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A regular polygon is an equilateral and equiangular polygon. The center of a regular polygon is the common center of its inscribed and circumscribed circles. A radius of a regular polygon is a segment joining its center to any vertex. A radius of a regular polygon is also a radius of the circumscribed circle. (Here, as for circles, we may use the word radius to mean the number that is the length of the radius. ) A central angle of a regular polygon is an angle included between two radii drawn to successive vertices. An apothem of a regular polygon is a segment from its center perpendicular to one of its sides. An apothem is also a radius of the inscribed circle. Thus for the regular pentagon shown in Fig. 10-1, AB BC CD DE EA and m/A m/B m/C m/D m/E. Also, its center is O, OA and OB are its radii; /AOB is a central angle; and OG and OF are apothems.



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Fig. 10-1

PRINCIPLE 1: PRINCIPLE 2: PRINCIPLE 3: PRINCIPLE 4:

If a regular polygon of n sides has a side of length s, the perimeter is p A circle may be circumscribed about any regular polygon. A circle may be inscribed in any regular polygon.

The center of the circumscribed circle of a regular polygon is also the center of its inscribed circle. An equilateral polygon inscribed in a circle is a regular polygon. Radii of a regular polygon are congruent.

PRINCIPLE 5: PRINCIPLE 6:





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Supplies A type of prepayment that is somewhat different from those previously described is the payment for of ce supplies or factory supplies. Assume that on April 1, $400 worth of supplies were purchased. There were none on hand before. This would increase the asset Supplies and decrease the asset Cash. At the end of April, when expense and revenue were to be matched and statements prepared, a count of the supplies on hand will be made. Assume that the inventory count shows that $250 of supplies are still on hand. Then the amount consumed during April was $150. The two entries are as follows: Apr. 1 Supplies Cash 30 Supplies Expense Supplies 400 400 150 150

A radius of a regular polygon bisects the angle to which it is drawn. Apothems of a regular polygon are congruent. An apothem of a regular polygon bisects the side to which it is drawn. For a regular polygon of n sides:

PRINCIPLE 7:

PRINCIPLE 8: PRINCIPLE 9:

PRINCIPLE 10:

360 1. Each central angle c measures n . (n 2)180 2. Each interior angle i measures . n 360 3. Each exterior angle e measures n .

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Thus for the regular pentagon ABCDE of Fig. 10-2, m/AOB and m/ABS 360 n 360 5 72 m/ABC m/ABC m/ABS (n 180 2) 180 n (5 2) 180 5 108

Fig. 10-2

Supplies Expense of $150 will be included in the April income statement; Supplies of $250 will be included as an asset on the balance sheet of April 30.

10.1 Finding measures of lines and angles in a regular polygon (a) Find the length of a side s of a regular pentagon if the perimeter p is 35. (b) Find the length of the apothem a of a regular pentagon if the radius of the inscribed circle is 21. (c) In a regular polygon of five sides, find the measures of the central angle c, the exterior angle e, and the interior angle i. (d) If an interior angle of a regular polygon measures 108 , find the measures of the exterior angle and the central angle and the number of sides.

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