6th Grade - Volume Of Prisms And Cylinders

 
     
 
     
 
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6th
Geometry
Volume of Prisms and Cylinders
Know and use the formulas for the volume of triangular prisms and cylinders (area of base x height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
Understand that volume is the measure of the amount inside a solid 3-D figure. Volume can be calculated by counting the number of cubic units needed to fill the 3-D object. Know how to find the volume of 3-D shapes by finding the area of the base and multiplying it by the height of the figure (B x h, where B represents the area of the base and h represents the height of the figure). Use the area formula for a rectangle to find the area of the base of a rectangular prism and multiply the result by the height of the figure. Understand that the formula V = l x w x h can be used to find the volume of a rectangular solid. Understand that the formula for the volume of a rectangular prism can be used to find the volume of a triangular prism. The area of a triangle is ½ the size of a rectangle, so the volume of a rectangular prism can be used and divided by 2 to find the volume of a triangular prism. To find the volume of a triangular prism use: V = lwh ÷ 2, where l and w can represent base and height of the base. Calculate the volume of a cylinder by using the area formula π r 2 and then multiplying the result by the height of the figure (V = π r 2 x h).
 

Sample Problems

(1)

Solve to find the volume of a rectangular prism with the length of 5 cm, width of 4 cm and height of 11 cm.

(220 cm3 )

(2)

Find the missing value for the rectangular prism, if:

Volume: 56 cm 2

Length: 4 cm

Width: 7 cm

Height: ______

(2 cm)

(3)

Find the volume of the triangular prism, where the triangular base has a base of 3 cm and height of 5 cm. The height of the prism is 10 cm.

(75 cm3 )

(4)

Find the volume of a cylinder with a circular base with the radius of 7 inches and a height of 9 inches. Round your answer to the nearest one.

(1385 in. 3 )


(5)

Find the volume of a cylinder with a circular base with the diameter of 10 inches and a height of 12 inches.

(942 in. 3 )

Learning Tips

(1)

Use solid, 3-D shapes around the house to find their bases (i.e. cans can represent cylinders and cereal boxes rectangular prisms). It’s important to understand that prisms always have only two bases. Also, these bases will have the same area. The rest of the shapes make up the sides of the figure. The sides of a prism are always rectangular or square. Practice finding and pointing to the bases and faces. This will be particularly important when looking at triangular prisms. Drawings of a 3-D shape can be turned in a variety of directions. For example, a can or a cereal box can be turned on its side. It is very important to be able to find the bases of a figure no matter which way the figure is placed.

(2)

3-D shapes from around the house can also be used to practice finding the volume of a figure. For example, the volume of a soup containter can be found by measuring the radius of the base (the measure from the center point to the edge of the circle) and the measure of the height of the can. If you found that the radius of the can lid was 6 cm and the height was 12 cm, you would input those numbers into the formula to solve to find volume. The same can be done for a box, using the rectangular prism formula, by measuring the length and width of the base and the height of the box.

(3)

It’s a good idea to review how to use a formula. Our basic formula for finding volume of any solid figure is B x h. In this formula the B stands for the measure of the area of the base and the h stands for the height of the figure. Your first step will always be to find the area of the base of the figure. The area formulas that will be required for sixth grade volume are: rectangle (A=lw), triangle (A=lw/2) and circle (A = π r 2) . The area for a rectangle will be the first step used to solve for the volume of a rectangular prism. The area for a triangle will be used as a first step to find the volume of a triangular prism. Lastly, the area of a circle will be used to calculate the volume of a cylinder. After finding the area of the base, you simply need to multiply that figure by the height of the solid. Finally, you will write your final answer in cubic units. Since we are measuring the inside of a solid shape, it is being measured in cubic units (3-D), and the units must be shown cubed, to the third power. Here are example of how solve to find the volume for each type of figure.

