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Aim: The Third Dimension: Volume – What is it?

Aim: The Third Dimension: Volume – What is it?. r = 1/2 s. Do Now: What is the area, in terms of p, of the shaded region in the figure below?. A. of sq. = s 2. s. A. of circle = p r 2. of shaded region = A. of sq. - A. of circle. A. of s. r. = s 2 - p ( 1/2s) 2.

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Aim: The Third Dimension: Volume – What is it?

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  1. Aim: The Third Dimension: Volume – What is it? r = 1/2 s Do Now: What is the area, in terms of p, of the shaded region in the figure below? A. of sq. = s2 s A. of circle = pr2 • of shaded region = A. of sq. - A. of circle A. of s. r. = s2 - p (1/2s)2 = s2 - p (1/4 s2) = s2 - 1/4s2 p

  2. 2 1 3 4 1 1 1 1 1 1 1 1 1 sq. un. 1 sq. un. 1 sq. un. 1 sq. un. 1 sq. un. 1 sq. un. Perimeter & Area Perimeter - the distance around a polygon P = 1 + 2 + 3 + 4 Area - the space inside a polygon - measured in square units units2, in2, miles2, etc.

  3. 4 1 1 1 2 1 cu. unit 1 3 3 3 Volume Volume - the measure of the space inside a polyhedron -measured in cubic units. 8 cubic units. V = Bh, where B = the area of the base 27cubic units. Polyhedron - three dimensional figure whose surfaces are polygons Edge – a segment that is the intersection of two faces Vertex – point where edges meet

  4. Volume Formulas Prism – a polyhedron with two congruent, parallel bases. The other faces are lateral faces. Prism B - area of the base h w l b Rectangular B = l • w Triangular B = 1/2 bh Rectagular prism – V = l  w  h A triangular prism is a solid whose base is a triangle.

  5. Prism h w l Rectangular B = l • w Model Problems 10 Find the Volume of the following polyhedron: 13 17 V = l • w • h = 17 x 13 x 10 = 2210 units3

  6. Prism h 20 w l Rectangular B = l • w 3 10 Model Problem What is the volume of a rectangular box whose base measures 3 units by 10 units and whose height is 20 units? Rectangular prism – V = l  w  h V = 10  3  20 = 600 units3

  7. Model Problem The volume of a rectangular box is 2,400 cu. units. The measurements of the base are 60 x 20. What is the height of the box? h = 2400 units2 20 w 60 l 2400 = 60 x 20 x h 2400 = 1200 x h 2 = h

  8. B - area of the base 30 b Triangular B = 1/2 bh Model Problem The base of a triangular prism has an area of 15 cm2 and a height of 30 cm. What is the volume of the triangular prism? 15 cm2 Triangular prism – V = B  h V = 15cm2 30 = 450 units3

  9. Cube e e e 10 10 10 Volume Formula Prism – a polyhedron with two congruent, parallel bases. The other faces are lateral faces. What is the volume of a cube with a side of 10 units?

  10. Cylinder Sphere r Volume Formulas

  11. Cylinder Model Problems A cylindrical tank has a radius of 10 ft. and a height of 25 ft. What is its volume? Give your answer in terms of  10 ft 25 ft

  12. Pyramid h B = area of base of pyramid Volume Formulas Pyramid – 3 dimensional figure with a single base and sides that are triangles. Triangular pyramid Pentagonal pyramid Rectangular pyramid

  13. 9 Find the volume of the pyramid with a height of 9. 10 15 Model Problems B = l • w

  14. h = 10 r = 5 Volume Formula Cone Cone – a pyramid with a circle for a base What is the volume of a cone whose height is 10 units and whose base has radius 5? Express in terms of 

  15. Model Problems The main tank at the Living Seas Aquarium at EPCOT Center in Florida is the largest enclosed tank in the world. It is a cylinder with diameter 203 ft. and height 25 ft. About how many million gallons of water does this tank hold? (1 gal. = 231 in3; 1728 in3 = 1 ft3) 203 ft. 25 ft. Radius is 1/2 the diameter r = 203  2 = 101.5 ft. h = 25’

  16. Model Problems The Great Pyramid at Giza, Egypt, was built about 2580 B.C. as a final resting place for Pharoah Khufu. At the time it was built, its height was about 481 ft. Each edge of the square base was 756 feet long. A. Find the surface area of the Great Pyramid, including its base. Height 481 ft. 756’ 756 ft B. Find the volume of the Great Pyramid.

  17. Model Problems Zia is planning to landscape her backyard. The yard is a 70ft-by-60ft rectangle. She plans to put down a 4- in. layer of topsoil. She can buy bags of topsoil at $2.50 per 3-ft3 bag, with free delivery. Or she can buy bulk topsoil for $25.00 per yd3, plus a $20 delivery fee. Which option is less expensive. Show your calculations and explanation. (1 yd3 = 9 ft3) 4” 60’ 70 feet

  18. Model Problems A cylinder has be cut out of the figure below. Find the volume of the remaining figure. Round your answer to the nearest tenth. 4” 6 in. 6 in. 6 in.

  19. Volume Formulas

  20. Volume Formulas

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