SOLID GEOMETRY

REVIEW PROBLEMS 13. A solid has a circular base of radius 15m. Find the volume of
1. The base of a prism is a rectangle with sides 5cm and 10cm. the solid if every section perpendicular to a certain diameter
2
If its lateral area is 480cm , find its altitude, total area and is an equilateral triangle.
volume.
2. The altitude of a parallelepiped is 20 and the base is a 14. A cylindrical tank 4m in diameter and 6m high is full of
rhombus with diagonals 10 and 16. Find the volume of the water. It is then tilted to a position enough for the water
parallelepiped. surface to cut the diameter at the base. (i) How much water
3. The dimensions of a rectangular parallelepiped are in the is retained in the tank? (ii)How much water be on the tank
ratio of 2:3:5. If its volume is 30000 cu.cm find its diagonal. when it is then in placed in the vertical position?
4. The diagonal of the cube is 25sqrt of 3. Find its volume and 15. If a wedge is cut from the cylindrical log of radius 3m by
total surface area. two planes, one perpendicular to the axis and the other
5. The lateral area of a right hexagonal prism is 360 cm 2. If the passing through a diameter of the section made by the first
distance between the bases of the prism is 15cm, determine plane and inclined to this plane at an angle of 45 0 , find the
its volume. volume of the wedge.
6. A trough having an equilateral triangle end sections has sides
0.3m and 0.6m long contains water. If the depth of water 16. The axes of two right circular cylinders of equal radii 9 cm
inside is half the depth of the trough, find its volume. Find each, intersect at right angles. Find the volume of the
also the area of wetted surface of the trough. common part of the cylinders.
7. Each side of a cube is increased by 1%. By what percent is
17. The volume of a regular hexagonal pyramid is 190 cm 3 . If
the volume and surface area of the cube increased?
the length of each base edge of the pyramid is 6 cm, find the
8. The volume of a cylinder is 100pi. If its lateral area is 20pi,
altitude, lateral and total surface area of the pyramid.
find total surface area.
9. A horizontal cylindrical tank has radius of 4m and length of 18. At what distance from the vertex of a pyramid of altitude
10m. If the depth of water is 2m, find the volume of water 4m must it be cut by a plane parallel to is base so that the
and the wetted surface inside the tank. two solids of equal weight will be formed.
10. Find the volume of the largest circular cylinder that can be
inscribed in a cube whose volume is 64 cu cm. 19. The regular pyramid of altitude 12cm is divided into 3 parts
by two planes parallel to the base at a distance of 4m and 8m
11. A truncated prism has a horizontal triangular base ABC, AB respectively from the vertex . (i) Find area of the two cutting
= 10 c, BC = 12 cm and CA = 8 cm. The vertical edges through planes if the area of the base of the pyramid 64cm 2. (ii)Find
A, B and C are 20 cm, 12 cm and 18 cm. Determine the the ratio of the volume of the smallest part to the volume of
volume of the prism. the biggest part of the pyramid.
12. A truncated prism has a square base. The vertical edges of
each corner are 4m, 5m, 6m, and 7m. If the volume of prism 20. A closed conical tank has a diameter of 2cm at the top and
3
is 8800 m , find the diagonal of the base. Find also the height of 6m contains water at a depth of 4m. Find the (i)
lateral area.
 volume of water inside the tank and (ii) wetted surface of the of the smaller spherical zone. (iii) Find the volume of
tank. smaller spherical segment.
21. A hole 10cm in diameter is to be punched out from a right 30. A sphere has a radius of 12cm. (i) How far from the center of
circular cone having a diameter of 16cm and height of 24cm. the sphere should a plane passed so that the ratio of the areas
Determine the volume punched out. of the two zones formed is 2:3? (ii) Compute the area of the
of the smaller spherical zone. (iii) Find the volume of
22. A circular piece of cardboard with a diameter of 1m will be smaller spherical segment.
made into a conical nut 40 cm high by cutting a sector off
and joining the edges to form a cone. Find angle subtended 31. A spherical cone has a chord of generating arc of its zone
by the sector removed equal to 5cm and the radius of zone is 4cm. Find the volume
and total surface area of spherical cone.
23. The volume of a fustum of a regular triangular pyramid is
135 m3. The lower has an edge of 9m. If the distance 32. Find the area of lune of spherical wedge of radius 12cm if
between the bases is 8m. Find the (i) upper base edge and the volume of wedge is 502.655cm3.
(ii) total surface of the frustum.
33. A pentagonal spherical pyramid of base angles 1000, 1300,
24. The volume of a frustum of a right circular cone is 1176 1350, 1600, and 1500. Find the (i) spherical excess, (ii)area of
𝜋cm3 and the distance between the bases is 18cm. If the the base and (iii) volume of the pyramid. The radius of the
product the radii is 60 cm2, find the radius of the upper base sphere is 10m.
and the lateral area of the frustum of a cone.
34. If the volume of a sphere inscribed in a regular tetrahedron is
25. Find the surface area and volume of sphere inscribed in a 86 cm3, find the total surface area and volume of regular
cube of volume 512 m3. tetrahedron. Find also its dihedral angle.
26. The corners of a cubical box touched a closed spherical shell 35. Find the total surface area and volume of regular octahedron
that encloses it. If the volume of the cubical box is 125m 3, if its edge is12cm. If a sphere is inscribed in it, find the
find the volume inside the shell not occupied by the box. radius of sphere.
27. A cylindrical tank of radius 1.5m and altitude 5m is filled
with water to depth of 2m. (i) If the spherical ball of radius
1m is dropped into the tank slowly, by how much is the level
of water raised?
28. If the surface area of the sphere is increased by 20%,
determine the percentage increase in diameter and volume.
29. A sphere has a radius of 12cm. (i) How far from the center of
the sphere should a plane passed so that the ratio of the areas
of the two zones formed is 2:3? (ii) Compute the area of the