Unit 3 Measuring Circles — Unit Plan

A circle consists of all of the points that are the same distance away from a particular point called the center of the circle.

A segment that connects the center with any point on the circle is called a radius. For example, segments $QG$, $QH$, $QI$, and $QJ$ are all radii of Circle 2. (We say one radius and two radii.) The length of any radius is always the same for a given circle. For this reason, people also refer to this distance as the radius of the circle.

Two circles labeled circle 1 and circle 2. Circle 1 has a center at P and the points A, C, F, B, D, and E lie on the circle. Diameters A B, C D, and E F are drawn. Circle 2 has a center at Q and the points G, H, I and J lie on the circle. Line segments Q G, Q H, Q I, and Q J are drawn.

A segment that connects two opposite points on a circle (passing through the circle’s center) is called a diameter. For example, segments $AB$, $CD$, and $EF$ are all diameters of Circle 1. All diameters in a given circle have the same length because they are composed of two radii. For this reason, people also refer to the length of such a segment as the diameter of the circle.

The circumference of a circle is the distance around it. If a circle was made of a piece of string and we cut it and straightened it out, the circumference would be the length of that string. A circle always encloses a circular region. The region enclosed by Circle 2 is shaded, but the region enclosed by Circle 1 is not. When we refer to the area of a circle, we mean the area of the enclosed circular region.

There is a proportional relationship between the diameter and circumference of any circle. That means that if we write $C$ for circumference and $d$ for diameter, we know that $C=kd$, where $k$ is the constant of proportionality.

The exact value for the constant of proportionality is called pi, and its symbol is $\boldsymbol\pi$. Some frequently used approximations for $\pi$ are $\frac{22} 7$, 3.14, and 3.14159, but none of these is exactly $\pi$.

A graph of a line in the coordinate plane with the origin labeled O. The horizontal axis is labeled “d” and the numbers 1 through 6 are indicated. The vertical axis is labeled “C” and the numbers 2 through 12, in increments of 2, are indicated. The line begins at the origin, slants upward and to the right, and passes through the point 1 comma pi.

We can use this to estimate the circumference if we know the diameter, and vice versa. For example, using 3.1 as an approximation for $\pi$, if a circle has a diameter of 4 cm, then the circumference is about $(3.1)\boldcdot 4 = 12.4$, or 12.4 cm.

The relationship between the circumference and the diameter can be written as

$\displaystyle C = \pi d$

The circumference of a circle, $C$, is $\pi$ times the diameter, $d$. The diameter is twice the radius, $r$. So if we know any one of these measurements for a particular circle, we can find the others. We can write the relationships between these different measures using equations:

$\displaystyle d = 2r$ $\displaystyle C = \pi d$ $\displaystyle C = 2\pi r$

If the diameter of a car tire is 60 cm, that means the radius is 30 cm, and the circumference is $60 \boldcdot \pi$, or about 188 cm.

If the radius of a clock is 5 in, that means the diameter is 10 in, and the circumference is $10 \boldcdot \pi$, or about 31 in.

If a ring has a circumference of 44 mm, that means the diameter is $44 \div \pi$, which is about 14 mm, and the radius is about 7 mm.

The circumference $C$ of a circle is proportional to the diameter $d$, and we can write this relationship as $C = \pi d$. The circumference is also proportional to the radius of the circle, and the constant of proportionality is $2 \boldcdot \pi$ because the diameter is twice as long as the radius. However, the area of a circle is not proportional to the diameter (or the radius). 

The area of a circle with radius $r$ is a little more than 3 times the area of a square with side $r$ so the area of a circle of radius $r$ is approximately $3r^2$. We saw earlier that the circumference of a circle of radius $r$ is $2\pi r$. If we write $C$ for the circumference of a circle, this proportional relationship can be written $C = 2\pi r$.

The area $A$ of a circle with radius $r$ is approximately $3r^2$.  Unlike the circumference, the area is not proportional to the radius because $3r^2$ cannot be written in the form $kr$ for a number $k$. We will investigate and refine the relationship between the area and the radius of a circle in future lessons.

If $C$ is a circle’s circumference and $r$ is its radius, then $C=2\pi r$. The area of a circle can be found by taking the product of half the circumference and the radius.

If $A$ is the area of the circle, this gives the equation:

$A = \frac12 (2\pi r) \boldcdot r$

This equation can be rewritten as:

$A=\pi r^2$

Remember that when we have $r \boldcdot r$ we can write $r^2$, and we can say “$r$ squared.”

This means that if we know the radius, we can find the area. For example, if a circle has a radius of 10 cm, then its area is about $(3.14) \boldcdot 100$, which is 314 cm².

If we know the diameter, we can figure out the radius, and then we can find the area. For example, if a circle has a diameter of 30 ft, then the radius is 15 ft, and the area is about $(3.14) \boldcdot 225$, which is approximately 707 ft².

The relationship between $A$, the area of a circle, and $r$, its radius, is $A=\pi r^2$. We can use this to find the area of a circle if we know the radius. For example, if a circle has a radius of 10 cm, then the area is $\pi \boldcdot 10^2$, or $100\pi$ cm². We can also use the formula to find the radius of a circle if we know the area. For example, if a circle has an area of $49 \pi$ m² then its radius is 7 m and its diameter is 14 m.

Sometimes instead of leaving $\pi$ in expressions for the area, a numerical approximation can be helpful. For the examples above, a circle of radius 10 cm has an area of about 314 cm². In a similar way, a circle with an area of 154 m² has a radius of about 7 m.

We can also figure out the area of a fraction of a circle. For example, the figure shows a circle divided into 3 pieces of equal area. The shaded part has an area of $\frac13 \pi r^2$.

A circle divided into three equal sections. From the center of the circle three line segments extend to a point on the circle. The line segment extending downward and to the right is labeled “r”. The upper left region of the circle is shaded.

Sometimes we need to find the circumference of a circle, and sometimes we need to find the area.

In both cases, the radius (or diameter) of the circle is all that is needed to make the calculation. The circumference of a circle with radius $r$ is $2\pi r$ while its area is $\pi r^2$. The circumference is measured in linear units (such as cm, in, km) while the area is measured in square units (such as cm², in², km²).