Regular Polygons and the Circle

10.1   Regular Polygons

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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, and are its radii; ∠AOB is a central angle; and and are apothems.

Fig. 10-1

10.1A   Regular-Polygon Principles

PRINCIPLE 1: If a regular polygon of n sides has a side of length s, the perimeter is p = ns.

PRINCIPLE 2: A circle may be circumscribed about any regular polygon.

PRINCIPLE 3: A circle may be inscribed in any regular polygon.

PRINCIPLE 4: The center of the circumscribed circle of a regular polygon is also the center of its inscribed circle.

PRINCIPLE 5: An equilateral polygon inscribed in a circle is a regular polygon.

PRINCIPLE 6: Radii of a regular polygon are congruent.

PRINCIPLE 7: A radius of a regular polygon bisects the angle to which it is drawn.

Thus in Fig. 10-1, bisects ∠ABC.

PRINCIPLE 8: Apothems of a regular polygon are congruent.

PRINCIPLE 9: An apothem of a regular polygon bisects the side to which it is drawn.

Thus in Fig. 10-1, bisects , and bisects .

PRINCIPLE 10: For a regular polygon of n sides:

Thus for the regular pentagon ABCDE of Fig. 10-2,

and

m∠ABC + m∠ABS = 180°

Fig. 10-2

SOLVED PROBLEMS

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.

Fig. 10-3

Solutions

10.2   Proving a regular-polygon problem stated in words

Prove that a vertex angle of a regular pentagon is trisected by diagonals drawn from that vertex.

Solution

Given: Regular pentagon ABCDE Diagonals and

To Prove: and trisect ∠A.

Plan: Circumscribe a circle and show that angles BAC, CAD, and DAE are congruent.

PROOF:

10.2   Relationships of Segments in Regular Polygons of 3, 4, and 6 Sides

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In the regular hexagon, square, and equilateral triangle, special right triangles are formed when the apothem r and a radius R terminating in the same side are drawn. In the case of the square we obtain a 45°-45°-90° triangle, while in the other two cases we obtain a 30°-60°-90° triangle. The formulas in Fig. 10-4 relate the lengths of the sides and radii of these regular polygons.

Fig. 10-4

SOLVED PROBLEMS

10.3   Applying line relationships in a regular hexagon

In a regular hexagon, (a) find the lengths of the side and apothem if the radius is 12; (b) find the radius and length of the apothem if the side has length 8.

Solutions

10.4   Applying line relationships in a square

In a square, (a) find the lengths of the side and apothem if the radius is 16; (b) find the radius and the length of the apothem if a side has length 10.

Solutions

10.5   Applying line relationships in an equilateral triangle

In an equilateral triangle, (a) find the lengths of the radius, apothem, and side if the altitude has length 6; (b) find the lengths of the side, apothem, and altitude if the radius is 9.

Solutions

10.3   Area of a Regular Polygon

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The area of a regular polygon equals one-half the product of its perimeter and the length of its apothem.

As shown in Fig. 10-5, by drawing radii we can divide a regular polygon of n sides and perimeter p = ns into n triangles, each of area . Hence, the area of the regular polygon is .

Fig. 10-5

SOLVED PROBLEMS

10.6   Finding the area of a regular polygon

(a)  Find the area of a regular hexagon if the length of the apothem is .

(b)  Find the area of a regular pentagon to the nearest integer if the length of the apothem is 20.

Solutions

Fig. 10-6

10.4   Ratios of Segments and Areas of Regular Polygons

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PRINCIPLE 1: Regular polygons having the same number of sides are similar.

PRINCIPLE 2: Corresponding segments of regular polygons having the same number of sides are in proportion. “Segments” here includes sides, perimeters, radii or circumferences of circumscribed or inscribed circles, and such.

PRINCIPLE 3: Areas of regular polygons having the same number of sides are to each other as the squares of the lengths of any two corresponding segments.

SOLVED PROBLEMS

10.7   Ratios of lines and areas of regular polygons

(a)  In two regular polygons having the same number of sides, find the ratio of the lengths of the apothems if the perimeters are in the ratio 5:3.

(b)  In two regular polygons having the same number of sides, find the length of a side of the smaller if the lengths of the apothems are 20 and 50 and a side of the larger has length 32.5.

