## 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 A**It is the area of the rectangle whose length is the length of the circumference of the bases with radius_{L}:**r**,**2πr**and its width is the height of the cylinder,**h**.

**A _{L}** =

**2πr · h**

**Area of the bases A**This is the sum of the areas of the bases with radius ._{B}:

**A _{B = }πr^{2}**

The **surface** **area of a cylinder **is:

**A _{T} = A_{L} + 2·A_{B} = 2πr**

**· h + 2· πr**

^{2}= 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 length
**2πr · 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 A**It is the area of the circular sector whose length is_{L}:**2πr**and its radius is,**g**.

**A _{L}** =

**πr · g**

**Area of the base A**This is the area of the base with radius ._{B}:

**A _{B = }πr^{2}**

The **area of a cone **is:

**A _{T} = A_{L} + A_{B} = πr**

**· g + · πr**

^{2}= 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.

The **surface area of sphere** of radius is :

**A =** **4 πr ^{2}**

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

- Names and parts of 3-dimensional figures
- Surface area of prisms and cylinders
- Surface area of pyramids and cones
- Volume and surface area of spheres (You will also find the volume of a sphere we will study in the next lesson)

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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