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Course: Grade 7 (TX) > Unit 1

Lesson 5: Circles and composite figures
Math
Grade 7 (TX)
Staging content
Circles and composite figures
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Diameter and circumference patterns

Google Classroom
People have long found circles powerful and fascinating. Let's join them in seeking patterns in how the parts of a circle relate.

How many side lengths does it take to go around a square?

Every square is a scale copy of every other square.
Complete the pattern in the table.
Square side lengthSquare perimeter
3cm
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cm (number)
10m
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m (number)
m units
units (expression with variable)

No matter what the side length of a square, we can use the side length to figure out the perimeter, because the measurements always have the same ratio.
What is the value of the ratio of a square's perimeter to its side length?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

How many diameters does it take to go around a circle?

Every circle is also similar to each other.
So the measurements in a circle would also form a constant ratio. A circle doesn't have sides, but it does have a diameter. Instead of a perimeter, we call the distance around a circle the circumference.
Suppose we have a square with a side length of 5cm. The perimeter is 20cm.
Predict the length of the circumference around the circle inside of the square.
This is ungraded. Guessing helps our brains warm up to new ideas.
Circumference ≈
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cm
The table has the approximate circumferences and diameters of the following circles, in millimeters.
Calculate the rate circumferencediameter for each circle.
You're welcome to split up the work with a friend and to use a calculator. This isn't about how fast you can divide.
CircleCircumferenceDiametercircumferencediameter (nearest hundredth)
A188.560
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
B125.740
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
C314.2100
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
D377.0120
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
E62.820
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
F251.380
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

We can wrap 3 entire diameters, plus a little bit more, around a circle.
Do you have circular plates, cans, steering wheels, or other objects in your community? Go measure some circles and tell us about the measurements and ratios in the comments.

Introducing π

Humanity has known about this ratio for nearly 4,000 years. The ancient Egyptians and Babylonians had approximations for the ratio, although not as accurate as we have today.
Archimedes, who lived in Europe and Africa at different times of his life, split a circle into lots and lots of triangles to estimate the circumference. In China, Zu Chongzhi calculated the ratio to a level of accuracy that it would take 800 years to outdo.
More recently, we've started to use the symbol π (pronounced "pie") to represent the ratio of a circle's circumference to its diameter. It's approximately equal to 3.14159, but the decimal part goes on forever, without ever ending or repeating. So it saves a lot of writing to use a symbol for the number.
The Welsh mathematician William Jones first used the symbol π itself in 1706.
We also have the symbol τ for the ratio of a circle's circumference to its radius. Since the diameter is twice as long as the radius, τ is 2 times as large as π.

Using π

For every circle in the whole world, π=circumferencediameter.
Now that we've found a pattern, let's try the circumference of the original circle again.
Estimate the length of the circumference around the circle to the nearest tenth of a centimeter.
Circumference ≈
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cm

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