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Volume of Cylinders, Cones & Spheres
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 How to find the volume of cones, cylinders, and spheres.
 How their volumes are related to one another.
 How this knowledge can help us bake a cake, take care of pets, and go for a run!

Discussion Questions

Before VideoWhat is volume? What units do you use for volume?
ANSWER
Volume measures the amount of space inside a 3D object. It is measured in cubic units.

42 cm^{3}. I used the formula volume equals length times width times height.
ANSWER
Pi is an irrational number. It is approximately 3.14. It is the ratio between the circumference and the diameter of a circle. It is represented by the symbol π.

A radius is a line from the center of a circle to its edge. A diameter is a line across a circle that passes through the center of the circle.

25π cm^{2} or about 78.5 cm^{2}


After VideoWhat is the formula for the volume of a cylinder? What does each part of the formula mean?ANSWER

V = πr^{2}h. r is the radius of the circular base of the cylinder and h is the height of the cylinder (the distance between the two circular ends).

V = [ggfrac]1/3[/ggfrac]πr^{2}h. r is the radius of the circle at the base and h is the distance between the center point of the circular base and the point at the top of the cone.

V = [ggfrac]4/3[/ggfrac]πr^{3}. If I cut the sphere exactly in half I could see a circle on the face. r is the radius of the circle at the very center of the sphere.

360π cm^{3} or about 1,130.4 cm^{3}.

The cylinder.



Vocabulary

Volume
DEFINE
The amount of space inside a 3D object.

Cone
DEFINE
A 3D shape with one circular base.

Cylinder
DEFINE
A 3D shape with two circular bases.

Sphere
DEFINE
A 3D shape in the shape of a ball.

Pi (π)
DEFINE
The ratio of the circumference of a circle to its diameter. This is always the same number: approximately 3.14. The actual number is an irrational number (a decimal that never ends or repeats).

Radius
DEFINE
The measurement from the center of a circle or sphere to its edge. The radius is half the diameter.

Diameter
DEFINE
The distance straight across a circle or sphere. The diameter always passes through the center point of the circle.

