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Rectangular Prisms

Rectangular Prisms. Information. How many cubes?. The following rectangular prism is made of interlocking cubes. How many cubes does it contain?. Divide the rectangular prism into layers. calculate the number of small cubes in a layer:. multiple the length by the width:. 3 × 4 =.

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Rectangular Prisms

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  1. Rectangular Prisms

  2. Information

  3. How many cubes? The following rectangular prism is made of interlocking cubes. How many cubes does it contain? Divide the rectangular prism into layers. calculate the number of small cubes in a layer: multiple the length by the width: 3 × 4 = 12 cubes find the number of cubes in the whole prism: multiple the number of cubes in one layer by the height: 12 × 3 = 36 cubes

  4. Volume of a rectangular prism The volume of a rectangular prism can be found by multiplying the area of the base by the height. The area of the base: = length × width = lw Volume of the prism: height, h = base × height width, w = lwh length, l The volume of a rectangular prism = length × width × height = lwh

  5. What is the volume?

  6. Volume of a cube How can the volume of a cube of side length s be found? The length, width and height of a cube are all the same. write the equation for the volume of a rectangle: =l× w × h = s× s × s substitute: = s3 combine terms: s The volume of a cube = (length of one edge)3 = s3

  7. What is the surface area? This cube is made from 125 cubes. Each of the smaller cubes is 1 cm long. What is the surface area of the larger cube? surface area of one side: 5 × 5 = 25cm2 multiply by the number of sides: 25 × 6 = 150cm2

  8. Surface area

  9. Surface area formula The surface area of a rectangular prism is the combined area of all its sides. l top and bottom + w 2 × lw+ front and back + h 2 × lh+ left and right side 2 × hw The surface area of a rectangular prism = 2lw + 2hw + 2lh

  10. Special rectangular prisms Rectangular prisms can have a variety of lengths and widths. They can range from short and fat to long and thin. Find a rectangular prism (with edges of integer lengths) that has a surface area of exactly 100cm2.

  11. Surface area of a cube How can the surface area of a cube of side length sbe found? write equation for surface area of one face: = l × w = s × s substitute: = s2 combine terms: multiple by the number of sides: = 6s2 s The surface area of a cube = 6s2

  12. Volume and surface area

  13. Length around the edges

  14. Length around the edges To find the length around the edges of a rectangular prism of length l, width w and height h, use the formula: Length around the edges of a rectangular prism h = 4l + 4w + 4h = 4(l + w + h) w l To find the length around the edges of a cube with side length l,use the formula: Length around the edges of a cube: = 12l

  15. Longest diagonal The Pythagorean theorem can be applied to 3D problems. What is the length of the longest diagonal ina rectangular prism measuring 5cm by 7cm by 8cm? CE identify the longest diagonal: We could also use AG, BHorDF. H G use the Pythagorean theorem to find the AC: E F AC2= CB2+AB2 AC2= 25 + 49 substitute values: AC= 8.60 cm 8cm evaluate: D C the Pythagorean theorem, to find the CE: 5cm CE2 = AE2 + AC2 A B 7cm CE2 = 82 + 74 substitute values: CE = 11.75 cm evaluate: (to nearest hundredth)

  16. Formula for the longest diagonal Show that the longest diagonal y in a rectangular prism measuring v by w by x is given by the formula: y = (v2 + w2 + x2) use Pythagorean theorem to find z: z2 = v2 + w2 use the Pythagorean theorem to find y: y y2 = z2 + x2 x substitute in the equation for z: z w y2 = (v2 + w2)+ x2 v y = (v2 + w2 + x2)

  17. Rectangular prism diagonals

  18. Longest diagonal A rectangular prism has side lengths of 8cm, 10cm and 11cm. Use the formula y = (v2 + w2 + x2) to find the length of the longest diagonal y to the nearest hundredth. Substitute the values into the formula: y = (82 + 102 + 112) y = (64 + 100 + 121) 10 cm y = 285 y= 16.88 cm (to the nearest hundredth) 8 cm 11 cm

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