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Warm up #3 Page 11 draw and label the shape 1. The area of a rectangular rug is 40 yd2. If the width of the rug is 10 yd, what is the length of the rug? 2. The perimeter of a square rug is 16yd. If the width of the rug is 4 yd, what is the length of the rug? 3. Jose wants new carpeting for his living room. His living room is an 9 m by 9 m rectangle. How much carpeting does he need to buy to cover his entire living room? 4. Patricia has a rectangular flower garden that is 10 ft long and 5 ft wide. One bag of soil can cover 10 ft2. How many bags will she need to cover the entire garden? A Prism Cylinder Cuboid Cross section Trapezoid Prism Triangular Prism Volume of Prism = length x Cross-sectional area Area Formulae r h Area Circle = πr2 a b Area Rectangle = Base x height h b Area Trapezium = ½ x (a + b) x h h b Area Triangle = ½ x Base x height Geometry Surface Area of Triangular and cuboid Prisms Surface Area Triangular prism – a prism with two parallel, equal triangles on opposite sides. h w l To find the surface area of a triangular prism we can add up the areas of the separate faces. Surface Area In a triangular prism there are two pairs of opposite and equal triangles. 8 cm A 2 cm B C 7 cm 5 cm We can find the surface area of this prism by adding the areas of the pink side (A), the orange sides (B), the green bottom (C) and the two ends (D). Surface Area We should use a table to tabulate the various areas. Example: Side 8 cm A 2 cm B C 7 cm 5 cm A B C D Total Area Number of Sides Total Area Surface Area We should use a table to tabulate the various areas. Example: 8 cm A 2 cm B C 7 cm 5 cm Side Area Number of Sides Total Area A 40 cm2 1 40 cm2 B C D Total Surface Area We should use a table to tabulate the various areas. Example: 8 cm A 2 cm B C 7 cm 5 cm Side Area Number of Sides Total Area A 40 cm2 1 40 cm2 B 10 cm2 1 10 cm2 C D Total Surface Area We should use a table to tabulate the various areas. Example: 8 cm A 2 cm B C 7 cm 5 cm Side Area Number of Sides Total Area A 40 cm2 1 40 cm2 B 10 cm2 1 10 cm2 C 35 cm2 1 35 cm2 D Total Surface Area We should use a table to tabulate the various areas. Example: 8 cm A 2 cm BD C 7 cm 5 cm Side Area Number of Sides Total Area A 40 cm2 1 40 cm2 B 10 cm2 1 10 cm2 C 35 cm2 1 35 cm2 D 7 cm2 2 14 cm2 Total Surface Area We should use a table to tabulate the various areas. Example: 8 cm A 2 cm BD C 7 cm 5 cm Side Area Number of Sides Total Area A 40 cm2 1 40 cm2 B 10 cm2 1 10 cm2 C 35 cm2 1 35 cm2 D 7 cm2 2 14 cm2 5 99 cm2 Total Surface Area Example: Now you try...find the surface area! B Side Area No of Sides 2m C 2m 11m 2m Area Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The top and the bottom of the cuboid have the same area. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The front and the back of the cuboid have the same area. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The left hand side and the right hand side of the cuboid have the same area. Formula for the surface area of a cuboid We can find the formula for the surface area of a cuboid as follows. Surface area of a cuboid = l h w 2 × lw Top and bottom + 2 × hw Front and back + 2 × lh Left and right side = 2lw + 2hw + 2lh Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. Can you work out the 5 cm surface area of this cuboid? 8 cm The area of the top = 8 × 5 = 40 cm2 7 cm The area of the front = 7 × 5 = 35 cm2 The area of the side = 7 × 8 = 56 cm2 Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. 8 cm 5 cm So the total surface area = 2 × 40 cm2 7 cm Top and bottom + 2 × 35 cm2 Front and back + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2 Chequered cuboid problem This cuboid is made from alternate purple and green centimetre cubes. What is its surface area? Surface area = 2×3×4+2×3×5+2×4×5 = 24 + 30 + 40 = 94 cm2 How much of the surface area is green? 48 cm2 Surface area of a prism What is the surface area of this L-shaped prism? 