Prisms

Page 1 Prisms A prism has two parallel faces, called bases, that are congruent polygons. The lateral faces are rectangles in a right prism, or parallelograms in an oblique prism. Lateral faces

Volume of a Prism B = Area of the base h = the height of the prism Page 1 Volume equals BIG B times h! B = Area of the base h = the height of the prism    

Rectangular Prisms 𝑉=𝐵ℎ 𝑆𝑖𝑛𝑐𝑒 𝐵=𝑙𝑤 𝑉=(𝑙𝑤)ℎ 𝑉=(7)(5)(4) 𝑉=140 𝑉=𝑙𝑤ℎ Page 2 A rectangular prism has rectangular bases and lateral edges perpendicular to the bases. Lets find the volume and surface area of the prism shown! 𝑉=𝐵ℎ 𝑆𝑖𝑛𝑐𝑒 𝐵=𝑙𝑤 𝑉=(𝑙𝑤)ℎ 𝑉=(7)(5)(4) 𝑉=140 𝑉=𝑙𝑤ℎ Useful Formula: Volume of a Rectangular Prism!

Examples Page 3 𝐴𝐵𝐶𝐷≅𝐸𝐹𝐺𝐻 𝐴𝐵𝐹𝐸≅𝐶𝐺𝐻𝐷 𝐴𝐸𝐻𝐷≅𝐵𝐹𝐺𝐶

𝑉=𝐵ℎ 𝐹𝑖𝑛𝑑 𝐵 𝑓𝑖𝑟𝑠𝑡, 𝑡ℎ𝑒𝑛 𝑠𝑢𝑏 𝑖𝑛 −𝑜𝑟− 𝑉=( 1 2 𝑙𝑤)ℎ 𝑉= 1 2 10 10 3 𝑉=150 Page 4 𝑉=𝐵ℎ Since B = a triangle, use the area formula for a triangle. 𝑉=( 1 2 𝑙𝑤)ℎ 𝑉= 1 2 10 10 3 𝐹𝑖𝑛𝑑 𝐵 𝑓𝑖𝑟𝑠𝑡, 𝑡ℎ𝑒𝑛 𝑠𝑢𝑏 𝑖𝑛 𝑉=150 𝐵= 1 2 𝑏ℎ 𝑉=𝐵ℎ −𝑜𝑟− 𝑉=(50)(3) 𝐵= 1 2 (10)(10) 𝑉=150 𝐵=50

𝑉=𝐵ℎ 𝑉=𝑙𝑤ℎ 𝑡𝑟𝑖𝑝𝑙𝑒 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ= 3𝑙 𝑡𝑟𝑖𝑝𝑙𝑒 𝑡ℎ𝑒 𝑤𝑖𝑑𝑡ℎ= 3𝑤 Page 4 𝑉=𝐵ℎ Since B = a rectangle, use the area formula for a rectangle. 𝑉=𝑙𝑤ℎ 𝑡𝑟𝑖𝑝𝑙𝑒 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ= 3𝑙 𝑡𝑟𝑖𝑝𝑙𝑒 𝑡ℎ𝑒 𝑤𝑖𝑑𝑡ℎ= 3𝑤 𝑡𝑟𝑖𝑝𝑙𝑒 𝑡ℎ𝑒 ℎ𝑒𝑖𝑔ℎ𝑡= 3ℎ 𝑁𝐸𝑊 𝑉𝑂𝐿𝑈𝑀𝐸: 𝑉=3𝑙∙3𝑤∙3ℎ 𝑉=27∙𝑙𝑤ℎ

Page 4 𝑉= 𝑒 3 𝑉= 8 3 𝑉=512

𝑉𝑜𝑙𝑢𝑚𝑒 𝑆ℎ𝑜𝑒 𝐵𝑜𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑇𝑟𝑖. 𝑃𝑟𝑖𝑠𝑚 𝑉=𝐵ℎ 𝑉=48ℎ 𝑉= 48 6 288=48ℎ 𝑉=288 Page 4 𝑉𝑜𝑙𝑢𝑚𝑒 𝑆ℎ𝑜𝑒 𝐵𝑜𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 𝑇𝑟𝑖. 𝑃𝑟𝑖𝑠𝑚 𝑉=𝐵ℎ 𝑉=48ℎ 𝑉= 48 6 288=48ℎ 𝑉=288 288=48ℎ 6=ℎ

