What do cylinders and spheres have in common

It is usually solved today using slicing techniques from integral calculus. Slicing techniques in calculus exploit the idea we saw when finding the volume of a prism or a cylinder. If we can find the volume of a typical slice of the solid, then, assuming the solid has uniform cross-section, we can add all the slices to find the volume. We can then integrate this to obtain the total volume. This method can be used very effectively to find the volume of solids which do not have uniform cross-section, and may have curved boundaries.

Those familiar with integral calculus will recognize the following formula for the volume of such a solid of revolution. A sphere of radius can then be formed by rotating this circle about the x -axis.

Thus, its volume is found by computing the integral. The portion of a right cone remaining after a smaller cone is cut off is called a frustum. Suppose the top and bottom of a frustum are circles of radius R and r , respectively, and that the height of the frustum is h , while the height of the original cone is H.

The volume of the frustum is by the difference of the volumes of he two cones and is given by. Similarly, it can be shown that the surface area of the frustum of a cone with base radii r and R and slant height s , is given by.

The concepts and insights developed in finding the formulas for areas and volumes are used in Physics and Engineering to find such quantities as the centre of mass and the moment of inertia of a solid body. Thus the development of volume formulas are important for students, as is the careful memorizing of the key formulas, such as the volume of a sphere.

Archimedes gave a geometric demonstration that the surface area of a sphere with radius r was equal to the area of the curved surface of the cylinder into which the sphere exactly fits. Take a hemisphere of radius and look at the area of a typical cross-section at height above the base. Also consider a cylinder of height h and radius r , with a cone of the same height and radius removed. The later Greek Mathematician Pappus AD discovered the following remarkable method for finding the volume of a solid of revolution generated by rotating a plane region with area A about a fixed axis.

The volume of the solid is equal to the product of the area A of the plane region and the distance travelled by its geometric centroid as it moves through one revolution. The centroid is another word for centre of mass. This method can be used to find the volume of a torus a Latin word meaning couch , which is the solid obtained by rotating a circle about a line external to the circle. Suppose that we rotate a circle of radius r about a vertical line whose distance from the centre of the circle is R.

Pappus also showed that the surface area of a solid of revolution is equal to the product of the perimeter of the plane region being rotated and the distance travelled by its geometric centroid as it moves through one revolution. Little further progress was made with volumes and surface areas until the development of the Calculus. The theory of several variable calculus enables more complicated volumes and surface areas to be calculated and the concept of area and volume to be generalised to higher dimensions.

Modern analysis is that branch of mathematics and studies and develops ideas from calculus. Using calculus, the theory of integration generalises the notions of area and volume. Measure Theory, a branch of modern analysis beginning the work of Henri Lebesgue, d.

In this appendix, we will show how to extend this result to any rectangular based pyramid. We will use the second principle to show that the formula holds for any square pyramid of height h. Take two square pyramids of height h , one with base square length 2 h and one with base square length 2 l. From our earlier discussion, we know that the volume of the first pyramid is. Let V 2 be the volume of the second pyramid.

Hence the area of the cross-section for the first pyramid is given by 4 a 2 but this equals 4 d 2. Thus the area of the cross-section is 4 b 2 but this equals 4 l 2 d 2. This gives us. Take slices at a distance d from the vertex in each pyramid. It remains to say that a similar method can be used to progress from a pyramid with rectangular base to one whose base is a regular polygon, although the technical details are more complicated and will not be given here.

Katz, Addison-Wesley, The ratios of the areas of the cross-sections taken at the same heights is 4x 2 : cd. Substituting in the above formula gives the result. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. Solution a The radius is 15 m. Although either method works, there are reasons students might choose one over the other. It might make more sense to students, however, since it describes the video they saw in that the volume of the sphere is the difference between the volumes of the cylinder and cone.

How good of an approximation do you think this is? Can you come up with a better one? Record and display student-created volume of a sphere approximation formulas for all to see.

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs. If we filled the cone and sphere with water, and then poured that water into the cylinder, the cylinder would be completely filled. That means the volume of the sphere and the volume of the cone add up to the volume of the cylinder.

Lesson 20 The Volume of a Sphere. Give students 1—2 minutes of quiet work time, followed by a whole-class discussion. Student Facing. Here is a method for quickly sketching a sphere: Draw a circle. Draw an oval in the middle whose edges touch the sphere. Student Response. Activity Synthesis. Identify students who discuss either method for calculating the volume of the sphere: subtract the volume of the cone from the volume of the cylinder. Arrange students in groups of 2.

Display for all to see: A sphere fits snugly into a cylinder so that its circumference touches the curved surface of the cylinder and the top and bottom touch the bases of the cylinder. Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem.

For example, ask students to color code the radii and heights in each of the figures and when calculating volume. Supports accessibility for: Visual-spatial processing. Use this routine to compare and contrast the different ways students calculated the volume of the sphere. Ask students to consider what is the same and what is different about each method used.

In this discussion, emphasize language used to help students make sense of strategies used to calculate the volume of the sphere. Design Principle s : Maximize meta-awareness; Support sense-making. But size is not common that the lip balm is smaller than the glue stick. Yes, except for all the spheres, pyramids, cones, rectangular boxes, egg shapes, cylinders, etc. The answer depends on information which has not been provided: the height of the cylinder and cone will affect their surface areas.

Log in. Math and Arithmetic. Study now. See Answer. Best Answer. Study guides. Algebra 20 cards. A polynomial of degree zero is a constant term. The grouping method of factoring can still be used when only some of the terms share a common factor A True B False. The sum or difference of p and q is the of the x-term in the trinomial. A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials.

J's study guide 1 card. What is the name of Steve on minecraft's name. Steel Tip Darts Out Chart 96 cards. Q: What do spheres and cylinders have in common? Write your answer Related questions. How are cylinders different from spheres? Why aren't cylinders and cones and spheres polygons? How are cylinders and spheres the same? How are cylinders and spheres alike?

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