(c)  In two regular polygons having the same number of sides, find the ratio of the areas if the lengths of the sides are in the ratio 1:5.

(d)  In two regular polygons having the same number of sides, find the area of the smaller if the sides have lengths 4 and 12 and the area of the larger is 10,260.

Solutions

10.5   Circumference and Area of a Circle

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π (pi) is the ratio of the circumference C of any circle to its diameter d; that is, π = C/d. Hence,

C = πd      or      C = 2πr

Approximate values for π are 3.1416 or 3.14 or . Unless you are told otherwise, we shall use 3.14 for π in solving problems.

A circle may be regarded as a regular polygon having an infinite number of sides. If a square is inscribed in a circle, and the number of sides is continually doubled (to form an octagon, a 16-gon, and so on), the perimeters of the resulting polygons will get closer and closer to the circumference of the circle (Fig. 10-7).

Fig. 10-7

Thus to find the area of a circle, the formula can be used with C substituted for p; doing so, we get

All circles are similar figures, since they have the same shape. Because they are similar figures, (1) corresponding segments of circles are in proportion and (2) the areas of two circles are to each other as the squares of their radii or circumferences.

SOLVED PROBLEMS

10.8   Finding the circumference and area of a circle

In a circle, (a) find the circumference and area if the radius is 6; (b) find the radius and area if the circumference is 18π; (c) find the radius and circumference if the area is 144π. (Answer both in terms of π and to the nearest integer.)

Solutions

(a)  r = 6. Then C = 2πr = 12π = 38 and A = πr² = 36π = 36(3.14) = 113.

(b)  C = 18π. Since C = 2πr, we have 18π = 2πr and r = 9. Then A = πr² = 81π = 254.

(c)  A = 144π. Since A = πr², we have 144π = πr² and r = 12. Then C = 2πr = 24π = 75.

10.9   Circumference and area of circumscribed and inscribed circles

Find the circumference and area of the circumscribed circle and inscribed circle (a) of a regular hexagon whose side has length 8; (b) of an equilateral triangle whose altitude has length (See Fig. 10-8.)

Solutions

Fig. 10-8

10.10 Ratios of segments and areas in circles

(a)  If the circumferences of two circles are in the ratio 2:3, find the ratio of the diameters and the ratio of the areas.

(b)  If the areas of two circles are in the ratio 1:25, find the ratio of the diameters and the ratio of the circumferences.

Solutions

10.6   Length of an Arc; Area of a Sector and a Segment

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A sector of a circle is a part of a circle bounded by two radii and their intercepted arc. Thus in Fig. 10-9, the shaded section of circle O is sector OAB.

A segment of a circle is a part of a circle bounded by a chord and its arc. A minor segment of a circle is the smaller of the two segments thus formed. Thus in Fig. 10-10, the shaded section of circle Q is minor segment ACB.

Fig. 10-9

Fig. 10-10

SOLVED PROBLEMS

10.11 Length of an arc

(a)  Find the length of a 36° arc in a circle whose circumference is 45π.

(b)  Find the radius of a circle if a 40° arc has a length of 4π.

Solutions

10.12 Area of a sector

(a)  Find the area K of a 300° sector of a circle whose radius is 12.

(b)  Find the measure of the central angle of a sector whose area is 6π if the area of the circle is 9π.

(c)  Find the radius of a circle if an arc of length 2π has a sector of area 10π.

Solutions

10.13 Area of a segment of a circle

(a)  Find the area of a segment if its central angle measures 60° and the radius of the circle is 12.

(b)  Find the area of a segment if its central angle measures 90° and the radius of the circle is 8.

(c)  Find each segment formed by an inscribed equilateral triangle if the radius of the circle is 8.

Solutions

See Fig. 10-11.

Fig. 10-11

10.14 Area of a segment formed by an inscribed regular polygon

Find the area of each segment formed by an inscribed regular polygon of 12 sides (dodecagon) if the radius of the circle is 12. (See Fig. 10-12.)

Fig. 10-12

Solution

10.7   Areas of Combination Figures

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The areas of combination figures like that in Fig. 10-13 may be found by determining individual areas and then adding or substracting as required. Thus, the shaded area in the figure equals the sum of the aeas of the square and the semicircle:

Fig. 10-13

SOLVED PROBLEMS

10.15 Finding areas of combination figures

Find the shaded area in each part of Fig. 10-14. In (a), circles A, B, and C are tangent externally and each has radius 3. In (b), each arc is part of a circle of radius 9.

