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whth52 CHapter 9 link = = =7.1 Surface area and Volume=

Surface area formula's
S = 2//L//w + 2wh + 2//L//h is for a retangular prism S = 6s² is for a [|cube] where S is the surface area, and s is the side lengths. This is a example of a rectangular prism.

Volume Formula's
V = //L//wh is for a rectangular prism, in which //L// is the length, w is the width, and h is the height. V = s³ is for a cube, where s is the sides.

Examples ofthe surface area and volume formula's.
Find the surface area and volume of a rectangular prism with the given dimentions. //L//= 5 w= 6 h=10

Find the surface area and volume of a cube with the given dimention. s= 20 (s is side lengths)

=7.2 Surface area and volume of prisms=

Altitude- a perpendicular line or segment or a polygon that is perpendicular to the vertex and the opposite base of the vertex. Height- is the Altitude's length of a polygon.

Surface area of a right prism
S= L+2B or S= hp + 2B Where S is the surface area, h is the height, B is the base area, and p is the perimeter of the [|prism].

Cavalieri's Principle
As long as the the surface areas and volumes of each penny and the cross sections and the bases are the same in both stacks of pennies, then even if one or both of the stacks is oblique they have the same volumes.

Volume of a prism
V = Bh where V is the volume, B is the base area, and h is the height.

=7.3 Surface area and volume of a pyramid= Pyramid- is a ployherdron that has a vertex that all of its faces (except the base) meet at the vertex. Base- a polygonal face that is an opposite face of a vertex of a pryamid. Lateral face- all the faces on a pyramid that are not the base or bases. Vertex of the pyramid- point where the lateral faces meet to form a pryamid. Base edge- the edge that each lateral face has, that is connected to the base of that pyramid. Lateral edge- the edges that connect that lateral faces in a pyramid. Altitude- a perpendicular line or segment or a polygon that is perpendicular to the vertex and the opposite base of the vertex. Height- is the Altitude's length of a polygon. Regular Pyramid- any pyramid that has a regular polygonal base, and the lateral edges are all equal/congruent isoloceles triangles. Oblique pyramid- any pyramid that does not have a regular polygonal base. Slant height- a lateral face's altitiude length

Surface area of a regular pyramid
S= L+2B or S= ½//L//p + B

Volume of a pyramid
V= 1/3 Bh where V is the volume, B is the base area, and h is the height.

=7.4 Surface area and volume of cylinders= Cylinder- an object that has 2 circular bases that are translated images of each other. Lateral surface- a cylinder's curved surface. Bases- the two circular regions of a cylinder. Altitude- a perpendicular line or segment or a polygon that is perpendicular to the vertex and the opposite base of the vertex. Height- is the Altitude's length of a polygon. Axis- a segment or point that connects the centers of the bases of a cylinder. Right Cylinder- a cylinder with an axis that is perpendicular to its base. Oblique Cylinder- any cylinder that does not have an axis that is perpendicular to its base.

Surface area of a Right Cylinder
S= L+2B or S= 2π rh+2π r² where L is the lateral surface, B is the base area, r is the radius, and h is the height.

Volume of a Cylinder
V= Bh or V= π r²h where V is the volume, B is the base area, r is the radius, and h is the height.

=7.5 Surface area and volume of cones= Cone- a 3-D shape that has a circular base and has a vertex opposite the base, and has a curved lateral surface. Base- the circular section on the bottom of the cone. Lateral surface- a cones curved surface. Altitude- a perpendicular line or segment or a polygon that is perpendicular to the vertex and the opposite base of the vertex. Height- is the Altitude's length of a polygon. Right cone- any cone that has an altitiude that interests at the center of its base. Oblique cone- any cone that does not have an altitiude that interests the base at its center. Slant height of a cone- the radius of a sector of any cone.

Surface area of a right cone
S= L+B or S= πr//L//+πr² where S is the surface area, L is the lateral area, B is the base area, r is the radius, and //L// is the slant height.

Volume of a cone
V= 1/3 Bh or V= 1/3πr²h where V is the volume, B is the base area, r is the radius, h is the height.

=7.6 Surface area and Volume of spheres= Sphere- any set of points that are equal in distance from the given point known as the center of the sphere. Annulus- is a circular or ring like structure, that is the region of space between two circles in a single plane that have the same center but have different radii.

Volume of a Sphere
V=4/3πr

Surface area of a sphere
S= 4πr²

Examples of surface area and volume of a sphere.
Find the volume of a sphere with the given radius. Radius = 60

Find the surface area of a sphere with a radius of 37.

More Examples
1. A triangluar prism has a height of 50. The triangular bases have side lengths of 5, 12, and 13. Find its volume. V= BH, V= (½ bh)H, V= (½ 5×12)50, V= 30×50, V= 1500 units³ 2. A triangular prism has a height of 77. The triangular bases have side lengths of 3, 4 and 5. It has a perimeter of 90. Find its surface area. SA= hp + 2B, SA= (77×90) + 2(½ 3×4), SA= 6930 + 12, SA= 6942 units² 3. A pyramid has a base edge length of 35 and its base is a square. Its height is 25. Find its volume. V= 1/3 Bh, V= 1/3 (35×35)(25), V= 1/3 (1225)(25), V= 1/3 (30625), V= 10208.333 units³ 4.A pryamid has a square base with a base perimeter of 40 and base edges of 10. Its slant height, //L//, is 60. Find its surface area. SA= ½//L//p + B, SA= ½ (60×40) + (10×10), SA= ½ (2400)(100), SA= 120,000 units² 5. A cylinder has a radius of 6 and a height of 10. Find its volume. V= π r²h, V= π (6)²(10), V= π (36)(10), V= 360π units³ 6.A cylinder has a radius of 60 and a height of 110. Find its surface area. SA= 2π rh+2π r², SA= 2π (60)(110) + 2π (60)², SA= 13200π + 7200π, SA= 20400π units² 7. A cone has a radius of 500 and a height of 70.Find its volume. V= 1/3πr²h, V= 1/3π(500)²(70), V=1/3π(250000)(70), V= 1/3π(17500000), V= 5833333.333π units³ 8.A cone has a radius of 4000 and a slant height of 500. Find its surface area. S= πr//L//+πr², S=π(4000)(500)+π(4000)², S= π(2000000)+π(16000000), S= 18000000π units² 9.V S 10.SA S