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Surface Area and Volume Formulas
(S:Surface area, L:length, w: width, h:height)
 * Regular prism:
 * Surface Area: S = 2Lw+2wh+2Lh
 * Volume: V = Lwh

(V: volume, L= length, w: width, h:height

Ex: A box has a height of 10, width of 2 and a length of 8. Find the surface area and Volume: S=2(8*2) + 2(2*10) + 2(8*10) S=32 + 40 + 160 S=232 units^2 (s= side width)
 * Cube:
 * Surface Area:S = 6s^2
 * Volume: V = s^



Fun Site: [|Interactive Surface Area]

Words to Know:**
 * __7.2 Surface Area and Volume of Prisms__
 * Altitude: Both end points are in the top and bottom bases of the prism.
 * Height: The length of the altitude of the prism.

(L:Lateral, B:Base, p:Perimeter, h:Height)
 * Surface Area of a Right Prism**
 * Surface Area: L+2B or hp+2B


 * Cavalieri's Principle**
 * If there are two solids with equal heights and a parallel plane cross section is formed, then the than the solids have equal volumes.

(V:volume, B:base area, h:height)
 * Volume of a Prism**
 * V=Bh

EX: What is the surface area of an Octagonal Prism with a height of 48in, a base length of 14in and an apotheom of 7in? B=1/2 ap= 1/2(14)(7)= 1/2(98)= 49 square inches. S= hp + 2B= 48(14*8) + 2(49)= 5474 square inches.

Base of the Octagonal Prism: Pyramid: A polyhedron made up of a base and three or more lateral faces. Base: A base is a polygon. Lateral Faces: Triangles sharing the same vertex. Vertex of the Pyramid: The very tip of the pyramid which is shared by all the lateral faces. Base Edge:The Lateral faces edge that is shared with the base. Lateral Edge: Two lateral faces that share the same edge. Altitude: The segment that is perpendicular from the vertex of the pyramid to the base plane. Height: Length of the altitude of the pyramid. Regular Pyramid: Has congruent isosceles triangles as faces. Slant Height:The latitude length of a lateral face of a regular pyramid.
 * __7.3 Surface Area Volumes of Pyramids__**
 * Words to know:**

(L:lateral base, B: base area, l: [|slant height], p: perimeter of the base)
 * Regular** **Pyramid**
 * Surface Area: L+B or 1/2 lp+B


 * Volume: V=1/3Bh

__**7.4 Surface Area and Volume of Cylinders Right Cylinder**__
 * Words to Know:**
 * Cylinder: A Prism with two circular bases connecting the lateral surface.
 * Lateral Surface: The area of a cylinder the is not a base.
 * Bases: The circular translated faces.
 * Altitude: The segment that connects the bases perpendicular to both planes.
 * Height:The length of the altitude.
 * Axis: A segment theat connects the two centers of the bases.
 * Right Cylinder: The axis of the cylinder must be perpendicular to both bases.
 * Oblique Cylinder: If the axis is NOT perpendicular to both bases.

(π:pi 3.14.....,r: radius)
 * Right Cylinder**
 * Surface Area: S=L+2B or S= π r h+2 π ^2.


 * Volume: V=Bh or V=πr^2

__**[|]

7.5 Surface Area and Volume of Cones**__
 * Words to Know:**
 * Cone: 3-D figure that has a circular base, a curved lateral surface, and a vertex
 * Base: Circular
 * Lateral Surface: Curved.
 * Vertex: Connects a single point to the base, other than in the plane of the base
 * Altitude: Perpendicular segment that connects the vertex to the plane of the base.
 * Height: The height of the altitude.
 * Right Cone: When the altitude intersects the center of the base of the cone.
 * Oblique Cone: When the altitude does NOT intersect the center of the base.


 * Cones**
 * Surface Area: S=L+B or S=π r l+π r^2
 * Volume: V=1/3Bh or V=1/3 π r^2

__**

7.6 Surface Area and Volume of Spheres:**__
 * Words to know:**
 * Sphere: When all the points in a space are the same distance from the radius, a given point in the center of the sphere.
 * Annulus: A ring-shaped figure in a cylinder.


 * Sphere**
 * Surface Area: S=4πr^2
 * Volume: V=4/3πr^3



More Help: [|Surface area] [|Help site] [|surface Area help] [|fun help video]

More Examples:

EX 1:Find the Volume of the following triangular prism: Base area= 1/2cm Height= 7cm

V= Bh V= -1/2*7 V= 3.5cm^3

EX 2: Find the surface area of a triangular prism with a lateral area of 16 units, and a base area of 12units:

S=L+2B S=16+(2*12) S= 40units^2

EX 3: Triangular Prism: Find the Volume of the following Pyramid:: Base area=5 inches height=3 inches Slant height= 4

Volume: V= 1/3 Bh V=1/3 5*3 V=1/3 (15) V= 5 inches ^3

EX 4:(Use Information from EX 3) Surface Area: S= 1/2*lp+B S=1/2*4p+5 -5 -5 -5=1/2*4p, -5(1/2)=1/2(1/2)*4p

EX 5: Find the volume of a cylinder with a diameter of 14cm and a height of 22cm: V= π r^2h r= 14/2, r= 7 V= π 7^2*2*22 V= π*2156 V= **6773.27376**cm^3

EX 6: Find the area of a cylinder with the same measurements as EX 5:

S=2π rh+2π r^2 S=2π7*22+2π7^2 S=2π154+2π49 S=967.61+307.88 S=1275.49cm^2

EX 7-8:A cone has a height of 10in, a radius of 6in and a slant height of 8:

7: Find the Volume: V=1/3π r^2h V=1/3π6^2*2*10 V=793.98in^3

8: Find the surface area: S=π rl+π r^2 S=π6*8+π6^2 S=π48+π36 S=263.89in^2

EX 9-10: A large sphere has a radius of 298mm:

9: Find the Volume: V=4/3π r^3 V=4/3π298^3 V=4/3π26,463,592 V=110,850,435mm^3

10: Find the surface area: S=4π r^2 S=4π298^2 S=4π 88,804 S=1,115,943.98mm^2