Surface Area & Volume Prism & Cylinders

A polyhedron (3D shape) has faces that are polygons (2D shape) - Poly means many; hedron means faces Two faces meet at an edge Three or more edges meet at a vertex Vertex Edge

A net is a two-dimensional figure that, when folded, forms a three-dimensional figure This is an animation showing the net on the last page to assist with the idea that it forms a cube

Nets of a Cube This is the answer.

Prism: A solid object that has two identical ends (bases) and all flat sides Discuss the idea of bases also refer to the previous slide and discuss how in a rectangular prism you can technically use any pare of parallel sides as the bases. Typically people view the base as the pair that can be considered the top and the bottom of the rectangular prism. Triangular Prism

Prism Names Rectangular prism Triangular prism Height Height Base Base

Name the prism

Rectangular Prism Base 2 Bottom Back Top Base 1 Front Top Back Side 2 Height (H) Bottom In this slide you begin the development of what is surface area. The shape will be broken up identify what shapes make up the outer surface of this shape. Also discuss what the shape is. It will break up into its rectangular parts as it does discuss the opposite sides and their relationship. Base (B) Length (L)

Example: Rectangular Prism Net

Surface Area - the total area of all the faces (surfaces) of an object. To find the surface area of an object we can add up the areas of the separate faces. Note: some faces have the same size and shape Surface Area of this prism: Add the areas of the two brown sides (A) to the two green sides (D) and to the two red sides (C). C D A w h l This is a rectangular prism. The l, w and h have different values.

Surface Area Find the surface area of the rectangular prism We should use a table to tabulate the various areas. Face Area Number of Sides Total Area A 12cm2 2 24cm2 D 15cm2 30cm2 C 20cm2 40cm2 TOTAL 6 94cm2 C D A 3 cm 5 cm 4 cm Area of rectangle = l x w

Volume 5 cm B B 2 cm B h h h The number of cubic units needed to fill the space occupied by a solid. 3 cm VOLUME OF A PRISM The volume V of a prism is the area of its base B times its height h. V = Bh Note – the capital letter stands for the AREA of the BASE not the linear measurement. Volume = Base area x height = (l x w) x height = (3 cm x 2 cm) x height = 6 cm2 x 5 cm = 30 cm3 Discuss why this is a unit cube and the volume vormula Area of rectangle = l x w

Prism Volume Example V = Bh = (4ft x 3ft)h = (12 ft2)h Find area of the base 1st = (4ft x 3ft)h = (12 ft2)h Multiply it by the height = (12 ft2) x 8 ft = 96 ft3 3 ft B 4 ft 8 ft Area of rectangle = l x w

What is the surface area and volume of the prism? SA rectangle = l x w SA (rect1) = 12 cm x 10 cm = 120 cm2 x 2 = 240 cm2 SA (rect2) = 12 cm2 x 22 cm = 264 cm2 x 2 = 528 cm2 SA (rect3) = 10 cm x 22 cm2 = 220 cm2 x 2 = 440 cm2 Total SA = SA (rect1) + SA (rect2) + SA(rect3) = 240 cm2 + 528 cm2 + 440 cm2 = 1208 cm2 10 cm B 12 cm 22 cm V = Bh = (12 cm x 10 cm) x h = (120 cm2) x 22 cm = 2640 cm3 Note Units

Example Nets For Triangular Prisms

Surface Area of a triangular prism 2 area of same triangle + 2 areas of 3 rectangles, which 2 are the same 15ft Area Triangle 1 & 2 = 𝑏ℎ 2 = 12 𝑓𝑡 𝑥 15 𝑓𝑡 2 = 180 ft2 2 = 90 ft2 Area Rect. 1 = l x w = 12 ft x 25 ft = 300 ft2 Area Rect. 2 & 3 = 25 ft x 20 ft = 500 ft2 SA=triangle1 + triangle2 + rectangle1 + rectangle2 + rectangle3 SA=90 + 90 + 300 + 500 + 500 SA = 1480 ft2

Volume of a triangular prism Area Triangles = 𝒃𝒉 𝟐 = 𝟏𝟐 𝒙 𝟏𝟓 𝟐 = 𝟏𝟖𝟎 𝟐 = 90 ft2 V= Bh Find area of the base = (90 ft2)h Multiply it by the height = (90 ft2) x 25 ft = 2250 ft3 15ft

Classwork Page 186-188 #4, 5, 6a, 7a Page 192-193 #4,6,8

Cylinder Bases Cylinder Discuss how prisms and cylinders have similar ways of looking at the base they are congruent parrallel and there are 2. Cylinder

SA of a Cylinder Formula for Area of Circle A=  r2 = 3.14 x 32 = 3.14 x 9 = 28.26 But there are 2 of them so 28.26 x 2 = 56.52 units squared Find the circumference to determine the length of the rectangle C =  x d = 3.14 x 6 (radius doubled) = 18.84 Now use that as your base. A = b x h = 18.84 x 6 (the height given) = 113.04 units squared Now add the area of the circles and the area of the rectangle together. 56.52 + 113.04 = 169.56 units squared The total Surface Area!

Volume of a Cylinder = (28.26)h = 169.56 unit3 V= Bh Formula for Area of Circle A=  r2 = 3.14 x 32 = 3.14 x 9 = 28.26 unit2 V= Bh Find area of the base = (28.26)h Multiply it by the height = (28.26) x 6 = 169.56 unit3

Calculate the SA and Volume of the cylinder Be sure you know the difference between a radius and a diameter! Formula for Area of Circle A=  r2 = 3.14 x 52 = 3.14 x 25 = 78.54 But there are 2 of them so 78.54 x 2 = 157.075 units squared V = Bh C =  x d = 3.1415 x 10 (radius doubled) = 31.15 Now use that as your base. A = b x h = 31.15 x 4.2 (the height given) = 130.83 units squared The radius of the cylinder is 5 m, and the height is 4.2 m; therefore B = A =  r2 = 3.14 · 52 = 78.54 m2 V = Bh V = 78.54 · 4.2 V = 329.9 m3 SA = 157.075 + 130.83 = 287.905 m2

Classwork Page 211-212 #4,8,9