Volume and Surface Area

1. Volume and Surface Area Make sure you have your mini-lesson paper in front of you. You will know you need to write something on the notes because it will be underlined.

2. height width length 2-D vs. 3-D Imagine placing a rectangular piece of paper on a table. Two dimensional objects have two dimensions or measurements. This rectangle has a length and a width. Now imagine what would happen if you piled hundreds of papers on top of one another. Three dimensional objects have three dimensions or measurements. This rectangular prism has length, width, and height.

3. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. The sides of a 3-D object are called faces. Three dimensional objects can be unfolded into flat surfaces (2-D shapes). Watch what happens when you unfold a cube. Each face becomes a flat 2-D shape.

4. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Cube Rectangular Prism Triangular Prism Cylinder We have already seen what a cube looks like unfolded. What do you think a rectangular prism, a triangular prism, and a cylinder look like unfolded?

5. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Cube Rectangular Prism Triangular Prism Cylinder What two dimensional shapes do you see in the unfolded objects? Rectangles (squares), triangles, and circles.

6. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Cube Rectangular Prism Triangular Prism Cylinder You already know how to find the areas of these shapes! Area of a rectangle = l x w Area of a triangle = (b x h)/2 Area of a circle = Õr2 Write these down under “Formulas Needed”

7. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. The most difficult part of finding surface area is keeping your work organized. Sometimes shading in areas or drawing a picture of the unfolded object helps. 3 cm 2 cm 5 cm

8. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Start with the front face. The area is 5 cm x 3 cm = 15 cm2 There are two surfaces with the same area. Don’t forget the surface on the back! 3 cm 2 cm The area of the front and back: 15 cm2 x 2 = 30 cm2 5 cm Next, the side face. Last, the top face. The area is 2 cm x 3 cm = 6 cm2 The area is 5 cm x 2 cm = 10 cm2 There are two surfaces with the same area. Don’t forget the surface on the side! There are two surfaces with the same area. Don’t forget the surface on the bottom! The area of both sides: 6 cm2 x 2 = 12 cm2 The area of the top and bottom: 10 cm2 x 2 = 20 cm2

9. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Start with the front face. The area is 5 cm x 3 cm = 15 cm2 There are two surfaces with the same area. Don’t forget the surface on the back! 3 cm 2 cm The area of the front and back: 5 cm 15 cm2 x 2 = 30 cm2 30 cm2 Next, the side face. Last, the top face. The area is 2 cm x 3 cm = 6 cm2 The area is 5 cm x 2 cm = 10 cm2 There are two surfaces with the same area. Don’t forget the surface on the other side! There are two surfaces with the same area. Don’t forget the surface on the bottom! The area of both sides: 6 cm2 x 2 = 12 cm2 12 cm2 The area of the top and bottom: Add all the areas: 10 cm2 x 2 = 20 cm2 20 cm2 + + = 62 cm2

10. Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. 3 cm 2 cm 5 cm The surface area of this rectangular prism is 62 cm2.

11. 2 cm 6 cm 4 cm 5 cm 3 cm Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. Drawing a picture of the unfolded object may make this easier. It doesn’t have to be perfect. 2 cm 3 cm 5 cm 6 cm Make sure you add the labels. 4 cm As you unfold the object in your mind, think carefully about which edges share the same measurements.

12. Starting with the area of the triangle: (4 cm x 5 cm)/2 = 10 cm2 There are two triangles: 10 cm2 x 2 = 20 cm2 2 cm 6 cm 4 cm 5 cm 3 cm Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. 2 cm 3 cm 5 cm 6 cm 4 cm Each of the rectangles has different dimensions. Area of the right rectangle: 6 cm x 2 cm = 12 cm2 Area of the middle rectangle: 6 cm x 4 cm = 24 cm2 Area of the left rectangle: 6 cm x 3 cm = 18 cm2 Add all the areas to find the surface area of the triangular prism: 20 cm2 + 12 cm2 + 24 cm2 + 18 cm2 = 74 cm2

13. 3 cm 5 cm Write this down under “Formulas Needed” Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. 3 cm Drawing a picture of the unfolded object may make this easier. It doesn’t have to be perfect. 5 cm Make sure you add the labels. Cylinders can be tricky because of the edge shared with the circle. Take a piece of paper and roll it into a tube/cylinder. What can be said about the edge of the paper that forms the circle on top and bottom? The length of the edge that forms the circle is the same as the circumference of the circle. Length of rectangle = circumference of circle Circumference of a circle = 2Õr

14. Starting with the area of the circle: 3.14 x 32 = 28.26 cm2 There are two circles: 28.26 cm2 x 2 = 56.52 cm2 3 cm 5 cm Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. 3 cm 5 cm We need to find the length (circumference of the circle) before we can find the area of the rectangle. Length = 2 x 3.14 x 3 = 18.84 cm Area of the rectangle: 18.84 x 5 = 94.2 cm2 Add all the areas to find the surface area of the cylinder: 56.52 cm2 + 94.2 cm2 = 150.72 cm2

15. Volume height width length Volume is the space a three dimensional object fills. We use area to find the space a two dimensional object fills, but a three dimensional object has one more dimension, the height. Let’s look at the growing rectangular prism again. The rectangular prism starts as a flat, 2 dimensional rectangle. Stacking on top of the base rectangle gives us the third dimension of height.

