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CHAPTER 6: AREA AND VOLUME OF SOLIDS 1) Area of solids a) Units of area Use your “Metric system conversions” chart. The main unit of area is the square metre (m2). It is the surface area of a square with 1 m sides. When going from one unit of area to the next unit (lowest or highest), you multiply or divide the measure by 100. E.g. 12 cm2 = _______________ m2 0.25 km2 = _______________ m2 1 258 dam2 = _______________ km2 0.048 dm2 = _______________ mm2 30 hm2 = _______________ cm2 Workbook, page 177, #1 and 2

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Remember that: Area of a triangle: A = b • h h 2 b h Area of parallelogram: A = b • h h b Area of rhombus: A = D • d D 2 d b Area of trapezoid: A = (B + b) h 2 h B

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Area of regular polygon: A = perimeter of polygon • apothem 2 A = P a 2 A = P a E.g. 2 A = (8 • 5) • 7 2 apothem 7 cm A = 140 cm2 5 cm Area of a disk: A = π r2 r

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b) Total area of a cube

The surface of a cube corresponds to 6 congruent squares therefore the total area of a cube is 6 times the area of 1 of those squares:

ATOT = 6s2 s c) Area of a prism The total area of a solid generally includes the area of the bases and the lateral area.

The lateral area is the sum of the areas of the lateral faces (“sides”) of the figure ⇒ all of the faces except for the bases.

h BASES LATERAL SURFACE b a

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To calculate the area of a right prism: AreaLATERAL = Perimeter of base height ALAT = Pb h AreaTOTAL = ALATERAL + (2 ABASE) ATOT = ALAT + 2Ab E.g. 4 3 Find the total area of the following prism: 8 5 ATOT = ALAT + 2Ab ALAT = Pb h ATOT = 96 cm2 + 2(6 cm2) ALAT = 12 cm 8 cm ATOT = 108 cm2 ALAT = 96 cm2 Pb = 3 + 4 + 5 Base is a triangle Pb = 12 cm Ab = b h 2

Ab = 3 4 2

Ab = 6 cm2 Workbook, pages 178-179, #3 to 13 (but not 4b)

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d) Area of a cylinder The net of a cylinder consists of a rectangle (lateral surface) and two discs (the bases). To calculate the area of a right cylinder: AreaLATERAL = Perimeter of base Height ALAT = Pb h

or because in a cylinder Pb = Cb = 2r

ALAT = 2rh

The total area is the sum of the lateral area and the areas of the 2 bases. AreaTOTAL = AreaLATERAL + 2 AreaBASE ATOT = ALAT + 2Ab

or, because in a cylinder Ab = r2

ATOT = 2rh + 2r2

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E.g. Find the total area of the following cylinder: 4 9 ATOT = ALAT + 2Ab ATOT = 2rh + 2r2 ATOT = 2(4)(9) + 2(4)2 ATOT 226.19 + 100.53 Use key on your calculator! ATOT 326.73 cm2 Keep all the decimals until the very end! or ATOT = 104 cm2 Workbook, page 181, #14 to 17

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e) Area of a pyramid h s s c The height of the pyramid is “h”, the slant height is “s”, and the measure of one side of the square base is “c”.

The slant height (s) is also the height of the triangles forming the lateral faces of the pyramid. Lateral area: We only s want the area in white! Pb

AreaLATERAL = Perimeter of base Slant height 2 ALAT = Pb s 2

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The total area of the pyramid is the sum of the area of the base and the lateral area. AreaTOTAL = AreaBASE + AreaLATERAL ATOT = Ab + ALAT ATOT = Ab + Pb s 2 E.g.

Find the total area of the following pyramid: 7 5 5 ATOT = Ab + ALAT ALAT = Pb s ATOT = 25 cm2 + 70 cm2 2 ATOT = 95 cm2 ALAT = 20 cm 7 cm 2 ALAT = 70 cm2 Base is a square Pb = 4 side = 4 c Ab = side2 = c2 Pb = 4 5 Ab = 52 Pb = 20 cm Ab = 25 cm2 Workbook, pages 182-183. #18 to 25

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f) Area of a cone The formula to find the lateral area of a cone is similar to that of a pyramid:

AreaLATERAL = Perimeter of base Slant height 2 ALAT = Pb s 2 Or, because in a cone Pb = Cb = 2r ALAT = 2r s 2 ALAT = rs The total area of a cone is the sum of the area of the base and the lateral area. AreaTOTAL = AreaBASE + AreaLATERAL ATOT = Ab + ALAT or, because in a cone Ab = r2

ATOT = r2 + rs

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Reminders: In a cone, the radius, the slant height and the height form a right triangle (Pythagorean Theorem): s2 = r2 + h2 c = 2r, so to find the radius: r = c 2 E.g. Find the total area of the following cone: 7

ATOT = Ab + ALAT ATOT = r2 + rs 3 ATOT = 32 + (3)(7) ATOT 28.27 + 65.97 Use key on your calculator! ATOT 94.25 cm2 Keep all the decimals until the very end! or ATOT = 30 cm2

Workbook, pages 184-185, #27 to 36

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g) Area of a sphere The area of a sphere is directly proportional to the square of the radius. ATOT = 4r2 Only one surface for a sphere! E.g. Find the area of a sphere with a diameter of 12 cm. r = d = 12 = 6 cm 2 2 ATOT = 4r2 ATOT = 4(6)2 Use key on your calculator! ATOT 452.39 cm2 or ATOT = 144 cm2 Workbook, pages 185-186, #37 to 42

