On the board, draw a square with a circle circumscribed inside it as a “template” for the first activity:
“Here, we have a circle drawn inside a square. The radius of the circle is labeled as r. What is the diameter of the circle?” (2r) If students struggle with using a variable such as r right away, begin the activity with a specific value chosen for r. “How long is one side of the square?” (2r) “And how long is the other side of the square?” (2r)
Make sure students know how to find the area of a square; it isn’t necessary that they determine the area of the square is 4r2.
“What is the area of the square if the radius of the circle is 1 unit?” (4 square units) “What do we know about the area of the circle compared to the area of the square?” (It has a lesser area than the area of the square.) Ask students to estimate the area of the circle.
It may help to divide the square into quadrants:
“What is the area of each quadrant?” Shade in one quadrant so students are clear about which part of the diagram you’re referring to. Students should recognize that it’s square unit. Ask students to estimate the area of the part of the circle in that quadrant. To help, ask, “Does the circle take up more than half the area in the quadrant?” (Yes) “Does it take up more than of the area of the quadrant?” Responses to this question will be mixed—it really takes up , or about 78.5%, of the area. Use as a guess.
“So, if the area of the entire square is 4 square units, and the area of the circle is the area of the square, approximately what is the area of the circle?” (3 square units)
“That’s a pretty good estimate. An estimate, to the nearest hundredth, is actually 3.14 square units. So an estimate that the area of the circle is about the area of the square is a pretty good one.” Write this information on the board:
- r = 1 unit, A ≈ 3.14 square units
“Is there anything special or unique about the number we got for the area of our circle with a radius of 1 unit?” Students should recognize that it is (approximately) π.
“So based on the radius of the circle, which was 1 unit, and the fact that the area is π, what do you think we have to do with the radius to determine the area?” Students should suggest that we multiply the two of them together; in other words, that A = πr.
“To test our guess, let’s pick a different value for the radius. Suppose the radius was 10 units. What would the area of the square be?” Make sure students remember that each side of the square measures 2r units (in this case, 20 units). The area is 400 square units. “So, if we think our circle has an area of about ¾ that of the square, what’s the approximate area of the circle?” (300 square units)
“Now, if we use our guess at a formula to find the area and multiply the radius of 10 units by π, what do we get for an area?” (31.4 square units) “Can this guess be correct, based on the fact that we know the area of the square is 400 square units?” (No)
“So, we need a little more information to make a better hypothesis of how to find the area. I’ll tell you that the actual area of the circle, rounded to the nearest whole number, is 314.” Write this information on the board:
- r = 10 units, A ≈ 314 square units
“Clearly, 314 is not equal to 10 times π. About how many times greater than π is 314?” (100.) “So, based on the fact that our area here is 100π, can anyone come up with a formula that uses the radius to get 100π?” Guide students toward recognizing that 100π = r2π.
Write the formula A = πr2 on the board.
“Let’s test out one more example. Suppose the radius of our circle is 8 units. What is the area of the square?” (256 square units) “And, based on our estimate that the circle has about ¾ the area of the square, what is the area of the circle?” (192)
“So, let’s test our formula, A = πr2, with a radius of 8 units and see if we get close to 192. What does the formula tell us our area should be?” (approximately 201) “Pretty close to our estimate—I think our formula works!”
Explain to students that when they said the area was 201 square units, they were approximating; even if they read all the digits on their calculator, it was still an approximation because the digits of π keep going forever. “So, it is more accurate to just write the answer to this problem, for example, as 64π. In other words, not multiplying the 64 by 3.14 gives us an exact answer.”
Using the formula, provide students with different values for radii of circles and ask for the area; also, provide them with different diameters and ask for the area. After students can handle the formula with some confidence, move on to Activity 2.
“So now you can—given the radius or the diameter—find the area of a circle. What if you know the area of a circle; can you work backwards to find the radius? Suppose there is a tarp that covers a circular swimming pool. You know the tarp has an area of 144π square feet. How can you determine the radius?”
Guide students toward realizing that they have to take the square root of 144. If students struggle with square roots, asking them, “What number squared equals 144?” or “What number times itself equals 144?” should help.
Point out to students that since the formula for area is πr2, we can write πr2 = 144π and cancel out the πs from each side.
“What if the area of the tarp is 615.44 square feet and we need to find the radius? What’s different about this problem than the previous one?” Students should note that the previous area had π in it (i.e., was expressed in terms of π) and this one does not.
“So, we can write πr2 = 615.44, but now what?”
Guide students to the understanding that they still need to get rid of the π and when it was “canceled” in the previous problem, we were really dividing both sides of the equation by π. “So, we still need to divide both sides of the equation by π, although we’ll divide both sides by 3.14, our approximation for π instead. After we divide both sides by π, what does our equation look like?” (r2 = 196) “So what is the radius of our tarp?” (14 feet)
Explain that the general process is to get r2 by itself by dividing both sides of the equation by π (or 3.14). Then it’s a matter of taking the square root of both sides.
“Now, what if the area of our tarp is 706.5 square feet and we want to know the diameter? What steps do we have to do differently or what additional steps do we need to perform?” Guide students toward the realization that the first part of the problem is the same—find the radius. Then, to find the diameter, double the radius.
Have students work in pairs to complete the Area Worksheet (M-7-4-2_Area Worksheet and KEY.docx). If students have already learned circumference, include the last column on the worksheet; otherwise, students should ignore it. Instruct students that their areas and circumferences should be expressed in terms of π whenever possible. If they’re given an area that is not in terms of π, they should use π = 3.14.
Have students turn in their completed worksheet at the end of class.
This lesson introduced the area of a circle using existing knowledge, incorporating a diagram, and using estimation skills to appeal to visual learners and those with an intuitive approach. Students then used the actual area formula, which is slightly more abstract, to appeal to students who enjoy computation and algebraic concepts.
Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.
- Routine: Finding the area of circles can be fit into a wide variety of other geometric topics throughout the year and, with simple numbers, can easily be part of future quizzes, entrance or exit tickets, and so on. Students can also explore and refine their understanding of (estimating) square roots in the context of the area of circles. Students may play the following game to refresh their circle calculation skills.
- Small Group: Students who need additional practice can be pulled into small groups and given the Area Worksheet (M-7-4-2_Area Worksheet and KEY.docx). Or students can get additional practice calculating areas of circles with the following game.
Students who may benefit from additional instruction will find the following Web site helpful.
- Expansion: For students who are ready for a challenge beyond the requirements of the standard. This lesson can be expanded to include ellipses, for which the formula is πab, where a and b are the lengths of the “long” and “short” radii of the ellipse. Students will discover that the formula for the area of a circle is really just a specific case of this where a = b = r and so, in a sense, a circle is just a specific type of an ellipse. Use the following Web sites as resources. Have students write a short explanation of how to find area of an ellipse. They can include some examples with calculations.
Answers may be checked using the following Web site.
The lesson can also be expanded to include three-dimensional objects involving circles (primarily cylinders and cones but could also include spheres because of the similarity in the formulas). Finding surface area and volume for these 3D objects involves finding the area of circles as well as the circumference of circles (for determining the surface area of a cylinder).