# 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.

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

Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. 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. The sides of a 3-D object are called faces.

Surface Area Surface area is the sum of all the areas on all the surfaces of a three dimensional object. CubeRectangular 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?

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

Surface Area CubeRectangular 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 =  r 2 Write these down under “Formulas Needed” Surface area is the sum of all the areas on all the surfaces of a three dimensional object.

Surface Area 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. 5 cm 2 cm 3 cm Surface area is the sum of all the areas on all the surfaces of a three dimensional object.

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

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

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

Surface Area 4 cm 6 cm 3 cm 2 cm 5 cm 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. 6 cm 2 cm 3 cm 4 cm 5 cm Make sure you add the labels. As you unfold the object in your mind, think carefully about which edges share the same measurements.

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

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. 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? 3 cm 5 cm 3 cm 5 cm 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 = 2  r 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 5 cm 3 cm 5 cm Starting with the area of the circle: 3.14 x 3 2 = 28.26 cm 2 There are two circles: 28.26 cm 2 x 2 = 56.52 cm 2 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 cm 2 Add all the areas to find the surface area of the cylinder: 56.52 cm 2 + 94.2 cm 2 = 150.72 cm 2

Volume Volume is the space a three dimensional object fills. 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. We use area to find the space a two dimensional object fills, but a three dimensional object has one more dimension, the height. length height width

Volume 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. To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H l w H

Volume 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. To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H l w 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

Volume 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. To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H l w 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 To find the volume of a cylinder: V = (area of a circle) x H V = (  r 2 ) x H H r

Volume Volume is the space a three dimensional object fills. Please write these formulas on your notes. To find the volume of a rectangular prism: V = (area of a rectangle) x H V = (l x w) x H l w 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 To find the volume of a cylinder: V = (area of a circle) x H V = (  r 2 ) x H H r

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

Volume 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 2 ft 5 ft 3 ft 1.) Find the area of the base triangle: 2.) Multiply the area of the base triangle by the height: (b x h)/2 = (3 ft x 2 ft)/2 = 3 ft 2 V = B x H = 3 ft 2 x 5 ft = 15 ft 3 Volume is a 3-D measurement, so the units need to be cubed.

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

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.

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

1.) Find the surface area. Does your work look something like this? 6 in 10 in 3 in 5 in 2 in 5 in 10 in 6 in 3 in 2 inTwo triangles: 5 in 2 x 2 = 10 in 2 Area of rectangle 1: 10 in x 6 in = 60 in 2 Area of rectangle 2: 10 in x 5 in = 50 in 2 Area of rectangle 3: 10 in x 3 in = 30 in 2 213 10 in 2 + 60 in 2 + 50 in 2 + 30 in 2 = 150 in 2 Surface area = 150 in 2 Area of triangle: (2 in x 5 in)/2 = 5 in 2

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

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

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

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

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

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

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