Solved Geometry problems for competitive examsFull description
Analytic Geometry
Plane Geometry
math
lp sg
Lemmas in olympiad geometry
developing thinking geometry about geomtry and the others..Descripción completa
Lemmas in olympiad geometryFull description
This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes 255 problems …Descrição completa
Solid State Physics
Solid State PhysicsDescripción completa
Descripción completa
Solved problemsDescription complète
Phonetic and Phonology in EnglishFull description
physics problems and solutionsFull description
SOLID GEOMETRY 3.
SOLID GEOMETRY
Volume of a prism with base B and altitude h:
Definition of Terms: Polyhedron – Polyhedron – a a solid bounded by planes. Regular Polyhedron – – polyhedron whose faces are congruent regular polygons and whose polyhedral angles are regular polygons and whose polyhedral angles are equal. There are only five regular polyhedron: Tetrahedron – Tetrahedron – 4 4 faces Hexahedron – Hexahedron – 6 6 faces Octahedron – Octahedron – 8 8 faces Dodecahedron – Dodecahedron – 12 12 faces Icosahedron – Icosahedron – 20 20 faces Cube – Cube – a a polyhedron whose six faces are all squares Rectangular Parallelepiped Parallelepiped – polyhedron – polyhedron whose six faces are all rectangles. Prism – – polyhedron of which two faces are equal polygon in parallel planes and the other faces are equal parallelogram. Pyramid – Pyramid – a a polyhedron of which one faces is a regular polygon and other faces are triangles which have a common vertex. Regular Prism – Prism – prism prism whose lateral edges are perpendicular to its bases. Regular Pyramid – – pyramid whose base is a regular polygon and whose altitude passes through the center of base. Slant height – height – altitude altitude of lateral faces. Section – – polyhedron is the plane figure formed by a plane passing through the solid. Frustum of Pyramid – Pyramid – section section of the pyramid between the base and a section parallel to the base. Cone – – a solid bounded by a conic surface and a plane intersecting all the elements. Sphere – Sphere – a solid bounded by a surface all points of which are equidistant from a point called center. Great circle – circle – the the intersection and a plane passing through the center. Small circle – – the intersection of a sphere and a plane not passing through the center. Quadrant – Quadrant – one-fourth one-fourth of a great circle. Zone – Zone – portion portion of sphere bounded by a spherical polygon and the plane of its sides. Lune – portion – portion of a sphere lying between two semi – circles – circles of great circle. Spherical pyramid – portion – portion of sphere bounded by a spherical polygon and the plane of its sides. Spherical Sector – – portion of sphere generated by the revolution of circular sector about any diameter of the circle of which the sector is apart. Spherical segment - portion of sphere included between two parallel lines. Spherical wedge – wedge – portion portion bounded by a lune and the planes of two great circles. Focus – Focus – a a solid formed by revolving a circle about a line not intersecting it.
Formulas in Solid Mensuration 1.
Volume : V s3
2.
4.
Pyramid with base B and altitude h: V
1 3
Bh
5.
Prismatoid. with bases b and B, midsection M and altitude h: 1 Prismoidal Formula: V hb B 4M 6
6.
Sphere of radius r or diameter D: Volume : V
4 3 r or V 3
Right circular cylinder with radius r and altitude h: Volume : V
r 2h
: S 2rh
Lateral Area
8.
1 3 D 6
: S 4r 2 or S D2
Surface Area
7.
