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=The Surface Area And Volume of Solid Figures=

The Surface Area and volume of solid figures, Maximizing and Minimizing the Surface Area of Solid Figures
To find the surface area and volume of a right rectangular prism, you use these equations: For surface area:

S=2lw+2lh+2wh
(S is surface area, l is length, h is height, and w is width.)

For volume:

V=lwh
(V is volume, l is length, h is height, and w is width)

How about we try an example. If you were making a box to fit in the corner of your room, to put all of your books or toys, and the demensions you want are 3 ft in length, a width of two feet and a height of two feet. To find to volume of your box and how many toys you could fit in, you use the equation for the volume of a rectangular prism, or V=lwh. You multiply 3 by 2 by 2 and get an answer of 12 ft³.

To find the surface area of a cube with sides:

S=6s²
(S is surface area, s is sides)

Here is an interesting example of a rectangular prism that you may have seen before.

These links will help your understanding of the concepts
[|Surface area of cubes and right rectangular prisms]

Surface Area and Volume of Prisms
Altitude of Prisms - A segment that has both of the endpoints in the base planes of a prism. It is perpendicular to both planes.\

The height of the prism is just the length of the altitude.

Surface area of a right prism, you can use two equations : or They are same exact equation, the first one already has a step done for you. hp is the same as L. When you multiply h and p, you are getting the lateral area, or L.
 * S = L + 2B**
 * S = hp+ 2B**


 * Cavalieri's Principle** - If there are two solid figures that are intersected by a plane, parallel to the bases, and both of the cross sections that are formed have equal areas, then the solids have equal volumes. ([|Who is this Cavalieri guy?] click the link for more info)


 * Volume of a Prism

V=Bh**

B is the base area and is the height. To find the volume you multiply the base area and the height.

Pyramid -
A polyhedron with a polygon **base** and three or more triangular lateral **faces**. The triangles all share one vertex, **called the vertex of the pryramid**. Each triangle also has one edge in common with the base, called **the base edge**. An lateral face's edge that is shared with another lateral face is called a **lateral edge**.

Altitude of a Pyramid
The perpendicular segment from the base plane to the vertex of the pyramid. The length of the altitude is the **height** of the pyramid.

A Regular Pyramid -
A pyramid with a regular polygon for a base with congruent isosceles triangles as lateral faces. The altitude of the regular pyramid intersects the base plane in the middle and the lateral faces are all the same. The length of the altitude of the regular prism is called the **slant height**.

The Surface Area of a Regular Pyramid -
There are two equations you can use : or S is the surface area, L is the lateral area of the pyramid, B is the base area, p is the perimeter and the slant height is l.
 * S = L+B**
 * S = ½lp+B**

So How Do You Find the **Volume of a Pyramid?** V is volume, B is base area, and h is height. Still don't get it? [|Try this link]
 * V = 1/3Bh**

Surface Area and Volume of Cylinders
So, **What is a Cylinder**? A Cylinder is a solid with a **lateral surface** that connects a ircular region with it's translation. The two circular regions are called the **bases** of the figure.


 * The Altitude of a Cylinder** - Much like other solid figures previosly mentioned, the altitude of a cylinder **is a segment with it's endpoints contianed on the base planes**. The segment **is also perpendicular** to the planes that hold the bases. The length of the altitude is the **height** of the prism.

If the Axis of the prism is perpendicular to the bases, that means that the cylinder is a **right cylinder**. If if the axis is not perpendicular, that means that the cyinder is an **oblique cylinder**.
 * The Axis of a Cylinder** - A segment that connects the two bases at their centers.

Now that you have all the parts down, let's learn the math that comes with it.

or The first equation represents the same exact things as the second one, but it uses letters instead of words. L = 2πrh (The Lateral Area) and B = 2πr² (The Base Area).
 * Surface Area of a Right Cylinder** - There are again two equations you can use:
 * S = L + 2B**
 * S = 2πrh + 2πr²**


 * Volume of a Cylinder** - You may use two equations:

or B is the base area, r is the radius, h is the height, when you multiply them together, you get the volume, or V.
 * V = Bh**
 * V = πr²h**

Cones, Their Surface Area and Volume
First, **What is a cone and what are it's properties?**

A cone is a **solid three-dimensional figure** that consists of a **circular base**, much like a cylinder, but instead of a lateral face connecting two bases, the **lateral area is curved** and there is **only one base**.

The lateral area connects the base to a point called the vertex of the cone.The altitude of the cone is a perpendicular segment that connects the vertex of the cone and the base plane, it is also perpendicular to that plane. To find the height of the cone, take the length of the altitude.


 * Right vs. oblique cones.**

In a **right** cone, the altitude will intersect the base at it's center. If it intersects the plane anywhere else the cone is **oblique**.

Now the most exciting part, The equations!


 * The Surface Area of Right Cones**

or
 * S = L + B**
 * S = πrl + πr²**

or V= **1/3πr²h**
 * Volume of a Cone**
 * V= 1/3Bh**

Volume equals one third of the base area times the height. So lets say you have a cone with a radius of 72 and a height of 56. The base area of the cone is multiplied by the height and then divided by three. if you followed the equation correctly, you should have come out with a volume of 304005.62.

L is the lateral area or πrl and B is base area, or πr², the area of the circular base. This link will help you understand and master the surface area and volume, Click [|here]

Surface Area and Volume of Spheres


What is a **Sphere**?

- A set of all points in a space which share a **common distance** to one point, the **center of the sphere**

When a plane cuts a cylinder and a sphere, you can prove that the cricular region of the sphere and the circular region of the cylinder (called the **annulus**) are both equal.

R is the radius of the sphere and v is the volume
 * Volume of a Sphere -**
 * V=4/3πr³**

4πr²
 * Surface area of a Sphere-**