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High School Mathematics
4.26 Solid Geometry Final Review - I

Important Points in Prism

1. An object having length breadth and height is called solid.
The side portion of a solid are called lateral surfaces.
A solid is a three dimensional object.

2. A prism is polyhedron with two identical ends such that the lines joining the corresponding vertices of the two ends are all parallel.
Example: brick, a cube, cuboid.

3. Right Prism: A prism whose bases are perpendicular to the lateral edges and all lateral faces are rectangles is called right prism.

4. A prism is named according to the shape of its base. If the base of a prism is a triangle, then the prism is called triangular prism.

5. Number of lateral surfaces of a right prism = Number sides of the base of the prism.

6. Total number of surfaces of a prism = Number of lateral surfaces + 2.

7. Number of edges of a prism = 3 * Number of sides.

8. Sum of the lengths of edges = Number of sides * height + twice the perimeter of the base. That is S = n*h + 2s.

9. Lateral surface of a prism = Perimeter of the base * height. That is, A = p * h.

10. If 'a' is the side of cube.
(i). The lateral surface area of the cube = 4a2.
(ii). Area of the base of a cube = a2.
(iii). The total surface area of a cube = 6a2.

11. If l is the length, b is the breadth and h is height of a cuboid.
(i). Lateral surface area of the cuboid = 2h(l+b).
(ii). The area of the base of cuboid = lb.
(iii). The total surface area = 2 (lb + lh + bh).

12. In triangular prism.
(i). The lateral surface area = ph.
(ii). The area of the base = Area of triangle. 13. If a,b and c respectively are the sides of triangle and 's' is its semi-perimeter then area of the triangle =[s(s-a)(s-b)(s-c)].

14. The area of a equilateral triangle = (3/4)a2. Where a is the side of the triangle.

15. The area of a right angled triangle = 1/2 * ab, Where a, b are the sides containing right angle.

16. The area of a right angled isosceles triangle = 1/2*a2.

17. The area of rhombus = 1/2 * d1 * d2. Where d1, d are the diagonals of rhombus.

18. Volume of cuboid = l * b * h cubic units.

19. Sum of the edges of a cuboid = 4(l + b + h) units.

20. Diagonal of a cuboid = (l2 + b2 + h2).

21. Volume of the cube = a3 cubic units.

Important Points in Cylinder

1. If the base of a right prism are circular it is called cylinder.

2. The cylinder belongs to the family of prisms.

3. The curved surface area is equal to = Perimeter of the base * height
= Circumference of the circle * height.
= 2 * pi * r * h.

4. Area of the base of a cylinder = Area of circle.
= pi * r2.

5. Total surface area of a cylinder = Curved surface area + 2 * base area.
= 2 * pi * r + 2 * pi * r2.
= 2 * pi * r(h+r).

6. Volume of a cylinder = base area * height
= pi * r2 * h.

7. A cycle tube is an example of a cylindrical shell which is neither a solid cylinder nor a hollow cylinder.

8. The ratio of curve surface area's of two cylinder of equal heights is equal to the ratio of their radii.

9. The ratio of the volumes of two cylinders of equal heights is equal to the ratio of squares of their radii.

10. The ratio of two curved surface area's of two cylinders of equal radii is equal to the ratio of their heights.

11. The ratio of the volumes of two cylinders of equal radii but of different heights is equal to the ratio of their heights.

12. If the volumes of two cylinders are equal, then the ratio of the squares of radii is inversely proportional to the heights of the cylinders.

13.The cross section by a vertical plane of a cylinder is a rectangle.

14. A right circular cylinder:
A geometrical object, it has two circular parallel plane ends. They are called the bases of the cylinder.

15. Axis of the cylinder:
The line segment joining the centres of the two bases. It is perpendicular to the bases. Hence it is named as right circular cylinder.

16. Lateral surface:
There is a curved (not flat) surface joining the two bases.

17. Hollow cylinder:
It is the figure in the space bounded by just the lateral surface of the cylinder.

18. Solid cylinder:
It is the region in the space formed by the two plane ends and the lateral surface of the cylinder.

19. Cylindrical shell:
The difference of two solid cylinders.

Important points in Cone

1. A cone is a pyramid with a circular base.

2. Solid cone: It is the figure in space bounded by the curved surface of the cone and its plane circular end.

3. Curved surface area of a cone = pi * r * l,
Where r = radius of the base, and l = slant height.

4. Slant height, l2 = h2 + r2.

5. Lateral surface area of a cone is equal to pi * r * l.

6. Total surface area of a cone = Lateral surface area + Area of base,
= pi * r* (l+r).

7.Volume of a cone = 1/3 * pi * r2 * h.

8. A circus tent will be generally of the form of hollow cylinder surmounted by a cone.

9. Base of a cone: A cone has a plane surface circular in shape as base.

10. Vertex of cone: It is one end of the cone very far from the base.

Important Points in Sphere

1. The solid obtained by rotating a semi circular plane with its diameter as axis is a sphere.

2. Any point on the surface of the sphere is equidistant from a fixed point in the interior of the sphere. This fixed point is called the centre of the sphere.

3. The line segment joining the centre of the sphere to any point on its surface is called the radius of the sphere.

4. As the sphere has only one surface namely curved surface its curved surface area and total surface area are not different.
Curved surface area = 4 * pi * r2

5. Volume of sphere = 4/3 * pi * r3

6. The volume of a sphere of radius r is equal to the volume of the cylinder whose base radius is also r and height (4/3)r.

7. A plane through the centre of the sphere divides the sphere into two equal parts and each is called a hemisphere.
curved surface area of the hemisphere = Curved surface area + Area of the plane circular surface
= 2 * pi * r2 + pi * r2
= 3 * pi * r2

8. A spherical shell can be regarded as the difference between two concentric spheres.

9. A spherical shell has a finite thickness which is the difference of the radii of the two solid spheres which determine it.

Directions: Choose the correct answer.
Q 1: If the lateral surface area of a cube is 64 Sq. cm. then a = _____.
5 cm.
8 cm.
6 cm.
4 cm.

Q 2: If a = 12 cm., b = 16 cm. and c = 20 are the sides of base of a triangular prism and h = 15 cm. is its height then lateral surface area = _____.
720 Sq. cm.
726 Sq. cm.
724 Sq. cm.
None of these

Q 3: if the total surface are of cube is 294 Sq. cm. then a = _____.
9 cm.
6 cm.
7 cm.
8 cm.

Q 4: If base area of cube is 36 Sq. cm. then perimeter of the base = _____.
26 cm.
28 cm.
24 cm.
22 cm.

Q 5: If a = 4 cm. then total surface area of the cube = ______.
96 Sq. cm.
92 Sq. cm.
90 Sq. cm.
94 Sq. cm.

Q 6: if a = 7 cm. the lateral surface area of a cube = _______.
None of these
194 Sq. cm.
192 Sq. cm.
196 Sq. cm.

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Question 8: This question is available to subscribers only!

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