Right circular cone with radius r and altitude h: Volume : V
1
r 2h
3 Lateral Area : S rl
(l = slant height)
9. R
Frustum of a right circular cone with base radii r and and altitude h, slant height l: 1 Volume : V h r 2 R 2 rR 3
Cube with edge s:
Surface Area
V Bh
: A
6s2
Lateral Area
: S lr R
Rectangular parallelepiped parallelepiped with edges a, b, c and diagonal D: Volume : V abc Surface Area : A 2ab ac bc Diagonal : D a2
b2
c2
Engr. RonaldoT. Tamayo
SOLID GEOMETRY
15. Ellipsiod 10. Frustum of a pyramid with bases b and B and altitude h: 1 Volume : V h b B bB 3 1 Lateral Area : V p P l 2 where l = slant height p = perimeter of base b P = perimeter of base B
4
V = /3 abc
For oblate spheroid 4 2 V = /3 ab
For prolate Spheriod 4 2 V = /3 a b
a
b c
16. Paraboloid 2
h
V = ½ a h
a
11. Area Z of a zone with altitude h on a sphere of radius R:
17. Ungula
h 2
V = 2/3 r h S = 2 rh
Z 2Rh
r
SOLID GEOMETRY ASSIGNMENTS
12. Spherical segment of one base and altitude h on a sphere of radius R: 1 Volume : V h2 3R h 3 Total Area
Total Area
: T 2Rh a2
b2
2.
A frustum of a cone have diameters of 5 cm and 2 cm at the lower and upper base respectively. What is the altitude in cm if its 3 volume is 50 cm ? A. 4.9 B. 4.3 C. 6 D. 3.4
3.
A prismatoid has its end areas 20 m and 2 30 m , respectively. What is its volume (in 3 m ) if the distance between end areas is 10 2 m and the area at the mid-section is 24 m A. 233.67 B. 223.67 C. 246.67 D. 243.33
4.
Find the volume in cu. m. of a pyramid having a square base 8 cm and altitude of 10 cm. A. 213 B. 354 C. 112 D. 51
5.
What is the volume of a spherical pyramid whose spherical excess is 100° and the radius of the sphere is 10 units? A. 512.78 B. 581.78 C. 145.44 D. 312.74
6.
The bases of a right prism are pentagons with each side 6 cm long. The bases are 12 cm apart. What is the volume of the prism? 3 A. 473.2 cm 3 B. 347.2 cm 3 C. 734.2 cm 3 D 743.2 cm
14. Spherical Cone = a spherical sector having only one conical surface 1 2 Volume : V ZR or V R 2h 6 3 where:
A prismatoid has its end areas 20 m and 2 30 m , respectively. What is its volume in cu. m if the distance between end areas is 10 m and the area at the mid section is 24 2 m? A. 243.33 B. 246.67 C. 233.67 D. 223.67
: T h4R h
13. Spherical segment of two bases with radii a and b and altitude h on a sphere of radius R: 1 Volume : V h 3a 2 3b 2 h2 6
2
1.
V = volume of spherical sector R = radius of the sphere h = altitude of the sphere Z = area of the zone which forms the base of the sector
2
Engr. RonaldoT. Tamayo
SOLID GEOMETRY
7.
If a solid steel ball is immersed in an eight cm. diameter cylinder , it displaces water to a depth of 2.25 cm. The radius of the ball is _________ A. 3 cm B. 9 cm C. 6 cm D. 12 cm
8.
Find the volume of a right circular cylinder whose lateral 2 2 area is 25.918 m and base area of 7.068 m 3 A. 19.44 m 3 B. 44.19 m 3 C. 91.44 m 3 D. 41.49 m
9.
