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Here are a few recommended readings before getting started with this lesson.
Heichi knows how to calculate the volume of a cylinder. He thinks that the volume of a cone can be found using cylinders. To do so, he makes a cone-shaped mold with a height and radius of r. Then, he fills it with sand and pours it into a cylinder with the same radius and height.
Before proceeding to the volume of a cone, the definition of a cone and its characteristics will be examined.
A cone is a three-dimensional solid with a circular base and a point, called the vertex or apex, that is not in the same plane as the base. The altitude of a cone is the segment that runs perpendicularly from the vertex to the base.
The length of the altitude is called the height of the cone. If the altitude intersects the base at the center, the cone is a right cone. In a right cone, the distance from the vertex to a point on the edge of the base is called the slant height of a cone.Considering Heichi's experiment, the formula for the volume of a cone will be one third of the volume of a cylinder with the same radius and height.
The volume of a cone is one third the product of its base area and its height.
The base area B is the area of the circle and the height h is measured perpendicular to the base.
V=31Bh
Since the base is a circle, its area depends on its radius. Therefore, the base area can also be expressed in terms of the radius r.
V=31πr2h
Number of Cylinders | Volume of Large CylinderSum of Volumes of Stacked Cylinders |
---|---|
4 | ≈0.469 |
16 | ≈0.365 |
64 | ≈0.341 |
256 | ≈0.335 |
1024 | ≈0.334 |
4096 | ≈0.333 |
∞ | 31 |
The Cathedral of Maringá, one of the tallest churches in the world, was designed in the form of a cone by José Augusto Bellucci.
The cathedral reaches 114 meters in height, excluding the cross. Furthermore, its circular base has a radius of 50 meters. Calculate its volume. Round the answer to the nearest cubic meter.The volume of a cone is one third of the product of its base area and height.
r=50, h=114
Calculate power
Multiply
b1⋅a=ba
Use a calculator
Round to nearest integer
Tadeo is learning how to make a traditional Chinese conical hat. He notices that the craftsman uses 20-centimeter bamboo sticks to make the framework.
If the radius of the base is 16 centimeters, find the volume of the hat. Round the answer to the nearest cubic centimeter.
Use the Pythagorean Theorem to find the height of the cone.
To find the volume of the conical hat, its height will be calculated first. Then, the formula for the volume of a cone will be used.
The distance from the vertex of the cone to its base is the height of the cone. Each stick if the frame represents the slant height of the cone. Therefore, the cone has a slant height of 20 centimeters and a radius of 16 centimeters.
These three segments form a right triangle, △ABC. From here, the height AB can be determined by using the Pythagorean Theorem.AC=20, BC=16
Calculate power
LHS−256=RHS−256
Rearrange equation
LHS=RHS
a2=a
r=16, h=12
Calculate power
Multiply
b1⋅a=ba
Use a calculator
Round to nearest integer
The diagram shows a traffic cone, which has a volume of 3128 cubic inches.
The height of the cone part is 28 inches. The prism below it is a square prism with side lengths of 14 inches and height of 1 inch. Find the radius of the cone. Write the answer to the nearest inch.The volume occupied by the traffic cone is the sum of the volume of the prism base and the volume of the cone part.
Vc=2932, h=28
LHS⋅3=RHS⋅3
Commutative Property of Multiplication
LHS/28π=RHS/28π
Rearrange equation
LHS=RHS
a2=a
Use a calculator
Round to nearest integer
For the Jefferson High Science Fair, Ali is thinking about a chemistry experiment in which he will need a cylinder with a radius of 20 centimeters and a height 81 centimeters, with a cone inside. The cylinder must be open on both ends, and the cone must have an open bottom.
To conduct the experiment, Ali needs to answer some questions first. Help him find the answers in order to win the first prize in the fair!
Ali will fill the cone with water. Therefore, the volume of the cone is needed. The height and radius of the cone are the same as the height and radius of the cylinder. Therefore, the height of the cone is 81 centimeters and its radius is 20 centimeters.
The volume of a cone is one third the product of π, the square of the radius, and the height.h=81, r=20
Calculate power
Multiply
Commutative Property of Multiplication
b1⋅a=ba
Calculate quotient
Ali will fill the part of the cylinder not occupied by the cone with foam. Therefore, the volume of this portion is needed. It is given that the height of the cylinder is 81 centimeters and that its radius is 20 centimeters.
The volume of a cylinder is the product of π, the square of the radius, and the height.Cancel out common factors
Simplify quotient
ba=b/10800a/10800
ba=a÷b
Convert to percent
Round to 1 decimal place(s)
Consider a right cone with radius r and slant height ℓ.
The surface area of a right cone is the sum of the base area and the lateral area. The area of the base is given by πr2 and the lateral area is πrℓ.
SA=πr2+πrℓ
Commutative Property of Multiplication
360∘θ⋅ℓ=r
For his experiment for the science fair, Ali plans to make the figure by himself.
The material to be used to create the figure costs $125 per square meter. What is the cost of the material for this object? Write the answer to two decimal places.The area of the lateral surface of a cone is the product of π, the radius, and the slant height. A cylinder's lateral surface area is twice the product of π, the radius, and the height.
The lateral areas of the cylinder and the cone will be calculated one at a time. Then, their sum will be multiplied by the cost per square meter.
h=0.81, r=0.2
Substitute values
Calculate power
Add terms
LHS=RHS
Rearrange equation
ℓ=0.6961, r=0.2
Commutative Property of Multiplication
Tiffaniqua's teacher gives her a piece of paper on which a circle and a sector are drawn. The paper is a square of side length 10 centimeters.
The teacher also gives the following set of information.
Help Tiffaniqua answer the following questions.