𝑉=𝐵ℎ 𝑉=( 1 2 𝑙𝑤)ℎ 48= 1 2 8 8 ℎ 48=32ℎ 1.5=ℎ Page 4 Since B = a triangle, use the area formula for a triangle. 𝑉=( 1 2 𝑙𝑤)ℎ 48= 1 2 8 8 ℎ 48=32ℎ 1.5=ℎ

𝑉=𝐵ℎ 𝑉=𝑙𝑤ℎ 𝑉= 𝑡+2 𝑡 3𝑡 𝑉= 𝑡+2 3 𝑡 2 𝑉=3 𝑡 3 +6 𝑡 2 Page 4 Since B = a rectangle, use the area formula for a rectangle. 𝑉=𝑙𝑤ℎ 𝑉= 𝑡+2 𝑡 3𝑡 𝑉= 𝑡+2 3 𝑡 2 𝑉=3 𝑡 3 +6 𝑡 2

Page 5

𝑉=𝐵ℎ 𝑉=𝑙𝑤ℎ 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ= 2𝑙 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡ℎ𝑒 𝑤𝑖𝑑𝑡ℎ= 2𝑤 Page 5 𝑉=𝐵ℎ Since B = a rectangle, use the area formula for a rectangle. 𝑉=𝑙𝑤ℎ 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ= 2𝑙 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡ℎ𝑒 𝑤𝑖𝑑𝑡ℎ= 2𝑤 𝑞𝑢𝑎𝑑𝑟𝑢𝑝𝑙𝑒 𝑡ℎ𝑒 ℎ𝑒𝑖𝑔ℎ𝑡= 4ℎ 𝑁𝑒𝑤 𝑉𝑜𝑙𝑢𝑚𝑒: 𝑉=2𝑙∙2𝑤∙4ℎ 𝑉=16𝑙𝑤ℎ

Page 5 𝑉=𝐵ℎ Since B = a rectangle, use the area formula for a rectangle. 𝑉=𝑙𝑤ℎ 𝑉= 6 3 4 𝑉=72

Page 6 𝑉=𝐵ℎ 𝑉=𝑙𝑤ℎ 𝑉= 4 6 12 Since B = a rectangle, use the area formula for a rectangle. 𝑉=288

Page 6 𝑉= 𝑒 3 𝑉= 𝑒 3 8= 𝑒 3 216= 𝑒 3 3 8 = 3 𝑒 3 3 216 = 3 𝑒 3 2=𝑒 6=𝑒

Page 6 𝑉=𝐵ℎ 42=𝐵∙3 14=𝐵

Page 6 𝑉= 1 2 𝑑 1 ∙ 𝑑 2 ∙ℎ 𝑉=𝐵ℎ 𝑉= 1 2 12 16 20 Since B = a rhombus, use the area formula for a rhombus that utilizes the diagonals. 𝑉=1920 𝐵 𝑟ℎ𝑜𝑚𝑏𝑢𝑠 = 1 2 𝑑 1 ∙ 𝑑 2

Page 6 Since B = a rectangle, use the area formula for a rectangle. 𝑉=𝑙𝑤ℎ 51840= 72 24 ℎ 8∙6480=51840 51840=1728ℎ 24 30=ℎ 72 𝑉=𝐵ℎ Minimum height is 30 inches

Homework Page 6 #2,4,5c,d,e,7,9,19

Page 6 𝑉= 𝑒 3 𝑉= 25 3 𝑉=15625

Page 6 𝑉=𝐵ℎ 𝑉= 𝑠 2 ℎ 𝑉= 7 2 ∙3 Since B = a square, use the area formula for a square. 𝑉=147

Page 6 𝑉= 𝑒 3 𝑉= 𝑒 3 𝑉= 𝑒 3 64= 𝑒 3 125= 𝑒 3 27= 𝑒 3 3 64 = 3 𝑒 3 3 125 = 3 𝑒 3 3 27 = 3 𝑒 3 4=𝑒 5=𝑒 3=𝑒

Page 6 𝑉=𝐵ℎ 9=18ℎ 9 18 = 18 18 ℎ 1 2 =ℎ

𝑉=𝐵ℎ 𝑉= 1 2 𝑑 1 ∙ 𝑑 2 ∙ℎ 60= 1 2 10∙12 ℎ 60=60ℎ 1=ℎ Page 6 𝑉= 1 2 𝑑 1 ∙ 𝑑 2 ∙ℎ 𝑉=𝐵ℎ 60= 1 2 10∙12 ℎ Since B = a rhombus, use the area formula for a rhombus that utilizes the diagonals. 60=60ℎ 1=ℎ

Page 6 𝑉=𝐵ℎ 𝑉=10𝐵ℎ The volume would be 10 times the original volume.