Rectangular Prism - (Base l = 5 m, w = 3m) (height, h = 10m)

1. V = B x h

2. V = (l x w) x h

Step 2 Explanation: l x w is substituted in for B because this is the formula used to find the area of the base.

3. V = 5 x 3 x 10

Step 3 Explanation: substitute in the value for each variable as given in the problem.

4. V = 15 x 10

Step 4 Explanation: Solve by using the order of operations. Since both operations are multiplication you move from left to right to solve.

5. V = 150 m3

Step 5 Explanation: 15 x 10 = 150, so this is our final calculation. However, you must not forget to write in the unit (m, meters) and cube it (put it to third power).

Triangular Prism – (BASE b = 4, h = 5) (height, h = 10)

1. V = B x h

Triangular prisms can be confusing because there will be two heights. The area of triangle is measured by the base x height of that triangle. This is similar to length x width of a rectangle. The two height may be confusing, but it is important to remember that all that you’re truly asked to do to solve for volume is multiply all three amounts and then divide by two.

2. V = (b x h † 2) x h

Step 2 Explanation: The B is replaced with the area formula for a triangle.

3. V = (4x5/2) x 10

Step 3 Explanation: All variables are replaced with their values from the original problem.

4. V = (20/2) X 10

5. V = 10 x 10

6. V = 100 cm3

Steps 4 – 6 Explanation: Solve by using the order of operations, moving left to right. Don’t forget to cube your units (cm) in the final answer.

Cylinder – (BASE radius – 10 ft) (height, h = 6 ft)

1. V = B x h

2. V = (π r 2) x h

Step 2 Explanation: The B is replaced with the area formula for a circle.

3. V = (3.14 x 10 2) x 6

Step 3 Explanation: The value of each variable is substituted into the problem. The symbol π has an estimated value worth 3.14 or 22/7. This is a constant and will always be used to solve for the area of a circle. The bolded values were taken from the original problem.

4. V = (3.14 x 100) x 6

Step 4 Explanation: The order of operations requires exponents to be solved before multiplication. 10 2 is requiring the solution of an exponent. Exponents are solved by multiplying the base number,10, by itself, so 10 x 10 = 100.

5. V = 314 x 6

Step 5 Explanation: The order of operations requires all numbers within grouping symbols, parenthesis to be solved. Hence 3.14 x 100 = 3.14.

6. V = 1,884 ft3

Step 6 Explanation: Solve by using the order of operations, moving left to right. Don’t forget to cube your units (ft) in the final answer.


(4)

You’ll need to be able to name a figure before you can find its volume. This is the only way to solve using the proper formulas. A cylinder is the easiest to name. A cylinder will always have two circular bases and a curved face. A rectangular prism will always have two rectangular bases and four rectangular faces. A triangular prism will always have two triangular bases and four rectangular faces. It is easiest to name a solid figure by finding the base shape. If the shape of the two bases triangles, this will give us the first name of the figure, triangular. We know its last name is prism because all prisms have two bases. Figures with one base and a point are pyramids. Sixth grades are not required by the Standards to calculate the volume of pyramids, so that will not be a focus here. However, it is helpful to understand the difference between the types of figures for upcoming grades.

(5)

Here are a few short cuts that can be memorized for finding volume.

Rectangular prism – multiply all three numbers together.

Triangular prism – multiply all three numbers together, then divide by 2.

Cube – put the side to the third power, multiply the side by itself three times

Extra Help Problems

(1)

Solve to find the volume of a rectangular prism, where:

l = 6 cm w= 5 cm h = 8 cm

(240 cm3 )

(2)

Solve to find the volume of a rectangular prism, where:

l = 12 in. w= 11 in. h = 16 in.

