SP about Rectangular Blocks

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Clemence Evans
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1 1 3D Measure Outcomes Recognise and draw the nets of prisms, cylinders, and cones. Solve problems about the surface area and volume of rectangular blocks, cylinders, right cones, prisms, spheres, and solids made from combinations of these.

2 2 This formula is not present in the F&T booklet SP about Rectangular Blocks A rectangular block (a.k.a. cuboid) has three dimensions at right angles to each other length, width, and height (breadth and/or depth may be used also). Opposite faces are equal rectangles. The volume, V of a rectangular solid is given by: V = l w h Dashed lines indicate edges we can t see usually on the back of a solid

3 3 SP about Rectangular Blocks The net of a solid is a 2D shape that can be folded to form the solid. You can make a net from a solid by breaking edges and unfolding faces until it lies flat. The area, A of the net is A = 2lw + 2wh + 2lh This is also the surface area of the cuboid.

4 4 SP about Rectangular Blocks Find the volume and surface area of the following rectangular blocks: V = lwh SA = 2lw + 2lh + 2wh

5 5 V = lwh SA = 2lw + 2lh + 2wh SP about Rectangular Blocks Find the missing measurements in this table: Length (cm) Width (cm) Height (cm) Volume (cm 3 ) Surface Area (cm 2 )

6 6 SP about Rectangular Blocks Find the volume of each of the following solids: V = lwh

7 7 This formula is found on pg 10 of F&T SP about Cylinders A cylinder has two important dimensions radius and height. The height should be perpendicular to the discs. The opposite discs are equal. The volume, V of a cylinder is given by: V = πr 2 h

8 8 The formula for curved surface area is on pg 10 of F&T SP about Cylinders The net of a cylinder shows two discs and a rectangle. The rectangle represents the curved surface of the cylinder, while the discs show the flat top and bottom. The curved surface area, CSA is given by: CSA = 2πrh The total area has two discs added to it: SA = 2πrh + 2πr 2

9 9 SP about Cylinders Find the volume, curved surface area, and total surface area of each of the following: 4 cm 10 cm V = πr 2 h CSA = 2πrh 8 cm 14 cm

10 10 Use π = 3.14 V = πr 2 h CSA = 2πrh SP about Cylinders Find the missing measurements in this table: Height (cm) Radius (cm) Volume (cm 3 ) CSA (cm 2 )

11 11 This formula is found on pg 10 of F&T SP about Cones A cone has three important dimensions radius, height, and slant height. The height should be perpendicular to the disc. The volume, V of a cone is given by: V = 1 3 πr2 h For our cones, the dimensions relate by Pythagoras theorem: l 2 = h 2 + r 2

12 12 The formula for curved surface area is on pg 10 of F&T SP about Cones The net of a cone shows a disc and a sector. The disc represents the flat top/bottom of the cone while the sector shows the curved surface. The curved surface area, CSA is given by: CSA = πrl The total surface area has a disc added to it: SA = πrl + πr 2

13 13 SP about Cones Find the volume, curved surface area, and total surface area of each of the following: l 2 = h 2 + r 2 V = 1 3 πr2 h CSA = πrl

14 14 Use π = 3.14 l 2 = h 2 + r 2 V = 1 3 πr2 h CSA = πrl SP about Cones Find the missing measurements in this table: Height (cm) Radius (cm) Slant height (cm) Volume (cm 3 ) CSA (cm 2 )

15 15 SP about Prisms A prism consists of two identical polygons connected on their sides by rectangles. The height of the prism should be perpendicular to the polygons. The volume of a prism is given by: V = Bh where B is the area of the polygons.

16 16 SP about Prisms The net of a prism shows two polygons and however many rectangles are required to match the polygon. The surface area is 2B plus the area of each rectangle.

17 17 SP about Prisms Find the volume and surface area of each of the following prisms: V = Bh

not have nets as there are no edges to break and nothing

18 18 SP about Spheres Spheres have only one dimension radius. The volume of a sphere is given by: V = 4 3 πr3 Spheres do not have nets as there are no edges to break and nothing to unfold. Their surface area is simply given by: SA = 4πr 2

19 19 SP about Spheres Find the volume and surface area of each of the following spheres: V = 4 3 πr3 CSA = 4πr 2 5cm 18cm

20 20 SP about Spheres Find the missing measurements in this table: V = 4 3 πr3 CSA = 4πr 2 Radius (cm) Volume (cm 3 ) Surface Area (cm

21 21 SP about Combinations When solids are joined together, their volumes simply sum. The surface area usually will need to be adjusted as the joined faces are no longer part of the surface. e.g. in the diagram on the right, the cone and cylinder are joined by a disc on each solid. These two discs are not part of the surface and are not counted for surface area.

22 OL P2 Q5 Cylinder V = πr 2 h CSA = 2πrh Cone V = 1 3 πr2 h CSA = πrl l 2 = r 2 + h 2 Prism V = Bh Sphere V = 4 3 πr3 CSA = 4πr 2 SP about Combinations A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. The diameter of the base of the cylinder is 3 cm and the height of the cylinder is 8 cm. The volume of the wax in the candle is 21π cm 3. i. Find the height of the candle. Nine of these candles fit into a rectangular box. The base of the box is a square. ii. Find the volume of the smallest rectangular box that the candles will fit into.

23 OL P2 Q1 Cylinder V = πr 2 h CSA = 2πrh Cone V = 1 3 πr2 h CSA = πrl l 2 = r 2 + h 2 Prism V = Bh Sphere SP about Combinations A solid object consists of a cylinder with hemispherical ends, as shown. The cylinder and hemispheres have the same radius. The volume of each hemisphere is 144π cm 3. i. Find the radius of each hemisphere. The total volume of the object is 720π cm 3. ii. Find d, the length of the object. V = 4 3 πr3 CSA = 4πr 2

24 OL P2 Q5 Cylinder V = πr 2 h CSA = 2πrh Cone V = 1 3 πr2 h CSA = πrl l 2 = r 2 + h 2 Prism V = Bh SP about Combinations A solid cylinder has a radius of 10 mm and a height of 45 mm. a) Draw a sketch of the net of the surface of the cylinder and write its dimensions on the sketch. b) Calculate the volume of the cylinder. Give your answer in terms of π. A sphere has the same volume as the cylinder. c) Find the surface area of the sphere. Give your answer in terms of π. Sphere V = 4 3 πr3 CSA = 4πr 2

25 2009 OL P2 Q1 Cylinder V = πr 2 h CSA = 2πrh Cone V = 1 3 πr2 h CSA = πrl l 2 = r 2 + h 2 SP about Combinations The volume of a sphere is 36π cm 3. i. Find the radius of the sphere.

25 OL P2 Q1 Cylinder V = πr 2 h CSA = 2πrh Cone V = 1 3 πr2 h CSA = πrl l 2 = r 2 + h 2 SP about Combinations The volume of a sphere is 36π cm 3. i. Find the radius of the sphere. When the sphere is fully immersed in a cylinder of water, the level of the water rises by 2.25 cm. ii. Find the radius of the cylinder. Prism V = Bh Sphere V = 4 3 πr3 CSA = 4πr 2

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth.

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth. Volume of Cylinders As with prisms, the area of the base of a cylinder tells the number of cubic units in one layer. The height tells how many layers there are in the cylinder. The volume V of a cylinder