Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Monthly Update sign up
Mailing List signup
Learning Math Home
Measurement Session 7: Solutions
Session 7 Part A Part B Homework
measurement Site Map
Session 7 Materials:

A B 


Solutions for Session 7, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | B9 | B10

Problem B1

The area of the figure is exactly the area of the circle, since no area has been removed or added, only rearranged.

<< back to Problem B1

<< back to Problem B1 (non-interactive version)


Problem B2

The length of the base is one-half the circle's circumference, since the entire circumference comprises the scalloped edges that run along the top and bottom of the figure, and exactly half of it appears on each side. The base length is C/2.

<< back to Problem B2

<< back to Problem B2 (non-interactive version)


Problem B3

Since the circumference is 2 • • r, where r is the radius, the base is half of this. The base length is • r.

<< back to Problem B3

<< back to Problem B3 (non-interactive version)


Problem B4

As the number of wedges increases, each wedge becomes a nearly vertical piece. The base length becomes closer and closer to a straight line of length • r (or half the circumference), while the height is equal to r. The area of such a rectangle is • r • r, or • r2.

<< back to Problem B4

<< back to Problem B4 (non-interactive version)


Problem B5

Here is the completed table:


Radius of Circle

Area of Radius Square

Area of Circle

Number of Radius Squares Needed




36 •

A little more than 3




16 •

A little more than 3




9 •

A little more than 3

<< back to Problem B5


Problem B6


In each case, it takes a little more than three radius squares to form the circle. If using approximations, it should always take around 3.14 of the squares to cover the circle.


The best estimate is somewhere between 3.1 and 3.2, which we know is roughly the value of .

<< back to Problem B6


Problem B7

The formula for the area of a circle is A = • r2. The activity helps one understand that a bit more than three times a radius square is needed to cover the circle. Namely, it illustrates why the formula is • r2.

<< back to Problem B7


Problem B8

Think about a circle with a radius equal to 1 (r = 1). The circumference and the area of this circle are as follows:

C = 2 • 1 • = 2
A = 12 =

Now double the radius to 2 units (r = 2). The circumference and the area of the new circle are as follows:

C = 2 • 2 • = 4
A = 22 = 4

The circumference of the new circle doubled, but the area is multiplied by a factor of 4 (the square of the scale factor). You can replace the 1 with any other number, or with a variable r, to see that this relationship will always hold.

<< back to Problem B8


Problem B9

Scale factor 3:
C = 2 • (3r) = 6 r
A = • (3r)2 = 9 r2

Scale factor 2/3:
C = 2 • (2/3r) = (4/3)r
A = • (2/3r)2 = (4/9)r2

Scale factor k:
C = 2 • (kr) = k(2r)
A = • (kr)2 = k2r2

As with other similar figures, the circumference (or perimeter) of the shape is multiplied by the scale factor, while the area is multiplied by the square of the scale factor. This is also evident in the formulas for each; the circumference formula involves r, while the area formula involves r2.

<< back to Problem B9


Problem B10

A reasonable approximation is 25 cm, but the margin of error will be larger than 0.2 cm. The actual area in square centimeters may be anywhere from (4.8)2 (lower limit) to (5.2)2 (upper limit). Since 4.82, or 23.04, and 5.22, or 27.04, are each about 2 units away from 52, the margin of error for the area is approximately 2 cm2.

<< back to Problem B10


Learning Math Home | Measurement Home | Glossary | Map | ©

Session 7 | Notes | Solutions | Video

Home | Catalog | About Us | Search | Contact Us | Site Map

  • Follow The Annenberg Learner on Facebook

© Annenberg Foundation 2013. All rights reserved. Privacy Policy