Let’s talk about circles—something you’re definitely going to see in your exams like the SAT, ACT, or even the GRE! A circle is a round shape, and all the points on its edge are equidistant from a fixed point at the center. You can think of it like the track at your school’s track and field event, or even a circular park you might jog around. The boundary of this circle is what we call the circumference. It’s basically the "perimeter" of the circle, the total distance around it.

Now, when we talk about the area of a circle, we’re referring to the space it covers or the region inside it. For example, if you were measuring the area of that same park, you’d be figuring out how much land the circle actually occupies. Understanding both the circumference and area of a circle is key to acing geometry questions in exams like the SAT, ACT, and others. The good news? You can easily calculate these by using formulas, and with some practice, you’ll be solving these types of problems like a pro!

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What is Circumference of a Circle?

Circumference of a circle or perimeter of a circle is the measurement of the path or the boundary that surrounds the circle whereas area of a circle is the measurement of the region occupied by it. A circle’s perimeter is the outcome of the constant term π and the diameter/radius of a circle. It is the 1D linear estimation of the boundary across any two-dimensional circular surface.

Terms used in Circumference of a Circle

• Diameter:  The diameter is the measurement across a circle by the centre, and it touches the two ends of the circle perimeter.
  The relationship between the diameter and the circumference is \(C=\pi D,\ \pi=\frac{C}{D}\)
• Centre: The centre is a point that is at a set range from any other point from the circumference.
• Radius: The radius of a circle is the length from the centre of a circle to any location on the circumference of the circle.
• Area of a Circle: The area of a circle is the region surrounded by the circle itself or the space incorporated by the circle.
  Formula: Area of circle=\(\pi r^2\), where r is the radius of the circle.
  \(=\pi r^{2\ }\ or\ \frac{\pi d^2}{4}\) in square units.
  Where \( \pi=\frac{22}{7}or\ 3.14\) and d is the diameter of the circle.

Formula of Circumference of a Circle

Circumference (or) perimeter of a circle (in terms of radius) = \(2\pi R\)

here, R is the radius of the circle, π is equal to 3.14 or \(\frac{22}{7}\)

Circumference (in terms of diameter) = \(\pi D\)

here, D is the diameter of the circle.

The circumference/perimeter is a linear value and its units are the same as the units of length. If we open a circle and generate a straight line out of it, then its range is the circumference. It is commonly measured in units, such as cm or unit m.

Learn about Geometric Shapes.

How to Find Circumference of a Circle?

Although the circumference of a circle is its length, it cannot be determined with the help of a scale-like it is normally done for other polygons like squares, triangles and rectangles. This is due to the curved shape of the circle. We can estimate the circle’s circumference with the below approaches.

Approach 1: We can track the path of the circle using the thread and label the points on the thread. The length of the thread later can be calibrated using a normal ruler. For example, consider if we are given a circular plate and its circumference is asked.

Step 1: Now using the above approach; we can take a thread and revolve it all around the circular plate.

Step 2: Next mark an initial and final point on the thread.

Step 3: Lastly measure the length of the thread (from initial to final point) using the scale for the circumference.

Approach 2: An accurate way of knowing the circumference of a circle in geometry is to calculate it. Therefore, to determine the circumference of a circle, we use the formula that involves the radius or the diameter of the circle and the value of Pi (π).

Starting if we are given the radius and  circumference is to be calculated then the process to be followed is:

Suppose the radius of any random circle is 28cm then we can obtain the circumference by using these steps:

Step 1: Check for the data given to us, here we are given the radius.

Step 2: Apply the formula: \(C=2\pi r\). (Here C=circumference, r=radius and π=22/7)

Step 3: Substituting the values in the formula we get the desired answer.

Step 4: \(C=2\pi r=2\times\frac{22}{7}\times28=176\text{ cm}\)

In a similar way, if we are given the diameter, we can follow the below steps.

Consider the diameter of any random circle is given as 7cm and the circumference is asked then:

Step 1: Check for the data given to us, here we are given the diameter.

Step 2: Apply the formula: \(C=\pi D\). (Here C=circumference, D=radius and π=22/7 or 3.14)

Step 3: Substituting the values in the formula we get the desired answer.

Step 4: \(C=\pi D=\frac{22}{7}\times 7=22\text{ cm}\)

Circumference to Diameter

Circumference to the diameter of a given circle is a ratio employed to specify the standard definition of Pi (π).

C= πd

Divide both sides by d.

C/d = π

If we divide both sides of the equation(C= πd) by the diameter(d), we will obtain a value that is roughly equal to π. Here C=circumference and d=diameter.

This implies that if the diameter is known to us we can obtain the circumference using the above relation.

Solved Examples of Circumference of a Circle

With all the knowledge of definition and formula, its time to practice some solved examples for the same:

Example 1: Find the Circumference of a circle whose radius is 21 cm.

Solution: Given: Radius of the circle\( =21\ cm.\)

⇒ Circumference of the Circle \(=\ 2\pi r\ =2\times\frac{22}{7}\times2\)

⇒ \(=2\times22\times3=132\ cm\)

Example 2: Find the area of a circle whose diameter is 14 cm.

Solution:Given: Diameter of the circle =14 cm, radius of the circle=\(\frac{14}{2}=7 \text{cm}\)

⇒ Using the area of Circle formula=\(\pi r^2\)

⇒ \(=\frac{22}{7}\times7\times7=154\ cm^2\)

⇒ or the area of the circle=\(\frac{\pi D^2}{4}=\frac{\left(\frac{22}{7}\times14\times14\right)}{4}=154\ cm^2\)

Example 3: Find the circumference of a circle whose diameter is 14 cm.

Solution: Given: Diameter of the circle =14 cm.

⇒ Circumference of the Circle=\(\pi D\ =\frac{22}{7}\times 14\ \text{cm}=44\ \text{cm}\)

Example 4: The circumference of a roller is 440 cm. Determine the radius of the roller.

Solution: Circumference of the roller=440cm

⇒Circumference of a circle=\(2\pi r\)

⇒ \(C=440=2\times\frac{22}{7}\times R\)

⇒ R=70 cm, and as Diameter=2r, Diameter=2* 70=140cm

Conclusion

To wrap it up, understanding the circumference and area of a circle is crucial for exams like the SAT, ACT, and others. By mastering the formulas and practicing with examples, you’ll feel confident when these questions pop up. Remember, the key is to apply the right formula based on whether you’re working with the radius or diameter. Keep practicing, and you’ll breeze through these problems on test day!