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Problem 493

# What is the measure of the interior angle of a regular triangle? A regular quadrilateral? A regular 10 -gon? A regular 2000 -gon?

Expert verified
The interior angle measures of the given regular polygons are as follows: Regular triangle (3 sides) - $$60°$$; Regular quadrilateral (4 sides) - $$90°$$; Regular 10-gon (10 sides) - $$144°$$; Regular 2000-gon (2000 sides) - approximately $$179.64°$$.
See the step by step solution

## Step 1: Regular Triangle (n=3)

To find the interior angle measure of a regular triangle, we will use the formula: $$Interior \ angle \ measure = \frac{(n-2) × 180°}{n}$$ $$= \frac{(3-2) × 180°}{3}$$ $$= \frac{1 × 180°}{3}$$ $$= 60°$$ So, the measure of the interior angle of a regular triangle is 60°.

## Step 2: Regular Quadrilateral (n=4)

To find the interior angle measure of a regular quadrilateral, we will use the formula: $$Interior \ angle \ measure = \frac{(n-2) × 180°}{n}$$ $$= \frac{(4-2) × 180°}{4}$$ $$= \frac{2 × 180°}{4}$$ $$= 90°$$ So, the measure of the interior angle of a regular quadrilateral is 90°.

## Step 3: Regular 10-gon (n=10)

To find the interior angle measure of a regular 10-gon, we will use the formula: $$Interior \ angle \ measure = \frac{(n-2) × 180°}{n}$$ $$= \frac{(10-2) × 180°}{10}$$ $$= \frac{8 × 180°}{10}$$ $$= 144°$$ So, the measure of the interior angle of a regular 10-gon is 144°.

## Step 4: Regular 2000-gon (n=2000)

To find the interior angle measure of a regular 2000-gon, we will use the formula: $$Interior \ angle \ measure = \frac{(n-2) × 180°}{n}$$ $$= \frac{(2000-2) × 180°}{2000}$$ $$= \frac{1998 × 180°}{2000}$$ $$= 179.64°$$ So, the measure of the interior angle of a regular 2000-gon is approximately 179.64°.

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