draw and label the shape Warm up #3 Page 11 draw and label the shape 1. The area of a rectangular rug is 40 yd 2. 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 ft 2. How many bags will she need to cover the entire garden?

A Prism Cylinder Cuboid Triangular Prism Trapezoid Prism Volume of Prism = length x Cross-sectional area Cross section

Area Formulae Area Circle = πr 2 r Area Rectangle = Base x height h b b h Area Triangle = ½ x Base x height h b Area Trapezium = ½ x (a + b) x h a

Geometry Surface Area of Triangular and cuboid Prisms

Surface Area  Triangular prism – a prism with two parallel, equal triangles on opposite sides. To find the surface area of a triangular prism we can add up the areas of the separate faces. l wh

Surface Area  In a triangular prism there are two pairs of opposite and equal triangles. 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). 7 cm 5 cm2 cm 8 cm A B C

Surface Area  We should use a table to tabulate the various areas. Example: SideAreaNumber of Sides Total Area A B C D Total 7 cm 5 cm2 cm 8 cm A B C

Surface Area  We should use a table to tabulate the various areas. Example: SideAreaNumber of Sides Total Area A40 cm 2 1 B C D Total 7 cm 5 cm2 cm 8 cm A B C

Surface Area  We should use a table to tabulate the various areas. Example: SideAreaNumber of Sides Total Area A40 cm 2 1 B10 cm 2 1 C D Total 7 cm 5 cm2 cm 8 cm A B C

Surface Area  We should use a table to tabulate the various areas. Example: SideAreaNumber of Sides Total Area A40 cm 2 1 B10 cm 2 1 C35 cm 2 1 D Total 7 cm 5 cm2 cm 8 cm A B C

Surface Area  We should use a table to tabulate the various areas. Example: SideAreaNumber of Sides Total Area A40 cm 2 1 B10 cm 2 1 C35 cm 2 1 D7 cm cm 2 Total 7 cm 5 cm2 cm 8 cm A B C D

Surface Area  We should use a table to tabulate the various areas. Example: SideAreaNumber of Sides Total Area A40 cm 2 1 B10 cm 2 1 C35 cm 2 1 D7 cm cm 2 Total5 99 cm 2 7 cm 5 cm2 cm 8 cm A B C D

Surface Area  Now you try...find the surface area! Example: C B SideAreaNo of Sides Area 2m 11m 2m

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. Surface area of a cuboid

We can find the formula for the surface area of a cuboid as follows. Surface area of a cuboid = Formula for the surface area of a cuboid h l w 2 × lw Top and bottom + 2 × hw Front and back+ 2 × lh Left and right side = 2 lw + 2 hw + 2 lh

To find the surface area of a shape, we calculate the total area of all of the faces. Can you work out the surface area of this cuboid? Surface area of a cuboid 7 cm 8 cm 5 cm The area of the top = 8 × 5 = 40 cm 2 The area of the front = 7 × 5 = 35 cm 2 The area of the side = 7 × 8 = 56 cm 2

To find the surface area of a shape, we calculate the total area of all of the faces. So the total surface area = Surface area of a cuboid 7 cm 8 cm 5 cm 2 × 40 cm × 35 cm × 56 cm 2 Top and bottom Front and back Left and right side = = 262 cm 2

This cuboid is made from alternate purple and green centimetre cubes. Chequered cuboid problem What is its surface area? Surface area = 2 × 3 × × 3 × × 4 × 5 = = 94 cm 2 How much of the surface area is green? 48 cm 2

What is the surface area of this L-shaped prism? Surface area of a prism 6 cm 5 cm 3 cm 4 cm 3 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 = 2 × = 110 cm 2

5 cm 6 cm 3 cm 6 cm 3 cm 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. 15 cm 2 18 cm 2 30 cm 2 18 cm 2 Then add the areas together to find the surface area. Surface Area = 126 cm 2

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. We can find the surface area of a cylinder by adding the areas of the two blue ends (A) and the yellow sides (B). B A

Surface Area  We can find the area of the two ends (A) by using the formula for the area of a circle.  A = π r 2 SideArea Number of Sides Total Area A B Total a B =10 5

Surface Area  Sketch cylinder and copy table. Work together to find the S.A. SideArea Number Sides Total Area

Surface Area  Assignment SideArea Number Sides Total Area A A 4m  Sketch cylinder and copy table. Calculate S.A.

