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“I do the very best I know how – the very best I can; and I mean to keep on doing so until the end.” -Abraham Lincoln

Plane and Solid Geometry

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Page 1: Plane and Solid Geometry

“I do the very best I

know how – the very

best I can; and I mean

to keep on doing so

until the end.”

-Abraham Lincoln

Page 2: Plane and Solid Geometry

Mathematics

Plane and Solid Geometry Plane and Solid Geometry Plane and Solid Geometry Plane and Solid Geometry

Page 3: Plane and Solid Geometry

A Polygon is a closed planefigure bounded by straightfigure bounded by straightline segments as sides.

Page 4: Plane and Solid Geometry

A Convex Polygon is a polygon inwhich no interior angle is greater than180 degrees.

A Concave Polygon is one havingat least one interior angle greater than180 degrees.

Page 5: Plane and Solid Geometry

QQQQ----1111A polygon whose interiorvertex angles are all less than 180degrees is:

A. concave C. convex

B. irregular D. regular

Page 6: Plane and Solid Geometry

Q-2 For any polygon the sum ofall the exterior angles is

A. 180˚ C. 360˚

B. 0˚ D. 90˚

Page 7: Plane and Solid Geometry

Q-3 Polygons are named accordingto their number of

A. diagonals C. edgesA. diagonals C. edges

B. exterior angles D. faces

Page 8: Plane and Solid Geometry

QQQQ----4444 How many sides are therein a regular icosagon?

A. 200 C. 1000A. 200 C. 1000

B. 20 D. 12

Page 9: Plane and Solid Geometry

Properties of regular polygon:

x

R r

Page 10: Plane and Solid Geometry

QQQQ----5555 How many sides have apolygon if the sum of its interiorangles equals the sum of itsexterior angles?

A. 4 B. 5

C. 6 D. 2

Page 11: Plane and Solid Geometry

QQQQ----6666 How many sides has a polygonif the sum of its interior anglesequals twice the sum of its exteriorangles?

A. 7 C. 4

B. 6 D. 5

Page 12: Plane and Solid Geometry

A. 16 C. 61

B. 20 D. 25

QQQQ----7777 How many diagonals are therein an octagon?

B. 20 D. 25

Diagonals = n (n - 3 ) / 2

Page 13: Plane and Solid Geometry

QQQQ----8888 How many diagonals can bedrawn from a 12 sided polygon?

A. 66 C. 54A. 66 C. 54

B. 48 D. 36

Page 14: Plane and Solid Geometry

x

Area and Perimeter of Regular Polygons:

R r

Page 15: Plane and Solid Geometry

Given apothem and number of sides:

2 180A nr tan

n =

180P 2nr tan = rP 2nr tan

n =

r

Page 16: Plane and Solid Geometry

Given R:

2nR 360A sin

2 n =

180P 2nR sin

n =

R R

Page 17: Plane and Solid Geometry

Given length and number of sides:

2nx 180A cot

4 n =

A cot4 n

=

X

Page 18: Plane and Solid Geometry

Given apothem and perimeter:

1A pr

2= 1

p semi perimeter2

= −

r = radius of inscribed circle (apothem)p = perimeter

Page 19: Plane and Solid Geometry

QQQQ----9999 A regular octagon isinscribed in a circle whose radiusis 12. Find the area of theoctagon.

A. 521.31A. 521.31

B. 351.27

C. 407.29

D. 351.25

R R

Page 20: Plane and Solid Geometry

QQQQ----10101010 Find the area of a regularhexagon whose sides measure 5 cm.

A. 64.95X = 5cm

B. 96.7

C. 47.6

D. 69.5

5cm 5cm

5cm

60O

60O60O

Page 21: Plane and Solid Geometry

QQQQ----11111111 The apothem of a regularnonagon is 10. Determine its area.

A. 227.43 C. 159.62

B. 327.57 D. 315.23

10

Page 22: Plane and Solid Geometry

A. 112.3 C. 125.4

QQQQ----12121212 Find the area of apentagram inscribed in a circle ofradius 10 cm.

