**Perimeter and Area of Inscribed Polygons**

**Perimeter Of Inscribed Regular Polygon**

Let *V* be a regular plane polygon having *n* sides, and whose vertices *P* _{1} , *P* _{2} , *P* _{3} , . . ., *P* _{n} lie on a circle of radius *r* (Fig. 4-11). Then the perimeter, *B* , of the polygon is given by the following formula when angles are specified in degrees:

*B* = 2 *nr* sin (180/ *n* )

If angles are specified in radians, then:

*B* = 2 *nr* sin ( *π* / *n* )

**Interior Area Of Inscribed Regular Polygon**

Let *V* be a regular polygon as defined above and in Fig. 4-11. The interior area, *A* , of the polygon is given by the following formula if angles are specified in degrees:

*A* = ( *nr* ^{2} /2) sin (360/ *n* )

If angles are specified in radians, then:

*A* = ( *nr* ^{2} /2) sin (2 *π* / *n* )

**Perimeter and Area of Circumscribed Polygons**

**Perimeter Of Circumscribed Regular Polygon**

Let *V* be a regular plane polygon having *n* sides whose center points *P* _{1} , *P* _{2} , *P* _{3} , . . ., *P* _{n} lie on a circle of radius *r* (Fig. 4-12). The perimeter, *B* , of the polygon is given by the following formula when angles are specified in degrees:

*B* = 2 *nr* tan (180/ *n* )

If angles are specified in radians, then:

*B* = 2 *nr* tan ( *π* / *n* )

**Interior Area Of Circumscribed Regular Polygon**

Let *V* be a regular polygon as defined above and in Fig. 4-12. The interior area, *A* , of the polygon is given by the following formula if angles are specified in degrees:

*A* = *nr* ^{2} tan (180/ *n* )

If angles are specified in radians, then:

*A* = *nr* ^{2} tan ( *π* / *n* )

**Inscribed and Circumscribed Polygons Practice Problems**

**PROBLEM 1**

What is the area of a regular octagon inscribed within a circle whose radius is exactly 10 units?

**SOLUTION 1**

Let’s use the formula for the area of an inscribed regular polygon, where angles are expressed in degrees:

*A* = ( *nr* ^{2} /2) sin (360/ *n* )

where *A* is the area in square units, *n* is the number of sides in the polygon, and *r* is the radius of the circle. We know that *n* = 8 and *r* = 10, so we can plug in the numbers and use a calculator as needed:

*A* = (8 × 10 ^{2} /2) sin (360/8)

= 400 sin 45°

= 400 × 0.7071

= 283 square units (approximately)

**PROBLEM 2**

What is the perimeter of a regular 12-sided polygon circumscribed around a circle whose radius is exactly 4 units?

**SOLUTION 2**

Let’s use the formula for the perimeter of a circumscribed regular polygon, where angles are expressed in radians:

*B* = 2 *nr* tan ( *π* / *n* )

where *B* is the perimeter, *n* is the number of sides in the polygon, and *r* is the radius of the circle. Consider *π* = 3.14159. We know that *n* = 12 and *r* = 4. We plug in the numbers and use a calculator, being sure the angle function is set for radians, not for degrees:

*B* = 2 × 12 × 4 tan ( *π* /12)

= 96 tan 0.261799

= 96 × 0.26795

= 25.72 units (approximately)

**PROBLEM 3**

How would you expect the perimeter of the circumscribed polygon in Problem 4-6 to compare with the perimeter of the circle around which it is circumscribed?

**SOLUTION 3**

It is reasonable to suppose that the perimeter of the polygon is slightly greater than the perimeter (or circumference) of the circle. Let’s calculate the circumference of the circle to see if this is true, and if so, to what extent. We use the formula for the circumference of a circle:

*B* = 2 *nr*

where *B* is the circumference and *r* is the radius. We know *r* = 4, and we can consider *n* = 3.14159. Thus:

*B* = 2 × 3.14159 × 4

= 25.13 units (approximately)

Practice problems for these concepts can be found at: Other Plane Figures Practice Test.

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