Volume & Surface Area

1. Volume & Surface Area

2. Volume & Surface Area • Objectives: • 7.2.02Solve problems involving volume and surface area of cylinders, prisms, and composite shapes. • Essential Question: • How can I use what I know about area to calculate volume and surface area of cubes, prisms, and cylinders?

3. Volume & Surface Area • Cube:a 3D shape with six square or rectangular sides, a block. • Rectangular Prism:a polyhedron that has two parallel and congruent bases that are rectangles; a 3D solid with six rectangular faces. • Triangular Prism:a polyhedron that has two parallel, congruent bases that are triangles; a prism whose faces are triangles. • Cylinder:a 3D figure that has two parallel congruent bases. • Volume:the measure of space occupied by a solid region. • Surface Area:the sum of the areas of all the surfaces (faces) of a three dimensional figure.

4. Volume & Surface Area • Triangular Pyramid:a polyhedron with a three-sided polygon for a base and triangles for its sides; a pyramid with a triangular base. • Square Pyramid:a polyhedron with a four-sided polygon for a base and triangles for its sides; a pyramid with a square base. • Sphere:a perfectly rounded 3D object such as a ball. • Cone:a 3D figure with one circular base.

5. Volume & Surface Area What is a 3D Figure: • What do they look like… In previous years you have studied 2D shapes like squares, rectangles, parallelograms, triangles, circles, and trapezoids. But now it is time to add a third dimension…

6. Volume & Surface Area What is a 3D Figure: • What do they look like…

7. Volume & Surface Area Some 3D Figures: • What do they look like… Cube Rectangular Prism Triangular Prism Cylinder

8. Volume & Surface Area Some Additional 3D Figures: • What do they look like… Triangular Pyramid Square Pyramid Sphere Cone

9. Volume & Surface Area Cubes: • What do they look like…

10. Volume & Surface Area Prisms: • What do they look like…

11. Volume & Surface Area Cylinders: • What do they look like…

12. Volume & Surface Area 3D Characteristics: • What makes what a what… A 3D shape with two parallel congruent polygon bases

13. Volume & Surface Area 3D Characteristics: • What makes what a what… A cylinder falls under its own category because its bases are not considered polygons

14. Volume & Surface Area 3D Characteristics: • What makes what a what… Others include pyramids and cones because they contain only one base. Their name is derived based on the shape of the base

15. Volume & Surface Area Important Volume Formulas: Rectangular Prism Triangular Prism Cube Cylinder V = s3 V = lwh V = ½bhw V = πr2h V = bhw

16. Volume & Surface Area Example 1: Cube • Find the volume of the cube whose sides measure 3 inches. Volume = s3 V = 3in x 3in x 3in V = 27in3

17. Volume & Surface Area Example 1: Rectangular Prism • Find the volume of the rectangular prism whose length is 5in, width is 9in, and height is 4in. Volume = lwh V = 5in x 9in x 4in V = 180in3

18. Volume & Surface Area Example 1: Triangular Prism • Find the volume of the triangular prism whose length is 6cm, width is 4cm, and height is 3cm. Volume = ½bhw V = ½(6cm)(3cm)(4cm) V = 36cm3

19. Volume & Surface Area Example 1: Cylinder • Find the volume of the cylinder whose height is 3ft and radius is 4ft. Volume = πr2h V = (3.14)(4ft)2(3ft) V = 150.72ft3

20. Volume & Surface Area Example 1: Rectangular Prism • Find the volume of the rectangular prism. V = lwh V = 4in x 6in x 5in V = 120in3

21. Volume & Surface Area Example 1: Rectangular Prism • Find the volume of the rectangular prism. V = lwh V = 5in x 7in x 11in V = 385in3

22. Volume & Surface Area Example 1: Triangular Prism • Find the volume of the triangular prism. Volume = ½bhw V = ½(15cm)(9cm)(4cm) V = 270cm3

23. Volume & Surface Area Example 1: Cylinder • Find the volume of the cylinder. Volume = πr2h V = (3.14)(3cm)2(12cm) V ≈ 339.3cm3