3 cm 3 cm 4 cm 6 cm To find the surface area of this shape we need to add together the area of the two L-shapes and the area of the 6 rectangles that make up the surface of the shape. Total surface area 5 cm = 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15 = 110 cm2 Using nets to find surface area Here is the net of a 3 cm by 5 cm by 6 cm cuboid Write down the area of each face. 6 cm 3 cm 18 cm2 3 cm 5 cm 15 cm2 30 cm2 15 cm2 3 cm 18 cm2 3 cm Then add the areas together to find the surface area. 6 cm 30 cm2 Surface Area = 126 cm2 Surface Area Cylinder – (circular prism) a prism with two parallel, equal circles on opposite sides. To find the surface area of a cylinder we can add up the areas of the separate faces. Surface Area In a cylinder there are a pair of opposite and equal circles. A B We can find the surface area of a cylinder by adding the areas of the two blue ends (A) and the yellow sides (B). Surface Area We can find the area of the two ends (A) by using the formula for the area of a circle. A = π 2 r Side A B a 5 Total Area Number of Sides Total Area Sketch cylinder and Surface Area copy table. Work together to find the S.A. Side Area Number Sides Total Area Assignment Surface Area Sketch cylinder and copy table. Calculate S.A. Side 4m AA Area Number Sides Total Area Volume Cylinder Area = π x r2 = π x 32 = π9cm2 3cm 5cm Volume = length x Area = 5 x π9cm2 = 5 x π x 9cm2 = 45 x π =45π Lets do these together. Find the volume. V = r h 2 16 Volume of a Cylinder The volume, V, of a cylinder is V = Bh = r2h, where B is the area of the base, h is the height, and r is the radius of the base. Volume Trapezoid Prism trapezoid Area = ½ x(a + b) x h = ½ x (6 + 2) x 5 = ½ x 40cm2 6cm 5cm = 20cm2 4cm 2cm Volume = length x area = 20x 4 = 80cm3 Volume Trapezoid Prism trapezoid Area = ½ x(a + b) x h = ½ x (8 + 3) x 4 =½x cm2 8cm 4cm = 20cm2 4cm 2cm Volume = length x area = 20x 4 = 80cm3 Geometry Volume of Rectangular and Triangular Prisms The same principles apply to the triangular prism. h b Volume To find the volume of the triangular prism, we must first find the area of the triangular base (shaded in yellow). Volume To find the area of the Base… Area (triangle) = b x h 2 h b This gives us the Area of the Base (B). Volume Now to find the volume… B h We must then multiply the area of the base (B) by the height (h) of the prism. This will give us the Volume of the Prism. Volume Volume of a Triangular Prism Volume (triangular prism) B h V = B x h Volume Together… Volume V = B x h Volume Together… Volume V = V= B x h (8 x 4) x 12 2 Volume Together… Volume V = B x h V = (8 x 4) x 12 2 V = 16 x 12 Volume Together… Volume V = B x h V = (8 x 4) x 12 2 V = 16 x 12 V = 192 cm3 Volume Your turn… Find the Volume Triangular Prism To find the volume of a triangular prism find the area of the triangular base and multiply times the height of the prism. The height will always be the distance between the two triangles. Volume Triangular Prism Cross-sectional Area = ½ x b x h =½x8x4 = .5 x 32 4cm 4.9cm = 16cm2 6cm 8cm Volume = length x CSA = 16 x 6 = 96cm3 Find the Volume of the Triangular Prism. 1 Area of Triangular Base 6 8 24 2 4 10 4 10 ! ! 8 Base x height 24 10 240 6 Volume Cuboid Cross-sectional Area = b x h =7x5 = 35cm2 5cm 10cm 7cm Volume = length x CSA = 10 x 35 = 350cm3 Ex. 1: Finding the Volume of a rectangular prism The box shown is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box? What is the volume of the box? VOLUMES OF PRISMS AND CYLINDERS Volume of a three-dimensional figure is the number of cubic units needed to fill the space inside the figure. 1cm How many 1cm3 cubes will fill the rectangular prism on the right Find the volume. 10 7 6 V Blw B(base) 7 10 B 70 V Bh V 98 h V 70 6 V 588 Volume of a Prism The volume, V, of a prism is V = Bh, where B is the area of the base and h is the height. Find the volume. 3 V=s V 9 9 in. 9 in. 3 V 999 V 729inches 3 9 in. Volume of a Cube The volume of a cube is the length of its side cubed, or V=s3 Volume of a cuboid We can find the volume of a cuboid by multiplying the area of the base by the height. The area of the base = length × width So, height, h Volume of a cuboid = length × width × height = lwh width, w length, l Volume of a cuboid What is the volume of this cuboid? Volume of cuboid = length × width × height 5 cm = 5 × 8 × 13 8 cm 13 cm = 520 cm3 Volume of a prism made from cuboids What is the volume of this L-shaped prism? 3 cm We can think of the shape as two cuboids joined together. 3 cm 4 cm Volume of the green cuboid = 6 × 3 × 3 = 54 cm3 6 cm Volume of the blue cuboid = 3 × 2 × 2 = 12 cm3 Total volume 5 cm = 54 + 12 = 66 cm3 Volume of a prism Remember, a prism is a 3-D shape with the same cross-section throughout its length. 3 cm We can think of this prism as lots of L-shaped surfaces running along the length of the shape. Volume of a prism = area of cross-section × length If the cross-section has an area of 22 cm2 and the length is 3 cm, Volume of L-shaped prism = 22 × 3 =66 cm3 Volume of a prism What is the volume of this prism? 12 m 7m 4m 3m 5m Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 = 72 m2 Volume of prism = 5 × 72 =360 m3