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Grade 9:  The Learning Equation Math

31.06 Volume

3icon.gif (4368 bytes)

Measurement

Refresher pp 68-9

 

Rectangular Prism Investigation

Cut a net for a rectangular prism from grid paper using sugar cubes to measure the dimensions:

4 units x 3 units x 2 units

Put the net together to form the rectangular prism, then use sugar cubes to build the same size structure.   What is the volume of both structures.

volume, area, base, height, pyramid, prism, cone, cylinder, radius, diameter, congruent

             

   

Learning Outcomes: 

The student will:

 

 

Geometric Shape

Surface Area

B = area of the base

P = perimeter of the base

Volume

B = area of the base

P = perimeter of the base

prism (general)

  • B = area of base
  • P = perimeter of base
  • h = height

 

   SA = 2B + Ph V = Bh

triangular prism

  • a = altitude
  • b = base
  • h = #height

 

   SA = 2B + Ph

SA = 2(1/2ab) + (b + c + d)h

SA = ab + (b + c + d)h

triangular_prism_sml.png (33744 bytes)

 V = Bh

V = 1/2 abh

triangular_prism.gif (1150 bytes)

rectangular prism

  • l = length
  • w = width
  • h = height

 

 SA = 2B + Ph

SA = 2(lw) + (2l + 2w)h

rectangular_prism_net.gif (1694 bytes)

V = lwh

rectangular_prism.gif (1540 bytes)

Regular square prism

  • s = side length
  • h = height

 

 SA = 2B + Ph

SA = 2(s2) + (4s)h

square_prism_sml.png (32923 bytes)

 V = Bh

V = s2h

square_prism_vol.png (28523 bytes)

regular pentagonal prism

  • a = apothem length
  • s = side length
  • n = #sides
  • h = height

 

 

SA = 2B + Ph

SA = 2(1/2ans) + nsh

SA = 2(1/2a)(5)s + 5sh

SA = 5as + 5sh

pent_net.gif (1423 bytes)

 V = Bh

V = 1/2ansh

V = 1/2a(5)sh

V = 5/2ash

pentagonal_prism.gif (1269 bytes)

regular hexagonal prism

  • a = apothem length
  • s = side length
  • n = #sides
  • h = height

 

 

SA = 2B + Ph

SA = 2(1/2ans) + nsh

SA = 2(1/2a)(6)s + 6sh

SA =6as + 6sh

hex_net.gif (1832 bytes)

 V = Bh

V = 1/2ansh

V = 1/2a(6)sh

V = 3ash

hexagonal_prism.gif (1406 bytes)

cube

  • s = side length

 

 SA = 2B + Ph

SA = 2(s2) + (4s)s = 6s2

cube_net_sml.png (28862 bytes)

 V = Bh

V = s3

cube.gif (1355 bytes)

regular pyramid (general)

  • B = area of base
  • s = side length
  • l = slant heigth

 

 SA = B + n(1/2sl)

l = slant height

 V = 1/3Bh

regular triangular pyramid

  • a = apothem length
  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

SA = B + n(1/2sl)

SA = 1/2as + (3)(1/2sl)

SA = 1/2as + 3/2sl

triangular_reg_pyramid_sml.png (40438 bytes)

l = slant height

V = 1/3Bh

V = 1/3(1/2 ab)h

V = 1/6 abh

tpyramid.gif (2377 bytes)
 

regular square pyramid

  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

  

SA = B + n(1/2sl)

SA = s2 + (4)(1/2sl)

SA = s2 + 2sl

square_pyramid_sml.png (37371 bytes)

l = slant height

 V = 1/3Bh

V = 1/3(b2)h

= 1/3b2h

spyramid2.gif (2815 bytes)

regular pentagonal pyramid

  • a = apothem length
  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

  

SA = B + n(1/2sl)

SA = 1/2a(5)s + (5)(1/2sl)

SA = 5/2as + 5/2sl

pentagonal_pyramid.gif (2204 bytes)

l = slant height

 V = 1/3Bh

V = 1/3(1/2anb)h

V = 1/6 anbh

pentagonal_pyramid_vol.png (29199 bytes)

regular hexagonal pyramid

  • a = apothem length
  • s = side length
  • n = #sides
  • l = slant height
  • h = height

 

  

SA = B + n(1/2sl)

SA = 1/2a(6)s + (6)(1/2sl)

SA = 3as + 3sl

hexagonal_pyramid_sml.png (40635 bytes)

l = slant height

 V = 1/3Bh

V = 1/3(1/2anb)h

V = 1/6 anbh

hexagonal_pyramid_vol.png (28342 bytes)

cylinder

  • B = area of base
  • P = perimeter of base
  • r = radius of circle
  • h = height

 

 

SA = 2B + Ph

SA = 2(p r2) + (2pr)h

cylinder_net_sml.png (33177 bytes)

V = Bh

V = p r2h

cylinder.GIF (1709 bytes)

cone

  • B = area of base
  • r = radius of circle
  • h = height

 

cone_sml.png (34352 bytes)

Explanation

V = 1/3 Bh

V = 1/3pr2h

cone.png (38273 bytes)

The volume of a cone is one third the volume of the corresponding cylinder.

sphere

r = radius of circle

SA = 4pr2

sphere.GIF (3975 bytes)

V = 4/3pr3

sphere_volume.gif (1432 bytes)

 

Prism

hexagonal_prism.gif (1406 bytes)

V = Bh

Pyramid

hexagonal_pyramid_vol.png (28342 bytes)

V = 1/3 Bh

The volume of a pyramid is one third the volume of the corresponding prism.

Cylinder

cylinder.GIF (1709 bytes)

V = Bh

Cone

cone.png (38273 bytes)

V = 1/3Bh

 The volume of a cone is one third the volume of the corresponding cylinder.

 

Review:

Interactive sums - Areas and Volume

 

Enrichment:

Rectangular Prism Analysis

Quadrangular Pyramid(1) -tricky

Quadrangular Pyramid(2)

 

 

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