16. Volume To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H H w l Volume is the space a three dimensional object fills. To find the volume of the three dimensional objects we are working with you use: V = B x H Where B is the area of the base shape.

17. Volume To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H To find the volume of a triangular prism: V = [area of a triangle] x H V = [(b x h)/2] x H H H b h w l Volume is the space a three dimensional object fills. To find the volume of the three dimensional objects we are working with you use: V = B x H Where B is the area of the base shape.

18. Volume To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H To find the volume of a triangular prism: V = [area of a triangle] x H V = [(b x h)/2] x H To find the volume of a cylinder: V = (area of a circle) x H V = (Õr2) x H H H r H b h w l Volume is the space a three dimensional object fills. To find the volume of the three dimensional objects we are working with you use: V = B x H Where B is the area of the base shape.

19. Volume To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H To find the volume of a triangular prism: V = [area of a triangle] x H V = [(b x h)/2] x H To find the volume of a cylinder: V = (area of a circle) x H V = (Õr2) x H H H r H b h w l Volume is the space a three dimensional object fills. Please write these formulas on your notes.

20. Volume Volume is a 3-D measurement, so the units need to be cubed. Volume is the space a three dimensional object fills. To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H 4 m 3 m 3 m 1.) Find the area of the base rectangle: l x w = (3 m x 3 m) = 9 m2 2.) Multiply the area of the base rectangle by the height: V = B x H = 9 m2 x 4 m = 36 m3

21. Volume Volume is a 3-D measurement, so the units need to be cubed. Volume is the space a three dimensional object fills. To find the volume of a triangular prism: V = [area of a triangle] x H V = [(b x h)/2] x H 5 ft 3 ft 2 ft 1.) Find the area of the base triangle: (b x h)/2 = (3 ft x 2 ft)/2 = 3 ft2 2.) Multiply the area of the base triangle by the height: V = B x H = 3 ft2 x 5 ft = 15 ft3

22. Volume Volume is a 3-D measurement, so the units need to be cubed. Volume is the space a three dimensional object fills. To find the volume of a cylinder: V = (area of a circle) x H V = (Õr2) x H 6 in 2 in 1.) Find the area of the base circle: Õr2= 3.14 x 2 in x 2 in = 12.56 in2 2.) Multiply the area of the base circle by the height: V = B x H = 12.56 in2 x 6 in = 75.36 in3

23. OK it is time to work on Part A. Remember to attempt to solve the problem first before you watch the solution on the following slide. Show all of your work! Use the formulas you have written on the notes. You may use a calculator for work on this mini-lesson with your home teacher’s permission.

24. 5 in 2 in 10 in 6 in 3 in Try the problem before moving to the next slide. 1.) Find the surface area.

25. Does your work look something like this? 1.) Find the surface area. 5 in 2 in 10 in 6 in 3 in Area of triangle: (2 in x 5 in)/2 = 5 in2 2 in Two triangles: 5 in2 x 2 = 10 in2 Area of rectangle 1: 10 in x 6 in = 60 in2 Area of rectangle 2: 10 in x 5 in = 50 in2 Area of rectangle 3: 10 in x 3 in = 30 in2 5 in 1 2 3 10 in 10 in2 + 60 in2 + 50 in2 + 30 in2 = 150 in2 Surface area = 150 in2 3 in 6 in

26. Try the problem before moving to the next slide. 2.) Find the surface area. 4 cm 8 cm

27. Does your work look something like this? 2.) Find the surface area. 4 cm 8 cm Area of circle: 3.14 x 4 cm x 4 cm = 50.24 cm2 Two circles: 50.24 cm2 x 2 = 100.48 cm2 Area of rectangle : 8 cm x 25.12 cm = 200.96 cm2 4 cm 25.12 cm 100.48 cm2 + 200.96 cm2 = 301.44 cm2 Surface area = 301.44 cm2 8 cm The length is the circumference of the circle: 2 x 3.14 x 4 cm = 25.12 cm

28. Try the problem before moving to the next slide. 3.) Find the volume. 2 m 5 m 6 m

29. Does your work look something like this? 3.) Find the volume. 2 m 5 m 6 m Area of base rectangle: 6 m x 5 m = 30 m2 Area of rectangle multiplied by the height : 30 m2 x 2 m = 60 m3 Volume = 60 m3

30. Try the problem before moving to the next slide. 4.) Find the volume. 5 in 2 in 10 in 6 in 3 in

31. Does your work look something like this? 4.) Find the volume. 5 in 2 in 10 in 6 in 3 in Extra information that you do not need to find the volume. Area of base triangle: (2 in x 5 in)/2 = 5 in2 Area of triangle multiplied by the height : 5 in2 x 10 in = 50 in3 Volume = 50 in3

32. The EndClick on the return button on your browser to go back to the class webpage.