A. 112.3 C. 125.4

B. 110.5 D. 117.3

10

Page 23: Plane and Solid Geometry

Parallelogram:Given Base and Altitude:

h

b

A = bhGiven diagonals:

Quadrilaterals:

bGiven diagonals:

θd1

d21 2

1A d d sin

2θ=

Given adjacent sides and included angle:

abA = ab sinθ θ

Page 24: Plane and Solid Geometry

Trapezoid:a

h

b

a bA h

2

+ =

Page 25: Plane and Solid Geometry

QQQQ----13131313 The diagonals of aparallelogram are 18 cm and 30 cmrespectively. One side of aparallelogram is 12 cm. Find the areaof the parallelogram.

A. 214 C. 361

B. 216 D. 108

of the parallelogram.

12θ

Page 26: Plane and Solid Geometry

A. 150 C. 164

QQQQ----14141414 Find the area of a trapezoidwhose median is 32 cm and whosealtitude is 6.

A. 150 C. 164

B. 142 D. 192

Page 27: Plane and Solid Geometry

Rhombus:

Given base and altitude:h

sA = hs

Given diagonals:

1 2

1A d d

2=

Given adjacent sides and included angle:

s

2A s sinθ=

Page 28: Plane and Solid Geometry

QQQQ----15151515 Find the area of a circleinscribed in a rhombus whoseperimeter is 100 in and whoselonger diagonal is 40 in.

A. 356.27 C. 452.39

B. 250.57 D. 549.65 25

25

25 25

θ

40

Page 29: Plane and Solid Geometry

QQQQ----16161616 The length of the side of arhombus is 5 cm. If its shorter diagonalis of length 6 cm. What is the area ofthe rhombus?

A. 24 cm2 C. 18 cm2

B. 14 cm2 D. 25 cm2

5

5

6

θ

Page 30: Plane and Solid Geometry

QQQQ----17171717 A rhombus is formed by tworadii and two chords of a circle ofradius 10 m. What is the area of therhombus?

A. 86.6 m2 C. 143.1m2 A. 86.6 m2 C. 143.1m2

B. 92.1 m2 D. 220. 1 m2

10

1060°

Page 31: Plane and Solid Geometry

General quadrilateral:

b

a

C

c

B

( )( )( )( ) 2A s a s b s c s d abcdcosθ= − − − − −

a b c ds

2

+ + +=a c

Ad D

A C B Dor

2 2θ + +=

Page 32: Plane and Solid Geometry

Cyclic Quadrilateral:

c

d1

b

d2

d

aPtolemy’s Theorem:

d1d2 = ac +bd

cBramaguptha’s formula:

( )( )( )( )A s a s b s c s d= − − − −

Page 33: Plane and Solid Geometry

Quadrilateral Circumscribing a Circle:

cb

a

quad

quad i

A abcd

A rs

=

=da ri

quad iA rs=

Page 34: Plane and Solid Geometry

QQQQ----18181818 Find the fourth side of aquadrilateral inscribed in a circle havingone of its sides equal to 20 m. as itsdiameter, and the other two sidesadjacent to the diameter are 8 m. and 12m., respectively.m., respectively.

A. 6.785 C. 8.785

B. 7.654 D. 9.864

Page 35: Plane and Solid Geometry

a 8=

c 12=

d ?=

1d2d

b 20=c 12=

Page 36: Plane and Solid Geometry

QQQQ----18181818 Find the fourth side of aquadrilateral inscribed in a circle havingone of its sides equal to 20 m. as itsdiameter, and the other two sidesadjacent to the diameter are 8 m. and 12m., respectively.m., respectively.

A. 6.785 C. 8.785

B. 7.654 D. 9.864

Page 37: Plane and Solid Geometry

QQQQ----19191919 The sides of a cyclicquadrilateral are a = 3 cm, b=3 cm, c=4cm and d=4 cm. Find the radius of thecircle that can be inscribed in it.

A. 2.71cm C. 1.51 cmA. 2.71cm C. 1.51 cm

B. 3.1 cm D. 1.71 cm

( )( )( )( )A s a s b s c s d= − − − −

Page 38: Plane and Solid Geometry

QQQQ----20202020 A right triangle is inscribed in acircle such that 1 side of the triangle is thediameter of the circle. If one of the acuteangles of the triangle measures 60 deg andthe side opposite that angle has length of 15,what is the area of the circle?