Fig. 10-14

Solutions

SUPPLEMENTARY PROBLEMS

  10.1. In a regular polygon, find

(10.1)

(a)  The perimeter if the length of a side is 8 and the number of sides is 25

(b)  The perimeter if the length of a side is 2.45 and the number of sides is 10

(c)  The perimeter if the length of a side is and the number of sides is 24

(d)  The number of sides if the perimeter is 325 and the length of a side is 25

(e)  The number of sides if the perimeter is and the length of a side is

(f)  The length of a side if the number of sides is 30 and the perimeter is 100

(g)  The length of a side if the perimeter is 67.5 and the number of sides is 15

  10.2. In a regular polygon, find

(10.1)

(a)  The length of the apothem if the diameter of an inscribed circle is 25

(b)  The length of the apothem if the radius of the inscribed circle is 23.47

(c)  The radius of the inscribed circle if the length of the apothem is

(d)  The radius of the regular polygon if the diameter of the circumscribed circle is 37

(e)  The radius of the circumscribed circle if the radius of the regular polygon is

  10.3. In a regular polygon of 15 sides, find the measure of (a) the central angle; (b) the exterior angle; (c) the interior angle.

(10.1)

  10.4. If an exterior angle of a regular polygon measures 40°, find (a) the measure of the central angle; (b) the number of sides; (c) the measure of the interior angle.

(10.1)

  10.5. If an interior angle of a regular polygon measures 165π, find (a) the measure of the exterior angle; (b) the measure of the central angle; (c) the number of sides.

(10.1)

  10.6. If a central angle of a regular polygon measures 58, find (a) the measure of the exterior angle; (b) the number of sides; (c) the measure of the interior angle.

(10.1)

  10.7. Name the regular polygon whose

(10.1)

(a)  Central angle measures 45°

(b)  Central angle measures 60°

(c)  Exterior angle measures 120°

(d)  Exterior angle measures 36°

(e)  Interior angle is congruent to its central angle

(f)  Interior angle measures 150°

  10.8. Prove each of the following:

(10.2)

(a)  The diagonals of a regular pentagon are congruent.

(b)  A diagonal of a regular pentagon forms an isosceles trapezoid with three of its sides.

(c)  If two diagonals of a regular pentagon intersect, the longer segment of each diagonal is congruent to a side of the regular pentagon.

  10.9. In a regular hexagon, find

(10.3)

(a)  The length of a side if its radius is 9

(b)  The perimeter if its radius is 5

(c)  The length of the apothem if its radius is 12

(d)  Its radius if the length of a side is 6

(e)  The length of the apothem if the length of a side is 26

(f)  Its radius if the length of the apothem is

(g)  The length of a side if the length of the apothem is 30

(h)  The perimeter if the length of the apothem is

10.10. In a square, find

(10.4)

(a)  The length of a side if the radius is 18

(b)  The length of the apothem if the radius is 14

(c)  The perimeter if the radius is

(d)  The radius if the length of a side is 16

(e)  The length of a side if the length of the apothem is 1.7

(f)  The perimeter if the length of the apothem is

(g)  The radius if the perimeter is 40

(h)  The length of the apothem if the perimeter is

10.11. In an equilateral triangle, find

(10.5)

(a)  The length of a side if its radius is 30

(b)  The length of the apothem if its radius is 28

(c)  The length of an altitude if its radius is 18

(d)  The perimeter if its radius is

(e)  Its radius if the length of a side is 24

(f)  The length of the apothem if the length of a side is 24

(g)  The length of its altitude if the length of a side is 96

(h)  Its radius if the length of the apothem is 21

(i)  The length of a side if the length of the apothem is

(j)  The length of the altitude if the length of the apothem is

(k)  The length of the altitude if the perimeter is 15

(l)  The length of the apothem if the perimeter is 54

10.12. (a)  Find the area of a regular pentagon to the nearest integer if the length of the apothem is 15.

(10.6)

(b)  Find the area of a regular decagon to the nearest integer if the length of a side is 20.

10.13. Find the area of a regular hexagon, in radical form, if (a) the length of a side is 6; (b) its radius is 8; (c) the length of the apothem is

(10.6)

10.14. Find the area of a square if (a) the length of the apothem is 12; (b) its radius is (c) its perimeter is 40.

(10.6)

10.15. Find the area of an equilateral triangle, in radical form, if

(10.6)

(a)  The length of the apothem is .

(b)  Its radius is 6.

(c)  The length of the altitude is 4.

(d)  The length of the altitude is

(e) The perimeter is

(f) The length of the apothem is 4.

10.16. If the area of a regular hexagon is find (a) the length of a side; (b) its radius; (c) the length of the apothem.

(10.6)

10.17. If the area of an equilateral triangle is find (a) the length of a side; (b) the length of the altitude; (c) its radius; (d) the length of the apothem.

(10.6)

10.18. Find the ratio of the perimeters of two regular polygons having the same number of sides if

(10.7)

(a)  The ratio of the sides is 1:8.

(b)  The ratio of their radii is 4:9.

(c)  Their radii are 18 and 20.

(d)  Their apothems have lengths 16 and 22.

(e)  The length of the larger side is triple that of the smaller.

(f)  The length of the smaller apothem is two-fifths that of the larger.

(g)  The lengths of the apothems are and 15.

(h)  The circumference of the larger circumscribed circle is times that of the smaller.

10.19. Find the ratio of the perimeters of two equilateral triangles if (a) the sides have lengths 20 and 8; (b) their radii are 12 and 60; (c) their apothems have lengths and (d) the circumferences of their inscribed circles are 120 and 160; (e) their altitudes have lengths 5x and x.

(10.7)

10.20. Find the ratio of the lengths of the sides of two regular polygons having the same number of sides if the ratio of their areas is (a) 25:1; (b) 16:49; (c) x²:4; (d) 2:1; (e) 3:y²; (f) x:18.

(10.7)

10.21. Find the ratio of the areas of two regular hexagons if (a) their sides have lengths 14 and 28; (b) their apothems have lengths 3 and 15; (c) their radii are and (b) their perimeters are 75 and 250; (e) the circumferences of the circumscribed circles are 28 and 20.

(10.7)

10.22. Find the circumference of a circle in terms of π if (a) the radius is 6; (b) the diameter is 14; (c) the area is 25π; (d) the area is 3π.

(10.8)

10.23. Find the area of a circle in terms of π if (a) the radius is 3; (b) the diameter is 10; (c) the circumference is 16π; (d) the circumference is π; (e) the circumference is

(10.8)

10.24. In a circle, (a) find the circumference and area if the radius is 5; (b) find the radius and area if the circumference is 16π; (c) find the radius and circumference if the area is 16π.

(10.8)

10.25. In a regular hexagon, find the circumference of the circumscribed circle if (a) the length of the apothem is (b) the perimeter is 12; (c) the length of a side is . Also find the circumference of its inscribed circle if (d) the length of the apothem is 13; (e) the length of a side is 8; (f) the perimeter is .

(10.9)

10.26. For a square, find the area in terms of π of the

(10.9)

(a)  Circumscribed circle if the length of the apothem is 7

(b)  Circumscribed circle if the perimeter is 24

(c)  Circumscribed circle if the length of a side is 8

(d)  Inscribed circle if the length of the apothem is 5

(e)  Inscribed circle if the length of a side is

(f)  Inscribed circle if the perimeter is 80

10.27. Find the circumference and area of the (1) circumscribed circle and (2) inscribed circle of

(10.9)

(a)  A regular hexagon if the length of a side is 4

(b)  A regular hexagon if the length of the apothem is

(c)  An equilateral triangle if the length of the altitude is 9

(d)  An equilateral triangle if the length of the apothem is 4

(e)  A square if the length of a side is 20

(f)  A square if the length of the apothem is 3

10.28. Find the radius of a pipe having the same capacity as two pipes whose radii are (a) 6 ft and 8 ft; (b) 8 ft and 15 ft; (c) 3 ft and 6 ft. (Hint: Find the areas of their circular cross-sections.)

(10.10)

10.29. In a circle, find the length of a 90° arc if

(10.11)

(a)  The radius is 4.

