior angles: 4500°Each interior angle: 166.67°Each exterior angle: 13.33°30-gon:Sum of interior angles: 5040°Each interior angle: 168°Each exterior angle: 12°35-gon:Sum of interior angles: 5940°Each interior angle: 169.71°Each exterior angle: 10.29°