A. 175.16 C. 235.62

B. 223.73 D. 228.61

Page 39: Plane and Solid Geometry

QQQQ----21212121 An engineer places his transitalong the line tangent to the circle atpoint A such that PA=200 m. Helocates another point B on the circleand finds PB=80 m. If a thirdportion C, on the circle lies alongportion C, on the circle lies alongPB, how far from point B will itbe?

A. 500 m. C. 480 m.

B. 450 m. D. 420 m.

Page 40: Plane and Solid Geometry

QQQQ----22222222 The radius of a circular sectoris 32 m. and the length of the circulararc is 200 m. Find the area of thesector.

A. 2300 C. 1600A. 2300 C. 1600

B. 3200 D. 2400

Page 41: Plane and Solid Geometry

QQQQ----23232323 The angle of a sector is 30degand the radius is 15cm. What is thearea of the sector in sq cm?

A. 59.8A. 59.8

B. 89.5

C. 58.9

D. 85.9

Page 42: Plane and Solid Geometry

Solid GeometrySolid Geometry

Page 43: Plane and Solid Geometry

QQQQ----24242424 A cylinder is circumscribed about aright prism having a square base onemeter on an edge. The volume of thecylinder is 6.283 cu. m. Compute itsaltitude.

A. 3 C. 5.4A. 3 C. 5.4

B. 4 D. 2.5

1

1

Page 44: Plane and Solid Geometry

QQQQ----25252525 The volume of a right prism is234 cu. m. with an altitude of 15 m.If the base of the prism is anequilateral triangle, find the lengthof the base edge.

A. 5 C. 6A. 5 C. 6

B. 10 D. 8

xx

xxx

x

h=15

Page 45: Plane and Solid Geometry

QQQQ----26262626 The volume of a truncated prism with anequilateral triangle as its horizontal base isequal to 3600 cu. cm. The vertical edges at eachcorners are 4, 6, and 8 cm., respectively. Findone side of the base.

A. 37.22 cm C. 25.34 cmA. 37.22 cm C. 25.34 cm

B. 15.64 cm D. 30.52 cm

86

4 xxx

Page 46: Plane and Solid Geometry

QQQQ----27272727 A cone and a cylinder have thesame height and the same volume.Find the ratio of the radius of thecone to the radius of the cylinder.

A. 1.732A. 1.732

B. 0.577

C. 0.866

D.1.414

Page 47: Plane and Solid Geometry

QQQQ----28282828 A cone is inscribed in a hemisphereof radius r. If the cone and thehemisphere share bases, find the volumeof the region inside the hemisphere butoutside the cone.

A. Pi r3 / 3 C. pi r3A. Pi r3 / 3 C. pi r3

B. 7pi r3 / 3 D. 4pi r3

Page 48: Plane and Solid Geometry

QQQQ----29292929 What is the volume of a pyramidwhose altitude is 16 cm. long and whosebase is enclosed by a rhombus whosesides are 6 cm. long and whose acuteangles are 30 degrees?

A. 64 cu. cm. C. 84 cu. cm.A. 64 cu. cm. C. 84 cu. cm.

B. 72 cu. cm. D. 96 cu. cm.

66

30°

Page 49: Plane and Solid Geometry

QQQQ----30303030 The lateral faces of a squarepyramid make an angle 60O with thebase. If the height of the pyramidis 5 square root of 3 m, find itslateral area.

A. 200 m2 C. 320 m2

B. 120 m2 D. 220 m2

Page 50: Plane and Solid Geometry

5 3x

L

x

x

x

x

Page 51: Plane and Solid Geometry

QQQQ----31313131 The lateral faces of a squarepyramid make an angle 60O with thebase. If the height of the pyramidis 5 square root of 3 m, find itslateral area.

A. 200 m2 C. 320 m2

B. 120 m2 D. 220 m2

Page 52: Plane and Solid Geometry

QQQQ----31313131 Two corresponding sides of 2similar polygons are 12 cm and 21 cm,respectively. If the perimeter of thesmall polygon is 60, find theperimeter of the big polygon.

