3D Measure – Outcomes

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. 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, of a rectangular solid is given by: This formula is not present in the F&T booklet Dashed lines indicate edges we can’t see – usually on the back of a solid

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, of the net is • This is also the surface area of the cuboid.

4. SP about Rectangular Blocks • Find the volume and surface area of the following rectangular blocks:

5. SP about Rectangular Blocks • Find the missing measurements in this table:

6. SP about Rectangular Blocks • Find the volume of each of the following solids:

7. 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, of a cylinder is given by: This formula is found on pg 10 of F&T

8. 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: • The total area has two discs added to it: The formula for curved surface area is on pg 10 of F&T

9. 10 cm 4 cm 8 cm 14 cm SP about Cylinders • Find the volume, curved surface area, and total surface area of each of the following:

10. SP about Cylinders Use • Find the missing measurements in this table:

11. SP about Cones • A cone has three important dimensions – radius, height, and slant height. • The height should be perpendicular to the disc. • The volume, of a cone is given by: • For our cones, the dimensions relate by Pythagoras’ theorem: This formula is found on pg 10 of F&T

12. 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, is given by: • The total surface area has a disc added to it: The formula for curved surface area is on pg 10 of F&T

13. SP about Cones • Find the volume, curved surface area, and total surface area of each of the following:

14. SP about Cones • Find the missing measurements in this table: Use

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: • where is the area of the polygons.

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 plus the area of each rectangle.

17. SP about Prisms • Find the volume and surface area of each of the following prisms:

18. SP about Spheres • Spheres have only one dimension – radius. • The volume of a sphere is given by: • Spheres do not have nets as there are no edges to break and nothing to unfold. Their surface area is simply given by:

19. SP about Spheres • Find the volume and surface area of each of the following spheres: 18cm 5cm

20. SP about Spheres • Find the missing measurements in this table:

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. SP about Combinations 2012 OL P2 Q5 • 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 21 cm3. • Find the height of the candle. • Nine of these candles fit into a rectangular box. The base of the box is a square. • Find the volume of the smallest rectangular box that the candles will fit into. • A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. Cylinder Cone Prism Sphere

23. SP about Combinations 2011 OL P2 Q1 • 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 144 cm3. Cylinder Cone Prism Sphere • Find the radius of each hemisphere. • The total volume of the object is 720 cm3. • Find , the length of the object.

24. SP about Combinations 2013 OL P2 Q5 • A solid cylinder has a radius of 10 mm and a height of 45 mm. • Draw a sketch of the net of the surface of the cylinder and write its dimensions on the sketch. • Calculate the volume of the cylinder. Give your answer in terms of . • A sphere has the same volume as the cylinder. • Find the surface area of the sphere. Give your answer in terms of . Cylinder Cone Prism Sphere

25. SP about Combinations 2009 OL P2 Q1 • The volume of a sphere is 36 cm3. • 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. • Find the radius of the cylinder. Cylinder Cone Prism Sphere