Apply Cavalieri's principle to determine when two solids must definitely have the same volume with this interactive video from the School Yourself Geometry series.

Content Standard(s):

Mathematics MA2015 (2016) Grade: 9-12 Geometry

35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [G-GMD1]

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.HS.35- Make a prediction about the volume of a container, the area of a figure, or the perimeter of a figure. Ex: how many cubes will go in one figure vs. another. Limit to cylinder, circle.

Mathematics MA2015 (2016) Grade: 9-12 Precalculus

38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

Mathematics MA2019 (2019) Grade: 8

30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.

Unpacked Content

Evidence Of Student Attainment:

Students:

Understand that the application of volume formulas and the relationship between these three formulas can be used in combinations when determining solutions involving real-world cylinders, cones, and spheres.

Teacher Vocabulary:

Radius

Pi

Volume

Cylinder

Cone

Sphere

Knowledge:

Students know:

The volume formulas for cylinders, cones, and spheres.

That 3.14 is an approximation of pi commonly used in these volume formulas.

That composite three dimensional objects in the real-world can be created by combining cylinders, cones, and spheres in part or whole.

Skills:

Students are able to:

Calculate the volume of cones, cylinders, and spheres given in real-world contexts. often times approximating solutions to a specified decimal place.

Identify the components of a composite figure as being portions of or whole cylinders, cones, and spheres.

Combine the results of calculations to find volume for real-world composite figures.

Understanding:

Students understand that:

the application of volume formulas and the relationship between these three formulas can be used in combinations when determining solutions involving real-world cylinders, cones, and spheres.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.8.30.1: Define formula, volume, cone, cylinders, spheres, and height.
M.8.30.2: Discuss the measure of volume and give examples.
M.8.30.3: Solve problems with exponents, with or without a calculator.
M.8.30.4: Recall how to find circumference of a circle, with or without a calculator.
M.8.30.5: Identify parts of a circle.
M.8.30.6: Calculate the volume of three-dimensional figures.
M.8.30.7: Solve real-world problems using the volume formulas for three-dimensional figures, with or without a calculator.

Prior Knowledge Skills:

Define volume, surface area, triangles, quadrilaterals, polygons, cubes, and right prisms.

Discuss strategies for solving real-world mathematical problems.

Recall formulas for calculating volume and surface area.

Identify the attributes of triangles, quadrilaterals, polygons, cubes, and right prisms.

Define diameter, radius, circumference, area of a circle, and formula.

Identify and label parts of a circle.

Recognize the attributes of a circle.

Apply the formula of area and circumference to real-world mathematical situations.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.8.30 Use the formulas for perimeter, area, and volume to solve real-world and mathematical problems (where volume problems are limited to finding the volume of cylinders and rectangular prisms).

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

Unpacked Content

Evidence Of Student Attainment:

Students:

(17a) Given a sphere,

Explain how surface area is the total area for the surface of a sphere, and that if we could "unroll" the sphere and show it as a rectangle, the rectangle would have a width that is equivalent to the diameter of the sphere. Its length would be the same as the circumference of the sphere.

Explain how we could find the volume of spheres by using pyramids., understanding the radius of the sphere would be the height of the pyramid.

Given a cylinder, explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

Given a pyramid or cone, explain that the shapes could be divided into cross-sections, and the area of the cross-sections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone.

(17b) Given a formula, explain how to solve for the missing linear dimension using opposite operations.

Teacher Vocabulary:

Dissection arguments

Principle

Cylinder

Pyramid

Cone

Ratio

Circumference

Parallelogram

Limits

Conjecture

Cross-section

Surface Area

Knowledge:

Students know:

Techniques to find the area and perimeter of parallelograms,
Techniques to find the area of circles or polygons

Skills:

Students are able to:

Accurately decompose circles, spheres, cylinders, pyramids, and cones into other geometric shapes.

Explain and justify how the formulas for surface area, and volume of a sphere, cylinder, pyramid, and cone may be created from the use of other geometric shapes.

Understanding:

Students understand that:

Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.

Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.17.1: Define Cavalieri's principle, circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone; oblique, radius, diameter, height, and base.
GEO.17.2: Compare surface areas of similar figures and volumes of similar figures to determine a relationship using dissection arguments, Cavalieri's principle, and informal limit arguments.
GEO.17.3: Compare the characteristics and volume of oblique and right solids.
GEO.17.4: Describe the properties of a given object (cylinder, pyramid, prism, and cone).
GEO.17.5: Identify the necessary characteristics of a given solid to solve for its volume and surface area(cylinder, pyramid, prism, and cone).
GEO.17.6: Calculate the surface area of three-dimensional figures (cylinder, pyramid, prism, and cone).
GEO.17.7: Calculate the volume of a cylinder, pyramid, prism, and cone.
GEO.17.8: Calculate the area of a circle.
GEO.17.9: Calculate the circumference of a circle.
GEO.17.10: Calculate the area of the base shape.
GEO.17.11: Identify the relationship of geometric representations to real-life objects.
GEO.17.12: Identify the base shape.

Prior Knowledge Skills:

Define three-dimensional figures, surface area, and nets.

Identify three-dimensional figures.

Evaluate how to apply using surface area of a three-dimensional figure to solving real-world and mathematical problems.

Draw nets to find the surface area of a given three-dimensional figure.

Recall how to calculate the area of a rectangle and triangle.

Select and create a three-dimensional figure using manipulatives.

Define diameter, radius, circumference, area of a circle, and formula.

Identify and label parts of a circle.

Recognize the attributes of a circle.

Apply the formula of area and circumference to real-world mathematical situations.

Define formula, volume, cone, cylinders, spheres, and height.

Discuss the measure of volume and give examples.

Solve problems with exponents.

Recall how to find circumference of a circle.

Identify parts of a circle.

Calculate the volume of three-dimensional figures.

Solve real-world problems using the volume formulas for three-dimensional figures.

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.10.17 Compare and contrast the volume of real-world geometric figures.