A. 105 cm C. 107 cm

B. 102 cm D. 103 cm

Page 53: Plane and Solid Geometry

QQQQ----32323232 If the edge of the cube isdecreased by 10%, by what percentis the surface area decreases?

A. 19% C. 89%

B. 81% D. 10%

Page 54: Plane and Solid Geometry

QQQQ----33333333 If the surface area of a sphereis increased by 30%, by whatpercent is the volume of the sphereincreased?

A. 51.8% C.61.7%A. 51.8% C.61.7%

B. 48.2% D. 30%

Page 55: Plane and Solid Geometry

QQQQ----34343434 A spherical wooden ball 15 cm. indiameter sinks to a depth of 12 cm. in acertain liquid. Find the area exposedabove the liquid.

A. 50 pi C. 45 pi

B. 25 pi D. 15 pi

12cm

Page 56: Plane and Solid Geometry

QQQQ----35353535 The volume of the two spheresis in the ratio 27:343 and the sumof their radii is 10. Find theradius of the smaller sphere.

A. 3A. 3

B. 5

C. 4

D. 6

Page 57: Plane and Solid Geometry

QQQQ----36363636 What is the area of a lunewhose angle is 85O on a sphere ofradius 30 cm.

A. 1,670.45 cm2

B. 2,670.35 cm2B. 2,670.35 cm2

C. 2,570.53 cm2

D. 1,670.35 cm2

Page 58: Plane and Solid Geometry

QQQQ----37373737 A lune has an area of 30 squaremeters. If the angle of the lune is 90degrees. What is the area of thesphere?

A. 110 sq. m. C. 120 sq. m.

B. 90 sq. m. D. 150 sq. m.

Page 59: Plane and Solid Geometry

QQQQ----38383838 Find the area of a spherical

triangle ABC, A=125°, B=73°,C=84° in a sphere of radius 30 cm.

A. 1562.4 cm2 C. 1602.2 cm2

B.1567.3 cm2 D. 1652.2 cm2

Page 60: Plane and Solid Geometry

PolyhedronsPolyhedrons

Page 61: Plane and Solid Geometry

PolyhedronsPolyhedrons

� Polyhedrons are solid whose faces are plane polygons

� Regular Polyhedrons – are polyhedrons � Regular Polyhedrons – are polyhedrons whose faces are regular polygons

Page 62: Plane and Solid Geometry

� Tetrahedron (4 equal faces)

1. Number of faces = 4

2. No. of vertices = 4

3. No. of edges = 63. No. of edges = 6

4. Total Area = √3a2

5. Volume = (√2/12)a3

6. Radius of inscribed sphere:

r = (√6/12)a

Page 63: Plane and Solid Geometry

� Hexahedron ( 6 equal faces)

1. Number of faces = 6

2. No. of vertices =8

3. No. of edges = 123. No. of edges = 12

4. Total Area = 6a2

5. Volume = a3

6. Radius of inscribed sphere:

r = a/2

Page 64: Plane and Solid Geometry

� Octahedron ( 8 equal faces)

1. Number of faces = 8

2. No. of vertices =6

123. No. of edges = 12

4. Total Area = (2√3)a2

5. Volume = (√2)/3 a3

6. Radius of inscribed sphere:

r = a/√6

Page 65: Plane and Solid Geometry

� Dodecahedron ( 12 equal faces)

1. Number of faces = 12

2. No. of vertices =20

303. No. of edges = 30

4. Total Area = 20.65 a2

5. Volume = 7.66 a3

6. Radius of inscribed sphere:

r = 1.11a

Page 66: Plane and Solid Geometry

� Icosahedron ( 20 equal faces)

1. Number of faces = 20

2. No. of vertices =12

303. No. of edges = 30

4. Total Area = 8.66 a2

5. Volume = 2.18 a3

6. Radius of inscribed sphere:

r = 0.76a

Page 67: Plane and Solid Geometry

QQQQ----39393939 FindFind thethe surfacesurface areaarea ofof regularregularicosahedronsicosahedrons whenwhen eacheach edgeedge isis ofoflengthlength 55..

A. 216.5A. 216.5 C.126.6C.126.6A. 216.5A. 216.5 C.126.6C.126.6

B. 261.5 B. 261.5 D.162.5D.162.5