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Lesson 10: Solids. Surface Areas. Revolution solids (III)

May 22, 2012

5. Solids of revolution

A solid of revolution is a solid figure obtained by rotating a flat figure around a line that lies on the same plane. The axis of revolution is the line about which the revolution takes place.

5.1. Cylinder

A cylinder is a solid of revolution generated by a rectangle that rotates around one of its sides.

Elements of a cylinder

Surface area of a cylinder

The net of a cylinder is formed by:

  • A rectangle whose length is the length of the circumference of the bases and its width is the height of the cylinder.
  • Two equal circles that are its bases.

The surface area of a cylinder can be calculated from its net. So it will be the sum of the lateral area and the areas of the two equal bases:

  • Lateral area AL: It is the area of the rectangle whose length is the length of the circumference of the bases with radius r, 2πr and its width is the height of the cylinder,h.

AL = 2πr · h

  • Area of the bases AB: This is the sum of the areas of the bases with radius . 

AB = πr2

The surface area of a cylinder is:

AT = AL + 2·AB = 2πr · h + 2· πr2 = 2πr (h + r)

5.2. Cone

The cone is a solid of revolution generated by a right triangle that rotates around one of its legs.

Elements of a cone

Calculation of the generatrix of a cone

 By the Pythagorean Theorem, the slant height or generatrix (or slant height, s) of the cone is equal to:

Surface area of a cone

 The net of a cone is formed by:

  • Circular sector of length2πr · h, being r the radius of the base, and radius g the generatrix of the cone.
  • A circle that is the base

As in other cases we work out the surface area from its net. So it will be the sum of the lateral area and the area of the base:

  • Lateral area AL: It is the area of the circular sector whose length is 2πr and its radius is,g.

AL = πr · g

  • Area of the base AB: This is the area of the base with radius . 

AB = πr2

The area of a cone is:

AT = AL + AB = πr · g + · πr2 = πr (g + r)

You can get the net of  a cone and a cylinder: http://www.senteacher.org/wk/3dshape.php

5.3. Sphere

A spherical surface is the surface generated by rotating a circle about its diameter.

A sphere is the region of the space that is inside a spherical surface.

A sphere can also be seen as the region of the space obtained by rotating a semicircle around its diameter.

 Elements of a sphere

The centre is the interior point that is equidistant to all points on the surface of the sphere.

Radius is the distance from the centre to any point on the surface of the sphere, .

The diameter is the distance from one point through the centre to another point . A diameter is twice the radius.

Surface area of a sphere

The surface area of sphere of radius is :

A = 4 πr2

In the following links you will find exercises about all the 3-D shapes we studied:

Finally, the following video will show you how to find the areas of a cylinder, a cone and a sphere. Improve your listening!


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