5cm 3cm Area = π x r 2 = π x 3 2 = π9cm 2 Volume = length x Area = 5 x π9cm 2 Volume Cylinder = 5 x π x 9cm 2 =45π = 45 x π

Lets do these together. Find the volume. Volume of a Cylinder The volume, V, of a cylinder is V = Bh =  r 2 h, where B is the area of the base, h is the height, and r is the radius of the base. V =  r 2 h 16

Volume Trapezoid Prism trapezoid Area = ½ x(a + b) x h = ½ x (6 + 2) x 5 Volume = length x area = 20x 4 = 80cm 3 2cm 4cm 6cm 5cm = ½ x 40cm 2 = 20cm 2

Volume Trapezoid Prism trapezoid Area = ½ x(a + b) x h = ½ x (8 + 3) x 4 Volume = length x area = 20x 4 = 80cm 3 2cm 4cm 8cm 4cm = ½ x cm 2 = 20cm 2

Geometry Volume of Rectangular and Triangular Prisms

Volume  The same principles apply to the triangular prism. To find the volume of the triangular prism, we must first find the area of the triangular base (shaded in yellow). b h

Volume  To find the area of the Base… Area (triangle) = b x h 2 This gives us the Area of the Base (B). b h

Volume  Now to find the volume… 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. B h

Volume  Volume of a Triangular Prism Volume (triangular prism) V = B x h B h

Volume  Together… Volume V = B x h

Volume  Together… Volume V = B x h V = (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 cm 3

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 = ½ x 8 x 4 Volume = length x CSA = 16 x 6 = 96cm 3 8cm 6cm 4cm 4.9cm =.5 x 32 = 16 cm 2

Find the Volume of the Triangular Prism. 6 8 ! !

Volume Cuboid 5cm 7cm 10cm Cross-sectional Area = b x h = 7 x 5 = 35cm 2 Volume = length x CSA = 10 x 35 = 350cm 3

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 1cm How many 1cm 3 cubes will fill the rectangular prism on the right Volume of a three-dimensional figure is the number of cubic units needed to fill the space inside the figure.

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. 6 Find the volume.

Volume of a Cube The volume of a cube is the length of its side cubed, or V=s 3 9 in. Find the volume. V=s 3

Volume of a cuboid We can find the volume of a cuboid by multiplying the area of the base by the height. Volume of a cuboid = length × width × height = lwh height, h length, l width, w The area of the base = length × width So,

Volume of a cuboid What is the volume of this cuboid? Volume of cuboid = length × width × height = 5 × 8 × 13 = 520 cm 3 5 cm 8 cm 13 cm

What is the volume of this L-shaped prism? Volume of a prism made from cuboids 6 cm 5 cm 3 cm 4 cm 3 cm We can think of the shape as two cuboids joined together. Volume of the green cuboid = 6 × 3 × 3 = 54 cm 3 Volume of the blue cuboid = 3 × 2 × 2 = 12 cm 3 Total volume = = 66 cm 3

Remember, a prism is a 3-D shape with the same cross-section throughout its length. Volume of a prism 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 cm 2 and the length is 3 cm, Volume of L-shaped prism =22 × 3 =66 cm 3 3 cm

Volume of a prism Area of cross-section = 7 × 12 – 4 × 3 =84 – 12 = Volume of prism = 5 × 72 =360 m 3 3 m 4 m 12 m 7 m 5 m 72 m 2 What is the volume of this prism?