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2) Area of decomposable solids a) Total area of a decomposable solid To calculate the area of a decomposable solid, separate it into solids such as a prism, a pyramid, a cone, a cylinder, a sphere… BASIC FORMULAS SOLIDS LATERAL AREA TOTAL AREA Right prisms ALAT = Pb h ATOT = ALAT + 2Ab ________________________________________________________________ Right cylinders ALAT = Cb h ATOT = ALAT + 2Ab ALAT= 2rh ATOT = 2rh +2r2

Cb = 2r = d Ab = r2 ________________________________________________________________ Right regular ALAT = Pb s ATOT = ALAT + Ab pyramids 2 ________________________________________________________________ Right cones ALAT = Cb s ATOT = ALAT + Ab

2 ALAT = rs ATOT = rs + r2

Cb = 2r = d Ab = r2 ________________________________________________________________ Spheres ALAT = ATOT = 4r2 Workbook, pages 188-189, #1 to 11

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3) Volume of solids a) Units of volume The main unit of volume is the cubic metre (m3). It is the space enclosed in a cube with 1 m sides. When going from one unit of volume to the next unit (lowest or highest), you multiply or divide the measure by 1000. Use your “Metric system conversions” chart. E.g. 12 cm3 = _______________ m3 0.25 km3 = _______________ m3 1 258 dam3 = _______________ km3 0.048 dm3 = _______________ mm3 30 hm3 = _______________ cm3 Workbook, pages 190-191, #1 to 8

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b) Units of capacity Units of capacity are used to measure the volume of a liquid in a container such as water, milk or gasoline. The main unit of capacity is the litre (L). Units of capacity are also units of volume. They can be changed into units of volume and vice versa.

1 m3 = 1 kL 1 dm3 = 1 L 1 cm3 = 1 mL

1 litre of water = 1 kg at STP Use your “Metric system conversions” chart. E.g. 5.4 L = _______________ cL 35 dL = _______________ L 12 dL = _______________ dm3 1 cL = _______________ mm3 0.25 hm3 = _______________ kL Workbook, pages 192, #9 to 16

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c) Volumes of cubes and prisms The volume of a cube with edge length “s” is: Vcube = s3 s The volume of a prism is equal to the product of the area of the base (Ab) and the height (h) of the prism. Vprism = Ab h E.g. Find the volume of the following prism: Vprism = Ab h 7 cm Vprism = 30 cm2 7 cm 3 cm 4 cm Vprism = 210 cm3 Base is a regular pentagon Pb = 5 side Ab = Pb a 2 Pb = 5 4 Ab = (5 4) 3 2 Pb = 20 cm Ab = 30 cm2 Workbook, pages 193-195, #17 to 30

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d) Volume of a cylinder The volume of a cylinder is equal to the product of the area of the base (Ab) and the height of the cylinder (h). Vcyl = Ab h or, because in a cylinder Ab = r2 Vcyl = r2h E.g. Find the volume of the following cylinder: 2cm 8cm Vcyl = r2h Vcyl = (2)2(8) Use key on your calculator! Vcyl 100.53 cm3 or Vcyl = 32 cm3 Workbook, pages 195-197, #31 to 43

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e) Volume of a pyramid

The volume of a pyramid is equal to one-third the product of the area of the base (Ab) and the height (h). Vpyr = Ab h 3 E.g. Find the volume of the following pyramid with a square base: 6 cm 4 cm 4 cm Vpyr = Ab h Base is a square 3 Ab = side2 = s2

Vpyr = 16 cm2 6 cm Ab = 42

3 Ab = 16 cm2 Vpyr = 32 cm3 Workbook, page 198, #44 to 51

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f) Volume of a cone The volume of a cone is equal to one-third the product of the area of the base (Ab) and the height (h). Vcone = Ab h 3 or, because in a cone Ab = r2 Vcone = r2h

3 E.g. Find the volume of the following cone: 7 cm 3 cm

Vcone = r2h

3 Vcone = (3)2(7) Use key on your calculator!

3 Vcone 65.97 cm3

or Vcone = 21 cm3

Workbook, page 199, #52 to 58

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g) Volume of a sphere The volume of a sphere with radius “r” is: Vsphere = 4r3 3 E.g. Find the volume of a sphere whose diameter is 9 cm.

r = d = 9 = 4.5 cm 2 2

Vsphere = 4r3

3 Vsphere = 4(4.5)3 Use key on your calculator! 3 Vsphere 381.70 cm3 or Vsphere = 121.5 cm3 Workbook, pages 200-202, #59 to 76

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4) Volume of decomposable solids a) Total volume of a decomposable solid To calculate the volume of a decomposable solid, separate it into solids such as a prism, a pyramid, a cone, a cylinder, a sphere… Workbook, pages 203-204, #1 to 8

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5) Missing measures of a solid When looking for the missing measure of a solid in a given situation: 1) translate the situation using the appropriate

formula; 2) replace, in the formula, each known measure by its

value; 3) find the unknown measure by solving the resulting

equation. E.g. Determine the radius of a cylinder if its volume is equal to 141.30 cm3 and its height is equal to 5 cm. Vcyl = r2h 141.30 = r2(5) 141.30 = r2(5) 5 5 8.9954… r2

89954. ... r 2

2.999 cm r

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E.g. The volume of a sphere is 87.11 cm3. Find the radius of that sphere. Vsphere = 4r3 3

87.11 = 4r3 3

3 x 87.11 = 4r3 x 3 3

261.33 = 4r3 4 4

65.3325 = r3

20.796 r3

3 3r 3 20.796

r 2.75 cm

Workbook, pages 205-208, #1 to 43 Page 175, #1 to 6 (Challenge 6) Pages 209-210, #1 to 14 (Evaluation 6)