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= = = = =This page will discuss Surface Area and Volume for Solid Figures.=

Link to [|Volume of Cones and Pyramids]



**__Surface Area and Volume__**
Objectives: -**//Exploring ratios of surface area to volume//** -**//Developing the concepts of maximizing volume and minimizing surface area//.**

__The Formulas for the Surface Area and Volume of a solid figure.__ S = Surface Area V = Volume L = Length W = Width H = Height s = Side

//-Retangular Prism// The formula for the **surface area** of a right retangular prism is: S = 2LW + 2WH + 2LH The formula for the **volume** of a right retangular prism is: V = LWH -//Cube// The formula for the **surface area** of a cube is: S = 6s² The formula for the **volume** of a cube is: V = s³

Two cereal boxes: Box A and Box B Find which cereal box has the greater surface area.
 * +Example** (surface area of a retangular prism)

__Dimensions: Box A__ Height = 8 inches. Width 1 = 4 inches. Width 2 = 5 inches. Apply the formula: S = 2LW + 2WH + 2LH So... Box A = 2(8)(5) + 2(4)(5) + 2(4)(8) = Total = 184 Surface area of Box A = 184

__Dimensions: Box B__ Height = 10 inches. Width 1 = 2 inches. Width 2 = 8 inches. Apply the formula: S = 2LW + 2WH + 2LH So... Box B = 2(10)(8) + 2(2)(8) + 2(2)(10) = Total = 232 Surface area of Box B = 232

Box B has the larger Surface area.
 * So...**

__Surface Area and Vol. of Prisms__


Objectives: - //**Define and use a formula for finding the surface area of a right prism.**// - //**Define and use a formula for finding the volume of a right prism.**// - //**Use Caalieri's Principle to develop a formula for the volume of a right oblique prism.**//

__Altitude Of a Prism:__ A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes. Example-
 * Terms**

__Height of a Prism:__ The length of an altitude. Example-


 * Right Prisms (Surface Area)**

2 parts:

1st: __Area of the Bases (Base Area)__ -The Bases of a right prism are congruent therefore the base area is twice that of one base or 2B where B = Base.

2nd: __Area of the Lateral faces (Lateral Area)__ -If given the sides (s1, s2, and s3) and height (H) of the base of a prism net then the lateral area is given by the formula: L (lateral Area) = s1H + s2H + s3 = H ( s1 + s2 + s3)
 * Because s1 + s2 + s3 is the perimeter of the base you can write L = HxP (or HP) where P is Perimeter.

The surface area (S) of a right prism with a Lateral Area (L), a base area (B), a perimeter (P), and height (H) is: S = L + 2B **or** S = HP + 2B

__Surface Area and Volume of Pyramids__
Objectives: - **//To use a formula for the surface area of a regular pyramid.//** - //**To use a formula for the volume of a pyramid.**//

__Pyramid:__ A polyhedron that consists of a base and 3 or more **lateral faces**. The lateral faces of a pyramid are triangles that come together in a single point or vertex called the **vertex of the pyramid**. Each of these lateral faces has one edge that it shares with the base. This is called the **base edge**. An edge in which two lateral faces meet is called a **Lateral Edge**.
 * Terms**

__Altitude of a Pyramid:__ The altitude of a pyramid is the distance from the vertex to the base, often shown as a perpendicular line from that vertex to the plane of the base. The length of this altitude is the **height** of the pyramid.

__Regular Pyramid:__ Has a regular polygon base and lateral faces that are congruent isosceles triangles. In this particular pyramid all its lateral faces are congruent and the altitude runs from the vertex of the pyramid to the center of the base plane.

__Slant Height:__ The altitude of a lateral face is called the slant height of a pyramid.

//Pyramids are named by their base so... A pyramid with a triangular base is called a triangular pyramid. One with a rectangular base is called a rectangular pyramid. etc...

