1 © 2012 AIMS Education Foundation

Surface Area of a SphereStudents will discover the formula for

calculating the surface area of a sphere and will use their findings to verify their estimation of the surface area of an orange or grapefruit.

Common Core State Standards for Mathematics*

• Makesenseofproblemsandpersevereinsolvingthem.(MP.1)

• Reasonabstractlyandquantitatively.(MP.2)• Constructviableargumentsandcritiquethe

reasoningofothers.(MP.3)• Modelwithmathematics.(MP.4)• Attendtoprecision.(MP.6)• Lookforandmakeuseofstructure.(MP.7)• Knowtheformulasforthevolumesofcones,

cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.(8.G.C)

You NeedOranges or grapefruits (see BeforeYouBegin1)

Metric rulers

Paper towels

Plastic serrated knife

Student page

Paper

Background Information One of the interesting features of a spheres is that, unlike its three-dimensional friends the cone and cylinder, it does not have faces and so it cannot be unfolded into a flat net. This presents a challenge for students understanding conceptually why the surface area is always 4πr2.

A sphere, like its two-dimensional relative the circle, is defined as the set of given points that are the same distance (r)from a given point in space (the center). The sphere just happens to be set in three-dimensional space rather than the circle’s two-dimensional plane. It is interesting, then, that while the area formula for a circle is πr2, the surface area formula for a sphere is exactly four times the area of a circle with radius r. This lesson invites students to make estimations and discoveries that help them connect what they already know about circles with this fascinating formula: surface area of a sphere = 4πr2.

Before You Begin 1. Each student pair or group will need an

orange or grapefruit. You will also need one for demonstration purposes.

2. Calculate the surface area of your orange or grapefruit. Use metric rulers to determine square centimeters.

3. Use a plastic serrated knife to cut the fruit in half when appropriate in the activity.

Do This 1. Show students your fruit and tell them

that you have figured out the surface area of the sphere. Tell them the surface area, including the units; discuss surface area definitions and units if needed.

2. Invite students to compare their fruits to yours.

Howdothesizesandshapescompare?Estimatethesurfaceareaofyourfruit.Recordtheestimationonthestudentpage.

3. Cut your demonstration fruit in half. Show the students the cross-section.

Whatisthecirclevocabularythatnamesthiscross-section? [Great circle of a sphere.]Whatpartoftheorangerepresentstheradius?…thesurfacearea?…thehemisphere?Recordanyimportantvocabularyonthestudentpage.(At this point, it is important that students understand that the radius of the great circle is also the radius of the sphere.)

4. Explain that each group will explore a method for finding the surface area of a sphere then use their findings to check their fruit surface area estimate.

5. Inform students that the method they will explore involves drawing the great circle of their sphere by placing the fruit on a piece of paper, holding a pencil perpendicular along the edge of the fruit, and tracing around it.

Howmanyofthesegreatcirclesdoyouthinkyoucouldfillwiththepeelsofyourfruit?Recordyourpredictiononthestudentpage.

6. Distribute paper. Have students draw as many great circles as they think they need, plus a few more—just in case.

SPHERESSPHERES“A peel ing”“A peel ing”

2 © 2012 AIMS Education Foundation

7. Direct students to begin to peel the fruit by tearing small nickel-sized pieces. As they tear the pieces, tell them to place them within the great circles they drew, being careful to fill the circles completely without overlaps or gaps. Once one great circle is filled, have them begin filling the next great circle.

Howmanytotalgreatcirclesdidyourpeelsfill?Ifyoufilledafractionofagreatcircle,simplyroundtothenearestcircle.Recordyourfindingsonthestudentpage.

8. Collect the results from each group and display them for the class.

Whatdoyounoticeabouttheseresults?Whydidgroupshavesimilarresultseventhoughthesizesoftheorangesdiffered?(The result should be close to four great circles filled. Some groups may have one more or one less. This presents an opportunity to discuss result variations.)

9. Have students make a conjecture about a formula for the surface area of a sphere using the information from the great circles they filled. If needed, ask students to remind each other how to find the area of a circle. They may record their conjecture on the student page. Discuss conjectures as a class.

10. Ask students to look at their previous estimation of the surface area of their fruit. Have them calculate the actual surface area of their fruit using a ruler and the surface area formula.

Ask These1. Why is the radius of the great circle the

same as that of your sphere? [The great circle is the plane of the sphere that passes through the center of the sphere. The radius is the line segment from the center of the circle or sphere to its perimeter.]

2. Why might some groups have filled three or five great circles instead of four? [Some pieces of the peel may have overlapped, making fewer than four filled great circles, or there may have been gaps between pieces of peel to make greater than four filled great circles.]

3. Why did groups fill approximately the same number of great circles even through their fruits were different sizes? [The proportion of the area of the great circle and the surface area of the sphere remains the same whether the fruit is big or small.]

4. Do you think your formula would calculate the surface area of any sphere? Why or why not?

* © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

3 © 2012 AIMS Education Foundation

VocabularyWord

What doesit mean?

How does it relateto my fruit?

Estimate the surface area of your fruit:

Record any important vocabulary inthe table.

Predict how many great circles yourpeels will fill: ________

Approximately how many great circlesdid your peels actually fill?

Using what you’ve learned, write theformula for the surface area of a sphereand explain in words what it means.

Surface area of a sphere = ___________

Use this formula to calculate the surface area of your fruit. How does it compare to your original estimation?

SPHERESSPHERES“A peel ing”“A peel ing”

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4 © 2012 AIMS Education Foundation

LEARNING

LEARNING

CONNECTING

CONNECTING

Ask These

1. Why is the radius of the great circle the same as that of your sphere?

SPHERESSPHERES“A peel ing”“A peel ing”

2. Why might some groups have filled three or five great circles instead of four?

3. Why did groups fill approximately the same number of great circles even through their fruits were different sizes?

4. Do you think your formula would calculate the surface area of any sphere? Why or why not?