# How do you find the side lengths of a polygon in a circle?

## How do you find the side lengths of a polygon in a circle?

Let O be the center of the circle, and let M be the midpoint of AB. Consider the right-angled triangle AOM; note that ∠AOM=360∘/10=36∘. Hence, by trigonometry, ¯AM/¯MO=¯AM/(d/2)=tan(36∘). Use that to solve for 2¯AM, which is the side length of the outer pentagon.

### How do you find the length of an inscribed angle?

By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the intercepted arc. The measure of the central angle ∠POR of the intercepted arc ⌢PR is 90°. Therefore, m∠PQR=12m∠POR =12(90°) =45°.

#### How do you find the measure of an arc with an inscribed angle?

**How do you find the side length of a polygon given the radius?**

Regular Polygon Formulas

- Side Length a. a = 2r tan(π/n) = 2R sin(π/n)
- Inradius r. r = (1/2)a cot(π/n) = R cos(π/n)
- Circumradius R. R = (1/2) a csc(π/n) = r sec(π/n)
- Area A. A = (1/4)na2 cot(π/n) = nr2 tan(π/n)
- Perimeter P. P = na.
- Interior Angle x. x = ((n-2)π / n) radians = (((n-2)/n) x 180° ) degrees.
- Exterior Angle y.

**How do you find the side length of a pentagon with the radius?**

If R is the radius, then the side length will be 2Rsin36∘ because the angle subtended by any side on to the center will be 72∘.

## What is the inscribed angle formula?

Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of the intercepted arc. That is, m∠ABC=12m∠AOC. This leads to the corollary that in a circle any two inscribed angles with the same intercepted arcs are congruent.

### What is the sides of an inscribed angle of a circle?

An inscribed angle is an angle whose vertex lies on a circle, and its two sides are chords of the same circle. On the other hand, a central angle is an angle whose vertex lies at the center of a circle, and its two radii are the sides of the angle.

#### Why is inscribed angle theorem true?

In a circle, the angle formed by two chords with the common endpoints of a circle is called an inscribed angle and the common endpoint is considered as the vertex of the angle….Inscribed Angle Theorem.

1. | What is Inscribed Angle Theorem? |
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4. | FAQs on Inscribed Angle Theorem |

**What is the formula to find the sides of a polygon?**

Answer: To find the number of sides of a polygon when given the sum of interior angles, we use the formula: Sum of interior angles = (n – 2) × 180, where n is the number of sides.