Volume
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageHOW DO YOU FIND THE VOLUME OF CYLINDERS, CONES & SPHERES?Volume is the space inside a 3D solid. Volume is measured in cubic units. Formulas are rules that are written using mathematical symbols that relate quantities. You have already used volume formulas to find the volume of rectangular prisms and cubes. You can also use volume formulas to find the volumes of cones, cylinders, and spheres.
To better understand how to find the volume of cylinders, cones, and sphere…
HOW DO YOU FIND THE VOLUME OF CYLINDERS, CONES & SPHERES?. Volume is the space inside a 3D solid. Volume is measured in cubic units. Formulas are rules that are written using mathematical symbols that relate quantities. You have already used volume formulas to find the volume of rectangular prisms and cubes. You can also use volume formulas to find the volumes of cones, cylinders, and spheres. To better understand how to find the volume of cylinders, cones, and sphere…LET’S BREAK IT DOWN!
Review Volume
You can find the volume of a rectangular prism by seeing how many cubic units fit inside it. The number of unit cubes that fit inside tells you the volume. You can also use a formula. The volume of a rectangular prism is length times width times height. If the length is 3 units, the width is 2 units, and the height is 4 units, the volume is 3 × 2 × 4 = 24 cubic units. Try this one yourself: A rectangular prism has length 5 units, width 6 units, and height 2 units. What is the volume of the rectangular prism?
Review Volume You can find the volume of a rectangular prism by seeing how many cubic units fit inside it. The number of unit cubes that fit inside tells you the volume. You can also use a formula. The volume of a rectangular prism is length times width times height. If the length is 3 units, the width is 2 units, and the height is 4 units, the volume is 3 × 2 × 4 = 24 cubic units. Try this one yourself: A rectangular prism has length 5 units, width 6 units, and height 2 units. What is the volume of the rectangular prism?Volume of a Cylinder
How can you find the volume of a cake in the shape of a cylinder? Think about this shape by comparing it to a rectangular prism. The formula for the volume of a rectangular prism is length times width times height. The length times the width is the area of the base. So, another way to think of this formula is the area of the base times the height. You can think of the volume of a cylinder in the same way: the area of the base times the height. The base is a circle. A circle has area πr^{2}. So, the formula for the volume of a cylinder is πr^{2}h. Amari’s cake is a cylinder with diameter 44 cm and height 50 cm. What is the volume of the cake? The radius is half the diameter, so the radius is 22 cm. Use the formula. 3.14 × 22^{2} × 50 = 75,988 cubic centimeters. Remember that when you use 3.14 to approximate pi, your answer is an approximation. Try this one yourself: A cylinder has height 20 cm and diameter 16 cm. What is the volume of the cylinder?
Volume of a Cylinder How can you find the volume of a cake in the shape of a cylinder? Think about this shape by comparing it to a rectangular prism. The formula for the volume of a rectangular prism is length times width times height. The length times the width is the area of the base. So, another way to think of this formula is the area of the base times the height. You can think of the volume of a cylinder in the same way: the area of the base times the height. The base is a circle. A circle has area πr2. So, the formula for the volume of a cylinder is πr2h. Amari’s cake is a cylinder with diameter 44 cm and height 50 cm. What is the volume of the cake? The radius is half the diameter, so the radius is 22 cm. Use the formula. 3.14 × 222 × 50 = 75,988 cubic centimeters. Remember that when you use 3.14 to approximate pi, your answer is an approximation. Try this one yourself: A cylinder has height 20 cm and diameter 16 cm. What is the volume of the cylinder?Volume of a Cone
How can you find the volume of a plastic cup in the shape of a cone? Compare the cone to a cylinder with the same diameter and same height. If you fill the cone with water and pour it into the cylinder, it takes exactly 3 cones full of water to fill the cylinder. The volume of the cone is one third the volume of the cylinder. Remember, the formula for the volume of a cylinder is πr^{2}h. So, the formula for the volume of a cone is [ggfrac]1/3[/ggfrac]πr^{2}h. Emily has a cone cup. The height is 10 cm and the diameter is 6 cm. What is the volume of the cup? The radius is half the diameter, or 3 cm. Substitute the numbers into the volume formula. The volume of the cone cup is approximately 94.2 cubic centimeters. Try this one yourself: A cone has height 14 cm and diameter 12 cm. What is the volume of the cone?
Volume of a Cone How can you find the volume of a plastic cup in the shape of a cone? Compare the cone to a cylinder with the same diameter and same height. If you fill the cone with water and pour it into the cylinder, it takes exactly 3 cones full of water to fill the cylinder. The volume of the cone is one third the volume of the cylinder. Remember, the formula for the volume of a cylinder is πr2h. So, the formula for the volume of a cone is [ggfrac]1/3[/ggfrac]πr2h. Emily has a cone cup. The height is 10 cm and the diameter is 6 cm. What is the volume of the cup? The radius is half the diameter, or 3 cm. Substitute the numbers into the volume formula. The volume of the cone cup is approximately 94.2 cubic centimeters. Try this one yourself: A cone has height 14 cm and diameter 12 cm. What is the volume of the cone?Volume of a Sphere
How can you find the volume of a sphere? Compare the sphere to a cone with the same radius and the same height. If you fill the cone with water and pour it into the sphere, it takes exactly 2 cones full of water to fill the sphere. The volume of the sphere is twice the volume of the cone. Remember, the formula for the volume of a cone was [ggfrac]1/3[/ggfrac]πr^{2}h. So, [ggfrac]1/3[/ggfrac]πr^{2}h + [ggfrac]1/3[/ggfrac]πr^{2}h is the volume of a sphere. You can make this formula simpler. The height of the sphere, h, is equal to the diameter, or 2r. Now you have [ggfrac]1/3[/ggfrac]πr^{2}(2r) + [ggfrac]1/3[/ggfrac]πr^{2}(2r). You can simplify this expression. Since you can multiply in any order, move the 2s to the front. Remember, r2 times r is r3. Now you have [ggfrac]2/3[/ggfrac]πr^{3} + [ggfrac]2/3[/ggfrac]πr^{3}. Add these parts together: [ggfrac]4/3[/ggfrac]πr^{3}. So, the formula for the volume of a sphere in its simplest form is [ggfrac]4/3[/ggfrac]πr^{3}. You could use another version of the formula, but this version is simplest. Use this formula to solve a problem. A fishbowl in the shape of a sphere has diameter 20 cm. What is the volume of the fishbowl? The radius of the fishbowl is half the diameter, so it is 10 cm. Substitute that number into the formula. The volume of the fishbowl is approximately 4,187 cm^{3} Try this one yourself: A sphere has diameter 18 cm. What is the volume of the sphere?
Volume of a Sphere How can you find the volume of a sphere? Compare the sphere to a cone with the same radius and the same height. If you fill the cone with water and pour it into the sphere, it takes exactly 2 cones full of water to fill the sphere. The volume of the sphere is twice the volume of the cone. Remember, the formula for the volume of a cone was [ggfrac]1/3[/ggfrac]πr2h. So, [ggfrac]1/3[/ggfrac]πr2h + [ggfrac]1/3[/ggfrac]πr2h is the volume of a sphere. You can make this formula simpler. The height of the sphere, h, is equal to the diameter, or 2r. Now you have [ggfrac]1/3[/ggfrac]πr2(2r) + [ggfrac]1/3[/ggfrac]πr2(2r). You can simplify this expression. Since you can multiply in any order, move the 2s to the front. Remember, r2 times r is r3. Now you have [ggfrac]2/3[/ggfrac]πr3 + [ggfrac]2/3[/ggfrac]πr3. Add these parts together: [ggfrac]4/3[/ggfrac]πr3. So, the formula for the volume of a sphere in its simplest form is [ggfrac]4/3[/ggfrac]πr3. You could use another version of the formula, but this version is simplest. Use this formula to solve a problem. A fishbowl in the shape of a sphere has diameter 20 cm. What is the volume of the fishbowl? The radius of the fishbowl is half the diameter, so it is 10 cm. Substitute that number into the formula. The volume of the fishbowl is approximately 4,187 cm3 Try this one yourself: A sphere has diameter 18 cm. What is the volume of the sphere?Relationship Between Cones, Cylinders, and Spheres
A restaurant serves smoothies in containers in three different shapes: cones, cylinders, and spheres. All three containers have the same radius and height. All of them are the same price. Which shape container should you order if you want to get the greatest amount of smoothie? Remember, 1 cylinder has the same volume as 3 cones. 1 sphere has the same volume as 2 cones. The cylinder container gives you the greatest amount of smoothie. Try this one yourself: A cylinder and a cone have the same radius. They hold the same amount of water. How many times as tall as the cylinder is the cone?
Relationship Between Cones, Cylinders, and Spheres A restaurant serves smoothies in containers in three different shapes: cones, cylinders, and spheres. All three containers have the same radius and height. All of them are the same price. Which shape container should you order if you want to get the greatest amount of smoothie? Remember, 1 cylinder has the same volume as 3 cones. 1 sphere has the same volume as 2 cones. The cylinder container gives you the greatest amount of smoothie. Try this one yourself: A cylinder and a cone have the same radius. They hold the same amount of water. How many times as tall as the cylinder is the cone? 
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