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

Mathematics

Plane and Solid Geometry

A Polygon is a closed plane line segments as sides.

A Convex Polygon is a polygon in which no interior angle is greater than 180 degrees.

A Concave Polygon is one having at least one interior angle greater than 180 degrees.

Q-1A polygon whose interior vertex angles are all less than 180 degrees is: A. concave

C. convex

B. irregular

D. regular

Q-2 For any polygon the sum of  all the exterior angles is

A. 180˚

C. 360˚

B. 0˚

D. 90˚

Q-3 Polygons are named according to their number of  . dia onals B. exterior angles

. ed es D. faces

Q--4 Q -4 How many sides are there in a regular icosagon? . B. 20

. D. 12

Properties of regular polygon:

 x

R

r

Q-5 How many sides have a polygon if the sum of its interior angles equals the sum of its exterior angles? A. 4

B. 5

C. 6

D. 2

Q-6 How many sides has a polygon if the sum of its interior angles equals twice the sum of its exterior angles? A. 7

C. 4

B. 6

D. 5

Q-7 7   How many diagonals are there in an octagon? A. 16

C. 61

B. 20

D. 25

Diagonals = n (n - 3 ) / 2

Q-8 How many diagonals can be drawn from a 12 sided polygon? A. 66

C. 54

B. 48

D. 36

Area and Perimeter of Regular Polygons:

 x

R

r

Given apothem and number of sides:

 180  A = nr  tan  n   2

=

 

180  n 

r

Given R: A=

nR 2 

360 

 sin  2  n 

R

 180  P = 2nR  sin  n  

R

Given length and number of sides:

nx  180  cot 4 n 2

A=

X

Given apothem and perimeter: A=

1 2

pr

1 2

p = semi − perimeter

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

Q-9 A regular octagon is inscribed in a circle whose radius is 12. Find the area of the octagon. . 521.31 B. 351.27  C. 407.29 D. 351.25

R

R

Q-10 Find the area of a regular hexagon whose sides measure 5 cm. A. 64.95

X = 5cm

B. 96.7  C. 47.6

5cm 60O

D. 69.5

60O

5cm

5cm 60O

Q-11 The apothem of a regular nonagon is 10. Determine its area. A. 227.43

C. 159.62

B. 327.57

D. 315.23         0         1

Q-12 Find the area of a pentagram inscribed in a circle of  radius 10 cm. A. 112.3

C. 125.4

B. 110.5

D. 117.3         0         1

Quadrilaterals: Parallelogram: Given Base and Altitude:

A = bh ven A=

1 2

h b

agona s: θ

d1d 2 sin θ  

d1 d2

Given adjacent sides and included angle:

A = ab sinθ

a θ  

b

Trapezoid: a

h

b

a+b A = h   2 

Q-13 The diagonals of a parallelogram are 18 cm and 30 cm respectively. One side of a parallelogram is 12 cm. Find the area of the arallelo ram. A. 214

C. 361

B. 216

D. 108

12

θ

Q-14 Find the area of a trapezoid whose median is 32 cm and whose altitude is 6. . B. 142

. D. 192

Rhombus: Given base and altitude:

h

A = hs

s

Given diagonals:

A=

1 2

d1d 2

Given adjacent sides and included angle:

A = s 2 sin θ  

s

Q-15 Find the area of a circle inscribed in a rhombus whose perimeter is 100 in and whose longer diagonal is 40 in. A. 356.27

C. 452.39

B. 250.57

D. 549.65

25

25 40

θ  

25

Q-16 The length of the side of a rhombus is 5 cm. If its shorter diagonal is of length 6 cm. What is the area of  the rhombus? A. 24 cm2

C. 18 cm2

B. 14 cm2

D. 25 cm2 6

5 θ  

5

Q-17 17  A rhombus is formed by two radii and two chords of a circle of  radius 10 m. What is the area of the rhombus? .

.

2

B. 92.1 m2

.

.

2

D. 220. 1 m2 10

60°

10

General quadrilateral: A=

s=

θ   =

( s − a )( s − b )( s − c )( s − d ) − abcd cos C

a +b+c+d

B

2 A+C 2

or

b c

a

B+ D 2

2

A d

D

θ  

Cyclic Quadrilateral: a

Ptolemy’s Theorem: d1

d1d2 = ac +bd

b

d d2

Bramaguptha’s formula:

A=

(s − a )(s − b )(s − c )(s − d )

Quadrilateral Circumscribing a Circle:

A quad

=

A

=

abcd rs

c

b

a

ri

d

Q-18 Find the fourth side quadrilateral inscribed in a circle one of its sides equal to 20 m. diameter, and the other two adjacent to the diameter are 8 m. ., .

of a having as its sides and 12

A. 6.785

C. 8.785

B. 7.654

D. 9.864

a

=

d

8

=

?

d2 d1 b = 20

c = 12

Q-18 Find the fourth side quadrilateral inscribed in a circle one of its sides equal to 20 m. diameter, and the other two adjacent to the diameter are 8 m. ., .

of a having as its sides and 12

A. 6.785

C. 8.785

B. 7.654

D. 9.864

Q-19 The sides of a cyclic quadrilateral are a = 3 cm, b=3 cm, c=4 cm and d=4 cm. Find the radius of the circle that can be inscribed in it. . 2.71cm B. 3.1 cm

A=

. 1.51 cm D. 1.71 cm

( s − a )( s − b )( s − c )( s − d )

Q-20 A right triangle is inscribed in a circle such that 1 side of the triangle is the diameter of the circle. If one of the acute angles of the triangle measures 60 deg and the 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

Q-21 An engineer places his transit along the line tangent to the circle at point A such that PA=200 m. He locates another point B on the circle and finds PB=80 m. If a third , PB, how far from point B will it be? A. 500 m.