(b)  The diameter is 40.

(c)  The circumference is 32.

(d)  The circumference is 44π.

(e)  An inscribed hexagon has a side of length 12.

(f)  An inscribed equilateral triangle has an altitude of length 30.

10.30. Find the length of

(10.11)

(a)  A 90° arc if the radius of the circle is 6

(b)  A 180° arc if the circumference is 25

(c)  A 30° arc if the circumference is 60π

(d)  A 40° arc if the diameter is 18

(e)  An arc intercepted by the side of a regular hexagon inscribed in a circle of radius 3

(f)  An arc intercepted by a chord of length 12 in a circle of radius 12.

10.31. In a circle, find the area of a 60° sector if

(10.12)

(a)  The radius is 6.

(b)  The diameter is 2.

(c)  The circumference is 10π.

(d)  The area of the circle is 150π.

(e)  The area of the circle is 27.

(f)  The area of a 2408 sector is 52.

(g)  An inscribed hexagon has a side of length 12.

(h)  An inscribed hexagon has an area of .

10.32. Find the area of a

(10.12)

(a)  60° sector if the radius of the circle is 6

(b)  240° sector if the area of the circle is 30

(c)  15° sector if the area of the circle is 72π

(d)  90° sector if its arc length is 4π

10.33. Find the measure of a central angle of an arc whose length is

(10.12)

(a)  3 m if the circumference is 9 m

(b)  2 ft if the circumference is 1 yd

(c)  25 if the circumference is 250

(d)  6π if the circumference is 12π

(e)  Three-eighths of the circumference

(f)  Equal to the radius

10.34. Find the measure of a central angle of a sector whose area is

(10.12)

(a)  10 if the area of the circle is 50

(b)  15 cm² if the area of the circle is 20 cm²

(c)  1 ft² if the area of the circle is 1 yd²

(d)  5π if the area of the circle is 12π

(e)  Eight-ninths of the area of the circle

10.35. Find the measure of a central angle of

(10.11 and 10.12)

(a)  An arc whose length is 5π if the area of its sector is 25π

(b)  An arc whose length is 12π if the area of its sector is 48π

(c)  A sector whose area is 2π if the length of its arc is π

(d)  A sector whose area is 10π if the length of its arc is 2π

10.36. Find the radius of a circle if a

(10.11 and 10.12)

(a)  120° arc has a length of 8π

(b)  40° arc has a length of 2π

(c)  270° arc has a length of 15π

(d)  30° sector has an area of 3π

(e)  36° sector has an area of

(f)  120° sector has an area of 6π

10.37. Find the radius of a circle if a sector of area

(10.12)

(a)  12π has an arc of length 6π

(b)  10π has an arc of length 2π

(c)  25 cm² has an arc of length 5 cm

(d)  162 has an arc of length 36

10.38. Find the area of a segment if its central angle is 60° and the radius of the circle is (a) 6; (b) 12; (c) 3; (d) r; (e) 2r.

(10.13)

10.39. Find the area of a segment of a circle if

(10.13)

(a)  The radius of the circle is 4 and the central angle measures 90°.

(b)  The radius of the circle is 30 and the central angle measures 60°.

(c)  The radius of the circle and the chord of the segment each have length 12.

(d)  The central angle is 90° and the length of the arc is 4π.

(e)  Its chord of length 20 units is 10 units from the center of the circle.

10.40. Find the area of a segment of a circle if the radius of the circle is 8 and the central angle measures (a) 120°; (b) 135°; (c) 150°.

(10.13)

10.41. If the radius of a circle is 4, find the area of each segment formed by an inscribed (a) equilateral triangle; (b) regular hexagon; (c) square.

(10.13 and 10.14)

10.42. Find the area of each segment of a circle if the segments are formed by an inscribed

(10.13)

(a)  Equilateral triangle and the radius of the circle is 6

(b)  Regular hexagon and the radius of the circle is 3

(c)  Square and the radius of the circle is 6

Fig. 10-15

10.43. Find the shaded area in each part of Fig. 10-15. Each heavy dot represents the center of an arc or a circle.

(10.15)

10.44. Find the shaded area in each part of Fig. 10-16. Each dot represents the center of an arc or circle.

(10.15)

Fig. 10-16