View Standards
**Standard(s): **
[MA2015] GEO (9-12) 5 :

[MA2015] GEO (9-12) 35 :

[MA2015] PRE (9-12) 38 :

[MA2019] REG-7 (7) 19 :

5 ) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [G-CO5]

[MA2015] GEO (9-12) 35 :

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]

[MA2015] PRE (9-12) 38 :

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

[MA2019] REG-7 (7) 19 :

19. Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections.

[MA2019] REG-8 (8) 30 : 30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.

[MA2019] GEO-19 (9-12) 16 : 16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

[MA2019] GEO-19 (9-12) 17 : 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.

Use the Pythagorean theorem to unroll a cone and find its surface area with this interactive video from the School Yourself Geometry series.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 35 :

[MA2015] PRE (9-12) 38 :

[MA2019] REG-8 (8) 30 :

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]

[MA2015] PRE (9-12) 38 :

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

[MA2019] REG-8 (8) 30 :

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

[MA2019] GEO-19 (9-12) 17 : 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.

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.

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]

[MA2015] PRE (9-12) 38 :

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

[MA2019] REG-8 (8) 30 :

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

[MA2019] GEO-19 (9-12) 17 : 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.

How can you find the volume of prisms that aren't rectangular? Learn how with this interactive video from the School Yourself Geometry series.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 35 : 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]

In this topic, Lesson 1 shows how finding the area of a curved figure can be approximated by rectangles and triangles. By refining the size of the rectangles and triangles, the approximation of the area becomes closer to the actual area. Students experience a similar process of approximation to estimate in Grade 8, Module 7, Lesson 14. The informal limit argument prepares students for the development of volume formulas for cylinders and cones and foreshadows ideas that students formally explore in calculus. This process of approximation is important to developing the volume formula of cylinders and cones. In Lesson 2, students study the basic properties of the area using set notation; Topic B will examine how the properties are analogous to those of volume.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 35 : 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]

[MA2019] GEO-19 (9-12) 16 :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.

[MA2019] GEO-19 (9-12) 16 :

16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

[MA2019] GEO-19 (9-12) 17 : b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

Students study the basic properties of two-dimensional and three-dimensional space, noting how ideas shift between the dimensions. They learn that general cylinders are the parent category for prisms, circular cylinders, right cylinders, and oblique cylinders, and study why the cross-section of a cylinder is congruent to its base. Next students study the explicit definition of a cone and learn what distinguishes pyramids from general cones, and see how dilations explain why a cross-section taken parallel to the base of a cone is similar to the base. Students revisit the scaling principle as it applies to volume and then learn Cavalieri’s principle, which describes the relationship between cross-sections of two solids and their respective volumes. This knowledge is all applied to derive the volume formula for cones, and then extended to derive the volume formula for spheres. Module 3 is a natural place to see geometric concepts in modeling situations. Modeling-based problems are found throughout Topic B and include the modeling of real-world objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit.