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

Mathematics

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

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

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.

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

A. concave C. convex

B. irregular D. regular

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

A. 180˚ C. 360˚

B. 0˚ D. 90˚

Q-3 Polygons are named accordingto their number of

A. diagonals C. edgesA. diagonals C. edges

B. exterior angles D. faces

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

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

B. 20 D. 12

Properties of regular polygon:

x

R r

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

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

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

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

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

B. 48 D. 36

x

Area and Perimeter of Regular Polygons:

R r

Given apothem and number of sides:

2 180A nr tan

n =

180P 2nr tan = rP 2nr tan

n =

r

Given R:

2nR 360A sin

2 n =

180P 2nR sin

n =

R R

Given length and number of sides:

2nx 180A cot

4 n =

A cot4 n

=

X

Given apothem and perimeter:

1A pr

2= 1

p semi perimeter2

= −

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

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

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

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

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

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

Trapezoid:a

h

b

a bA h

2

+ =

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

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

Rhombus:

Given base and altitude:h

sA = hs

Given diagonals:

1 2

1A d d

2=

Given adjacent sides and included angle:

s

sθ

2A s sinθ=

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

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

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°

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 + +=

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= − − − −

Quadrilateral Circumscribing a Circle:

cb

a

quad

quad i

A abcd

A rs

=

=da ri

quad iA rs=

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

a 8=

c 12=

d ?=

1d2d

b 20=c 12=

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

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= − − − −

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

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.

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

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

Solid GeometrySolid 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

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

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

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

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

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°

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

5 3x

L

x

x

x

x

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

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

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%

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%

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

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

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

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.

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

PolyhedronsPolyhedrons

PolyhedronsPolyhedrons

� Polyhedrons are solid whose faces are plane polygons

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

� 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

� 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

� 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

� 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

� 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

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