Example - Finding the surface area of a regular square pyramid.// Slant Height: l Lateral Area: L Base Area: B Base edge length: s Given area of each Triangle: ½sl The surface area is the sum of the lateral areas and the base area. S = L + B Because 4s is the perimeter of the base: Using: S = L + B The Base area is equal to s² because the area of a square (square pyramid) is the height multiplied by the base. Since every side is the same it is the same variable s. So... The equation is written: S = 4( ½sl ) + s²

The surface area S, of a regular pyramid with lateral area L, base area B, perimeter of the base p, and the slant height l, is S = L + B or S = ½ l p + B
 * __Surface Area of a Regular Pyramid:__**

//Example - Finding the surface area of a regular octagonal pyramid.// Base Edge: 7 feet. Slant Height: 3 feet. -Find the area of the octagonal pyramid. -Find how much it will cost to cover the entire pyramid if paper costs $4.00 per square foot. Remember there are 8 sides to an octagon. Equation.. L = ½lp = ½ ( 3 ) ( **8** x 7) = 84 square feet. Then... 84 square feet x $4.00 = $336.00

Equation V = ¹/3 Bh
 * __Volume of a Pyramid__**

//Example - finding the volume of a regular square pyramid//

A rectagular pyramid that has a base edge of 40 in. and a height of 14 in. so.. V = ¹/3 (40) (14) V = 186.6 is approx. the volume of the pyramid.

= =

__Volume and surface area of Cylinders__
//Objectives - **Using the formula for the surface area of a right cylinder.** - **Using a formula for the volume of a cylinder.**//

__Cylinder__: A solid that consists of a circular region and its translated image on a parallel plane, with a **lateral surface** connecting the circles. __Bases__: The bases of a cylinder are the two circular regios; one top, one bottom. __Altitude__: The altitude of a cylinder is a segment that has endpoints in the planes containing the bases and is perpendicular to both planes. __Height__: The height of a cylinder is the length between the two circular bases or regions (as mentioned before). __Axis__: The axis of a cylinder is the segment that joins the center of the two bases. If the axis of the cylinder is perpendicular to the bases then the cylinder is a **right cylinder**. If the axis is not perpendicular to the bases of the cylinder then it is an **oblique cylinder**.
 * Terms**


 * __Surface areas and volumes of a right cylinders__**

-The surface area, or S, of a right cylinder with a lateral area, L, base area B, radius r, and height, h, is determined by the equation: (note: pi = 3.14) S = L + 2B or S + 2(pi)rh + 2(pi)r²

-The volume, or V, of a right cylinder with a radius, r, height, h, and base area, B, is determined by the equation: V = Bh or V = (pi)r²h

__Cones: Surface area and volume__
//Objectives - **Defining and useing the formula for the surface area of a cone.** - **To defining and use the formula for the volume of a cone.**//

__Cones__: Three dimensional figures that consist of a base, lateral surface, and a vertex. --//__Base__//: Bottom circular surface of the cone. --//__Lateral Surface__//: connects the base to a single point not on the plane of the base. --//__Vertex__//: What connects the base to a single point not located on the plane of the base.
 * Terms**

(note: pi = 3.14) S = L + B or S (pi)rl + (pi)r²
 * Surface area of a right cone**: The surface area, S, with a lateral area L, base of area, B, radius r, and slant height l is dertemined by the equation(s):

V = ¹/3Bh or V = ¹/3(pi)r²h.
 * Volume of a right cone**: The volume, V, of a cone with a radius r, height h, and base area B, is determined by the equation(s):

__Spheres: Surface area and Volume__
//Objectives// - **//Defining and using the formula for the surface area of a sphere.//** - **//Defining and using the formula for the volume of a sphere.//** __Sphere__: The set of all points in space that are the same distance, r, from a given point also known as the center of the sphere.
 * Terms**

__**Volume of a sphere:**__ The volume of a sphere with a radius r is determined by the equation: V = 4/ 3(pi)r³