24. Volume & Surface Area Independent Practice: Volume • Find the volume of each 3d shape. • 1. 2. • 3. 4.

25. Volume & Surface Area Independent Practice: Volume • Answers. • 1. 27 m3 2. 1260 cm3 • 3. 27 m3 4. 336 mm3

26. Volume & Surface Area • Find the volume of the block. • A wooden block has a single hole drilled entirely through it. What is the volume of the block? Round to the nearest hundredth. The block is a rectangular prism with a cylindrical hole. To find the volume of the block, subtract the volume of the cylinder from the volume of the prism.

27. Volume & Surface Area • Find the volume of the block. • A wooden block has a single hole drilled entirely through it. What is the volume of the block? Round to the nearest hundredth. The volume of the box is about 72 – 9.42 = 62.58 cubic centimeters.

28. Volume & Surface Area 12 in. 12 in. 12 in. • Find the volume of the cube. • A moving company has boxes of various sizes for packing. The smallest box available has the dimensions shown below. Find the volume of a larger box that is 3 times as large. Answer

29. Volume & Surface Area • Find the volume of the cylinder. • A jumbo-size can of tomato soup is about 3 times the size of a standard-sized can of soup. The standard can has the dimensions shown. Find the surface area and volume of the jumbo-size can. Answer

30. Volume & Surface Area Formula Summary: Rectangular Prism Triangular Prism Cube Cylinder V = s3 V = lwh V = ½bhw V = πr2h V = bhw

31. Volume & Surface Area So What’s The Difference: • Now that we have studied volume it is time to move on to surface area…another important concept dealing with 3D shapes: • When you think about the words • SURFACE AREA • What comes to mind?

32. Volume & Surface Area Surface Area & Nets: • When thinking about surface area we need to be able to break down the 3D solid by its faces…for instance: • If took the above cube and cut along the edges we could open the solid and see that there a total of 6 squares (we call these faces) – the figure on the right is called a net

33. Volume & Surface Area Surface Area & Nets: • Each solid has its own set of faces or NET:

34. Volume & Surface Area Important Surface Area Formulas: Rectangular Prism Cube SA = 6s2 SA = 2lw + 2lh + 2hw

35. Volume & Surface Area Important Surface Area Formulas: Triangular Prism Cylinder SA = 2(½bh) + lw1 + lw2 + lw3 SA = 2πr2 + 2πrh

36. Volume & Surface Area Example 1: Cube • Find the surface area. SA = 6s2 SA = 6(4in)2 SA = 6(16in2) SA = 96in2

37. Volume & Surface Area Example 1: Rectangular Prism • Find the surface area. SA = 2lw + 2lh + 2wh SA = 2(15)(9) + 2(15)(7) + 2(9)(7) SA = 270mm2+ 210mm2 + 126mm2 SA = 606mm2

38. Volume & Surface Area Example 1: Triangular Prism • Find the surface area. SA = 2(½bh) + lw1 + lw2 + lw3 SA = 2(½)(4.5)(3) + (6x3.75) + (6x3.75) + (6x4.5) SA = 13.5in2 + 22.5in2 + 22.5in2 + 27in2 SA = 85.5in2

39. Volume & Surface Area Example 1: Cylinder • Find the surface area. SA = 2πr2+ 2πrh SA = 2(3.14)(3mm)2 + 2(3.14)(3mm)(8mm) SA = 56.52mm2 + 150.72mm2 SA = 207.24mm2

40. Volume & Surface Area • Camping. • A family wants to reinforce the fabric of its tent with a waterproofing treatment. Find the surface area, including the floor, of the tent below. Remember, a triangular prism consists of two congruent triangular faces and three rectangular faces.

41. Volume & Surface Area • Camping. • A family wants to reinforce the fabric of its tent with a waterproofing treatment. Find the surface area, including the floor, of the tent below. bottom left side right side two bases SA = 29 + 36.54 + 36.54 + 29 = 131.08

42. Volume & Surface Area HOMEWORK