(2,112 in. 3 )

(3)

Solve to find the volume of a rectangular prism, where:

l = 17 ft w= 4 ft h = 2 ft

(136 ft3 )

(4)

Solve to find the volume of the rectangular prism. Round your answer to the nearest tenth.

l = 4.5 mm w= 3 mm h = 4.25 mm

(57.38 mm3 )

(5)

Solve to find the volume of the rectangular prism. Write your answer in fraction form.

l = 5 ½ cm w= 2 ¼ cm h = 10 cm

(123-3/4 cm3 )

(6)

Solve to find the volume of a triangular prism, where:

b = 5 m h = 11 m H = 10 m

(275 m3 )

(7)

Solve to find the volume of a triangular prism, where:

b = 6 m h = 3 m H = 17 m

(153 m3 )


(8)

Solve to find the volume of a triangular prism, where:

b =12 m h = 4 m H = 7 m

(168 m3 )

(9)

Solve to find the volume of a triangular prism, where:

b = 9.25 m h = 11.5 m H = 10.4 m

(553.15 m3 )

(10)

Solve to find the volume of a triangular prism, where:

b = 7 ½ m h = 4 ½ m H = 5 m

(168 ¾ m3 )

(11)

Solve to find the volume of a cylinder, where:

r = 4 in. h = 5 in.

(251.2 in. 3 )

(12)

Solve to find the volume of a cylinder, where:

r = 14 ft. h = 12 ft.

(7,385.28 ft3)

(13)

Solve to find the volume of the cylinder. Round your answer to the nearest tenth.

r = 2.2 mm h = 4.5 mm

(68.4 mm )

(14)

Solve to find the volume of a cylinder, where:

r = 8 1/3 cm h = 4 cm

(873 1/63 cm3 )

(15)

Solve to find the volume of a cylinder, where:

d= 12 yd h = 5 yd

(565.2 yd3 )

(16)

Solve to find the volume of a cylinder, where:

d= 7 yd h = 12 yd

(461.58 yd3 )

(17)

Solve to find the volume of the cylinder. Round your anwer to the nearest whole number.

d= 2 ½ yd h = 6 ½ yd

(32 yd3 )


(18)

Sandie has a box with two triangular bases and 4 rectangular faces. How would Sandie find the volume of the box?

(She would need to measure the base and height of the triangle on one base. Then she would need to measure the height of the box along one of the rectangular faces. After she found all her measures, she would multiply the base x height and then divide by 2. Her last step would be to multiply her quotient by the height of the box.)

(19)

Gary wants to know how many 1 cubic inch blocks could fit in his shoebox. What formula could help Gary find this?

(lxwxh)

(20)

Melissa has an empty coffee can that she’d like to fill. If the bases are circular with a radius of 12 cm. How could Melissa find the amount she could fill the can with?

(She would need to use the formula V = (π r 2) x h. To find V, she would replace with 3.14, r with 12 and h with the measure of the height of the can. She would begin by squaring 12, then, multiply the answer by 3.14. Lastly, she would multiply the product by the height of the can.)

(21)

Alex is building an igloo with ice blocks. Each block is 30 cm x 15 cm x 40 cm. What is the volume of each block of ice?

(18,000 cm 3)

(22)

Timmy has a fish tank shaped as a cylinder. The radius of the tank is 3 ft. The height of the tank is 4 feet. What is the volume of the tank?

(113.04 ft. 3 )

(23)

Nina’s mom bought a block of swiss cheese that is cut into a triangular solid. The base and height of the triangle is 8 cm. The height of the cheese block is 4 cm. How much cheese did she buy?

(128 cm3 )

(24)

The volume of a flower vase with a circular base is 452.16 cubic inches. Write a formula to show the height of the vase, if the radius is 6 inches. Use the formula to find the height of the vase.

(V = r 2) x h; 452.16 in. 3 = 3.14 x 62 x h; h =4)

(25)

A moving box has a volume of 300 cubic inches. The length and width of the base of the box are 10 inches. Write an equation that can be used to find the height of the box. Use the formula to solve to find the height.

(V=lwh; 300 = 10 x 10 x h; h = 3)

 

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