Geometry

GEOMETRY

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a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids broadly : the study of properties of given elements that remain invariant under specified transformations.

CONE -A solid or hollow object which tapers from a circular or roughly circular base to a point. -A cone is a shape with a circular base and smooth curved sides ending in a point at the top.

Sample Problem 251. A conical vessel with a horizontal bottom base is filled with water to 2/3 of its altitude. What is the ratio of the volume of water to the volume of the vessel

252. a right conical solid is cut into two parts by a plane parallel to the base. If the two parts have equal volumes and the whole cones altitude is 45 cm., Find the distance of the cutting plane from the base.

253. A right circular cone has a lateral area of 386 cm2. Its diameter is one half of its altitude. What is the diameter?

254. The base diameter of a right circular cone is 18 cm. If the lateral area is 516.4 cm2. Find its volume in m3.

259. A closed conical vessel with vertical axis and horizontal base uppermost holds liquid at a depth of 2m. and the altitude is 3m. If the cone were inverted such that the horizontal base is lower most, how high would the liquid stand?

275. A right circular cone, with vertical axis and with base uppermost, is surmounted by a hemisphere. If the volume of the cone and the hemisphere are equal, what is the angle at the vertex of the cone included between two slant height at the same plane?

277. A piece of thin card board in the form of a sector of a circle of radius 36 cm. is rolled into the cone. Find the volume of the cone if the angle of the sector is 45 degree.

281. A cone was formed by rolling a thin of sheet in the form of a sector of a circle 72 cm. in diameter with a central angle of 210 degree. What is the volume of the cone?

283. A cone has an axis of 30 cm. Inclined at 60 degree with the horizontal base. The horizontal base has a diameter of 20 cm. What is the volume of the cone in cm3 ?

PYRAMID A

pyramid is a polyhedron that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex. These triangular sides are sometimes called the lateral faces to distinguish them from the base.

Sample Problem 298. A regular triangular pyramid has an altitude of 9 m. and a volume of 187.06 cu.m. What is the base edge in meters?

Sample Problem 299. The base edge of a regular triangular pyramid is 12 m. If the altitude is 9m., What is the volume in cu.m.?

Sample Problem 300. A regular hexagonal pyramid has a volume of 280.59 cu.m. and an altitude of 9 m. What is the base edge in meters?

Sample Problem 311. A triangular pyramid is cut by a plane parallel to its base. The area of the triangle of intersection is only one fourth of the base area. If the altitude of the pyramid is 30 cm., find the distance of the vertex of the pyramid to the cutting plane.

Frustum the portion of a cone or pyramid which remains after its upper part has been cut off by a plane parallel to its base, or which is intercepted between two such planes. -

Volume of a frustum of a Cone

Sample Problem 293. The frustum of a right circular cone has an upper base radius of 2 m. and a lower base radius of 1 m. If the distance between bases is 4 m., What is the volume of the frustum of the cone in cu.m.

Sample Problem 296. The frustum of a right circular cone has an altitude of 7 m. and volume of 51.31 cu.m. If the upper base diameter is 2 m., what is the lower base diameter in meter

Sample Problem 297. What is the lateral area of the frustum of a right circular cone whose base diameters are 2 and 4 m. if its altitude is 5 m.?

Sample Problem 321. The frustum of a regular triangular pyramid has an equilateral triangle for each bases. The lower and upper edges are 9 m. and 3 m. respectively. If the volume is 118.2 cu.m., how far apart are the bases?

Sample Problem 320. The volume if the frustum of a regular triangular pyramid is 135 cu.m. The lower base is an equilateral triangle with an edge of 9 m. The upper base is 8 m above the base edge. What is the upper base edge?

Sample Problem 327. The altitude of the frustum of a regular rectangular pyramid is 5 m. The volume is 140 cu.m. and the upper base is 3m. X 4m.. Find the dimensions of the lower base in m.

Sample Problem 334. A lateral edge of the frustum of a regular pyramid is 1.8 m. long. The lower base is 2.4m by 2.4m. Square while the upper base is 1m. X 1m. Square. What is the volume of the frustum?

337. A lateral edge of the frustum of a regular pyramid is 1.8 m. long. The lower base is 2.4 m. x 2.4 m. while the upper base is 1m. X 1m. Square. What is the lateral area of the frustum and its total surface area.

Hexahedron (CUBE)

Surface Area

Sample Problem 339. One edge of a regular hexahedron is 24 cm long. Find the ratio of its volume to its total surface area

Sample Problem 342. If each edge of a cube is decrease by 50%, how much (percentage) will the volume decrease?

Sample Problem 345. To double the volume of a cube, by how much (percentage) should the edge be increase

347. If the edge of a cube is decreased by 50%, by how much (percentage) will the surface area decrease

Sample Problem 353. The corners of conical box touch the spherical shell that encloses it. If the volume of the box is 27,000 cu.m. What is the volume of the space outside the box but inside the sphere?

CYLINDER

Sample Problem 354. A 53.4 sq.m. sheet metal was used to construct a cylindrical tank with open top. The tank height is 1.5 times its diameter. Find the diameter.

Sample Problem 358. The lateral surface of a circular column is covered with 5.65 sq.m. of wall paper. If the column is 3 m. high, find its diameter in cm.

Sample Problem 360. A circular cylindrical tank having a base diameter of one meter and a length of 2 m. is partly filled with water. When its axis is horizontal the water is 2/3 of the diameter deep. How deep will the water be when its axis is vertical?