Surface Area of a Sphere

Lesson 3: Surface Area of Spheres

Materials:

Instructional Objectives:

To investigate and derive the formula for finding surface areas of a sphere
To use multiple methods of investigation and insight
To enable students to work cooperatively with group members and communicate mathematical concepts

NCTM/ISBE Standards covered:

analyze properties of two and three dimensional objects
establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others

NCTM Reasoning and Proof Standard
NCTM Problem Solving Standard
NCTM Communication Standard
ISBE Solving Problems Application Standard
ISBE Communicating Application Standard
ISBE Working on Teams Application Standard
ISBE 7.A.4b – Apply formulas in a wide variety of theoretical and practical real-world measurement applications
ISBE 9.B.4 – Recognize and apply relationships within and among geometric figures
ISBE 9.C.4b – Construct and communicate convincing arguments for geometric situations

Lesson:

Begin class by introducing the first activity to the students (wrapping the basketball).
Place the students in groups of 3-4 students and pass out the necessary materials.
Allow the students 15-20 minutes to experiment with wrapping the ball and to play with how much paper is necessary to accomplish this task. Encourage them to attempt to figure out what the formula for surface area of a sphere might be. (Hopefully they should figure out that the amount of paper needed to cover the ball is the circumference of the ball times the diameter of the ball, which equals 4*p*r². This is because the paper is the same height as the diameter of the ball and the same length as its circumference).
Bring the class back together and discuss the groups’ findings and ideas about surface areas of spheres. From this, lead into the derivation of the surface area formula.

This can be done using the concepts from the activity (hopefully that SA = circumference * diameter), which follows.

Surface area of the paper is:

Length * width = circumference * diameter

= (2pr)*d

= (2pr)*2r

= 4pr²

Also it can be done using the idea that the surface area of a sphere is the same as the curved surface area of a cylinder, when the cylinder has the same diameter and with a height the same length as the diameter of the sphere.

Curved surface area of cylinder = 2prh

= 2pr(2r)

= 4p r²

Check that students understand these derivations of the formula for surface areas of a sphere and allow them to ask any questions they might have.
Assign homework.