The volume of a cube is reduced by how much if all sides are halved A. 1 /8 B. 6 / 8 C. 5 / 8 D. 7 / 8
10. If 23 cu.m. of water is poured into a conical vessel, it reaches a depth of 12 cm. How much water must be added so that the depth reaches 18 cm ? A. 95 cu.cm. B. 54.6 cu.cm. C. 100 cu.cm. D. 76.4 cu. cm. 11. A regular triangular pyramid has an altitude of 9 m and 3 a volume of 187.06 m . What is the base edge? A. 42 m B. 12 m C. 24 m D. 21 m 12. A cylindrical tank lying horizontally , 0.90 m in diameter and 3 m long is filled to a depth of 0.60 m. How many gallons of gasoline does it contains? A. 250 B. 300 C. 350 D. 270 13. The surface area of a sphere is 4πr 2. Find the percentage increase in its diameter when t he surface area is increase by 21%? A. 5% B. 15% C. 10% D. 20% 14. Determine the volume of a regular tetrahedron whose side is 3 m 3 A. 2.183 m 3 B. 8.132 m 3 C. 3.182 m 3 D. 1.382 m 15. A right circular conical vessel is constructed to have a volume of 100,000 liters. Find the diameter if depth is to be 1.25 times the diameter A. 6.736 m B. 8.24 m C. 5.85 m D. 2.75 m 16. If the edge of a cube is increased by 30%, by how much is the surface area increased? A. 44 % B. 96 % C. 88 % D. 69 %
17. A right circular cone with an altitude of 8 cm is divided into two segments. One is a smaller circular cone having the same vertex with volume equal to ¼ of the bigger cone. Find the altitude of the smaller cone A. 21 cm B. 12 cm C. 30 cm D. nota 18. A conical vessel has a height of 24 cm and a base diameter of 12 cm. It holds water to a depth of 18 cm above its vertex. Find the volume of its content 3 A. 381.7 cm 3 B. 831.7 cm 3 C. 138.7 cm 3 D. 318.7 cm 19. The diameter of sphere and the base of a cone are equal. What percentage of that diameter must the cone’s height be so that both volumes are equal ? A. 50 % B. 150% C. 100 % D. 200% 20. If the volume of the sphere is 345cc’ solve for its diameter A. 8.7 m B. 8.7 mm C. 8.7 cm D. 8.7 km 21. A cubical container that measures 2 inches on a side is tightly packed with 8 marbles and is filled with water. All 8 marbles are in contact with the walls of the container and the adjacent marbles. All marbles are of the same size. What is the volume of water in the container? A. 4.21 cu. in. B. 2.51 cu. In C. 0.381 cu. in. D. 3.81 cu.in 22. A cone was formed by rolling a thin sheet of metal in the form of a sector of a circle 72 cm in diameter with a central angle of 210° . What is the volume of the cone? 3 A. 31,504 cm 3 B. 15,304 cm 3 C. 13,504 cm 3 D. 51,304 cm 23. A frustum of a cone whose upper base is 15 cm in diameter and lower base is 10cm in diameter with an altitude of 25 cm is filled with water. It is wished that the water shall be transferred to a spherical container. Determine the diameter of the spherical container. A. 11.90 cm B. 29.08 cm C. 18.11 cm D. 23.90 cm 24. The space diagonal of a cube is 4 times the square root of 3 meters. Find the volume of the cube A. 16 cu.m B. 64 cu.m C. 48 cu.m D. 86 cu.m 25. An artificial lake, 5 m deep, is to be dug in the form of a frustum of an inverted pyramid. The level bottom is 8 m by 80 m and its top is 10 m by 100 m. How many cubic meters of earth are to be removed? 3 A. 6,047 m 3 B. 4,067 m 3 C. 7,064 m 3 D. 7,046 m
Engr. RonaldoT. Tamayo
SOLID GEOMETRY
26. A sector of radius 6 cm. and central angle of 60 degrees is bent to form a cone . Find the volume of the cone in cu. cm A. 6.19 B. 5.9 C. 18.58 D. 10.23 27. A solid steel ball immersed in an 8cm diameter cylinder displaced water to a depth of 2.25 cm. The radius of the ball is _______ A. 3 cm. B. 1.5 cm C. 9 cm D. 2.0 cm 28. What is the surface area of a sphere whose volume is 3 36 m ? 2 A. 52.7 m 2 B. 25.7 m 2 C. 72.5 m 2 D. 25.7 m 29. What is the area of the zone of a spherical segment 3 having a volume of 1,470.265 m if the diameter of the sphere is 30 m? 2 A. 556.487 m 2 B. 568.457 m 2 C. 487.565 m 2 D. 565.487 m 30. The volume of two spheres is in the ratio of 27 : 343 and the sum of their radii is 10. Find the radius of the smaller sphere A. 6 B. 5 C. 3 D. 4