C. 480 m.

B. 450 m.

D. 420 m.

Q-22 The radius of a circular sector is 32 m. and the length of the circular arc is 200 m. Find the area of the sector. . 2300 B. 3200

. 1600 D. 2400

Q-23 The angle of a sector is 30deg and the radius is 15cm. What is the area of the sector in sq cm? A. 59.8 B. 89.5 C. 58.9 D. 85.9

Q-24 A cylinder is circumscribed about a right prism having a square base one meter on an edge. The volume of the cylinder is 6.283 cu. m. Compute its altitude.

.3

. 5.4

B. 4

D. 2.5

1 1

Q-25 The volume of a right prism is 234 cu. m. with an altitude of 15 m. If the base of the prism is an equilateral triangle, find the length of the base edge. A. 5

C. 6

B. 10x

x x

h=15 x

x

D. 8

Q-26 The volume of a truncated prism with an equilateral triangle as its horizontal base is equal to 3600 cu. cm. The vertical edges at each corners are 4, 6, and 8 cm., respectively. Find one side of the base.

. 37.22 cm

. 25.34 cm

B. 15.64 cm

D. 30.52 cm

6 4

x

8 x

Q-27 27 A cone and a cylinder have the same height and the same volume. Find the ratio of the radius of the cone to the radius of the cylinder. A. 1.732 B. 0.577  C. 0.866 D.1.414

Q-28 A cone is inscribed of radius r. If the hemisphere share bases, of the region inside the outside the cone.

.

r

3

B. 7pi r3 / 3

in a hemisphere cone and the find the volume hemisphere but

.p r D. 4pi r3

Q-29 What is the volume of a pyramid whose altitude is 16 cm. long and whose base is enclosed by a rhombus whose sides are 6 cm. long and whose acute angles are 30 degrees?

A. 64 cu. cm.

C. 84 cu. cm.

B. 72 cu. cm.

D. 96 cu. cm.

6

Q-30 The lateral faces of a square pyramid make an angle 60O with the base. If the height of the pyramid is 5 square root of 3 m, find its lateral area. A. 200 m2

C. 320 m2

B. 120 m2

D. 220 m2

L

5 3 x

x x

x

Q-31 The lateral faces of a square pyramid make an angle 60O with the base. If the height of the pyramid is 5 square root of 3 m, find its lateral area. A. 200 m2

C. 320 m2

B. 120 m2

D. 220 m2

Q-31 Two corresponding sides of 2 similar polygons are 12 cm and 21 cm, respectively. If the perimeter of the small polygon is 60, find the perimeter of the big polygon. A. 105 cm

C. 107 cm

B. 102 cm

D. 103 cm

Q-32 If the edge of the cube is decreased by 10%, by what percent is the surface area decreases?

A. 19%

C. 89%

B. 81%

D. 10%

Q-33 If the surface area of a sphere is increased by 30%, by what percent is the volume of the sphere increased? . 51.8%

.61.7%

B. 48.2%

D. 30%

Q-34 A spherical wooden ball 15 cm. in diameter sinks to a depth of 12 cm. in a certain liquid. Find the area exposed above the liquid.

A. 50 pi

C. 45 pi

B. 25 pi

D. 15 pi

12cm

Q-35 The volume of the two spheres is in the ratio 27:343 and the sum of their radii is 10. Find the radius of the smaller sphere. . 3 B. 5 C. 4 D. 6

Q-36 What is the area of a lune whose angle is 85O on a sphere of  radius 30 cm. A. 1,670.45 cm2 B. 2,670.35 cm2 C. 2,570.53 cm2 D. 1,670.35 cm2

Q-37 37  A lune has an area of 30 square meters. If the angle of the lune is 90 degrees. What is the area of the sphere? A. 110 sq. m.

C. 120 sq. m.

B. 90 sq. m.

D. 150 sq. m.

Q-38 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

Polyhedrons

Polyhedrons 

Polyhedrons are solid whose faces are plane polygons eg a o y e ons – a e o y e ons whose faces are regular polygons



Tetrahedron (4 equal faces) 1.

Number of faces = 4

2.

No. of vertices = 4

.

.

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.

o. o e ges =

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

3.

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

3.

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

3.

No. of edges = 30

4.

Total Area = 8.66 a2

5.

Volume = 2.18 a3

6.

Radius of inscribed sphere: r = 0.76a