Area is one of the most basic geometric concepts. Area refers to the amount of surface area an object has. For shapes such as triangles, squares, and circles, area is calculated by how much surface they have on the inside and outside.
Area can be defined in many different ways. One way to define area is how many units of a given measurement you need to fill the shape. For example, one square has an area of one square unit, so a square has an area of one square unit per side.
The average person uses the term “circle” when referring to any kind of circle or any other kind of shape that is circular in nature. However, geometry distinguishes between these two types of shapes: a circle and a sphere.
This article will discuss the differences between these two shapes and how to find the area of a circle as a function of its circumference.
The area of a circle is proportional to its circumference
A circle is a plane figure formed by a curve line called a circle
arc. The circle arc is defined by its radius and the length of the curve itself.
The circumference of a circle is the distance around the circle, which is equal to
the length of the radius. The area of a circle is given by square of the radius.
Use the formula for the area of a circle
The area of a circle is measured in square units, usually meters or kilometers.
Because the circle is defined by its radius, the circumference can be used to define the area. The area is given by the formula: A= πr².
The constant π (pi) is a number that defines the circumference of a circle. It is defined as the length of a line segment equal to the circumference of a circle with diameter one.
The notation for π is Pi and it is an archaic Greek letter denoting perimeter. Many calculators and programs recognize Pi as a variable and have it built in.
Break down the formula for the area of a circle
To write the area of a circle as a function of its circumference, you need to first write the formula for the circumference of a circle as a function of its radius.
The formula for the circumference of a circle is πd, where d is the diameter and π is pi, which is approximately 3.14.
Then, under that column, write the column for the radius. The radius is half of the diameter, so you can insert that as its own row.
Under that, write the column for the number of circles fit inside the given circle. For example, if your given circle has a circumference of 10 meters, then 100 meters worth of circles could fit inside it- one centimeter worth of circles fits inside it ten times.
Under that, write down how many centimeters worth of circles fit inside your given circle ten times– so 1×10=10. Put that as the tenth row.
Determine what variables you want to use for your function
In this case, the variable you will use is the area of the circle and the variable you will tie it to is the circumference of the circle. The relationship you will show is that the area of a circle is equal to its circumference multiplied by its radius.
The area of a circle is pi (r²) where r is the radius and pi (p) is approximately 3.14. Circle radius can be measured in many units, such as inches, meters, or centimeters. The unit does not matter when calculating area using this formula.
The circumference of a circle is 2πr or πd, where r is the radius and d is the diameter. The diameter of a circle is twice its radius so that part of this equation can be ignored if using only the circumference in calculations.
Write down all the values you found using the formula for the area of a circle
Now that you know the basic formulas, let’s put them to use! You should know how to write the area of a circle as a function of its circumference, so let’s start there.
The general formula for the area of a circle is A=ρr², where ρ is the radius of the circle and r is the circumference. You can change these to different variables if you like, but remember that you have to keep the same variable on both sides of the equality sign.
Now, let’s find some areas using this formula. For this first one, we will use some common numbers. Let’s say we have a circle with a radius of 2 units and a circumference of 10 units. The area of this circle would be A=2×10²=202.
Now, let’s try one with more digits in the radius- 2.−4 Units. The area would be A=2.−4+4×10−6+6²=20000−8+8. That one took some digging, but we got there!
Let’s do one more- this time with decimal places in the radius- 0.11/1000th sup >unit. The area would be A=(0.111/1000th )×10−6 +6)=0.(01)=0.1(1/1000) sup > unit. span >That one was not so simple! But we got it.
Put all your values into a function and solve for one of your variables using algebra
Now let’s solve for the area of a circle as a function of its circumference. First, write the equation for the area of a circle as a function of its circumference.
In this case, we are solving for A, so put A on the left side of the equals sign. Next, multiply both sides by r2.
πr2 = A × r2
Subtract r2 from both sides to get an empty variable. This will help you solve for A when you know r. Then, divide both sides by π. You now have solved for A!
Try it out yourself with some circles and see if you get the same answer.
Check your answer using your function
Now that you can calculate the area of a circle as a function of its circumference, you can check your answer by calculating the area of a circle using the formula for its area and its circumference.
If you did it correctly, your two answers will match!
Circumference is an important element in geometry, and it will come up again as you learn more about shapes. For now, just be confident in your ability to calculate the area of a circle using this method.
The next section will discuss how to write the equation for the radius of a circle as a function of its diameter. This is another essential skill for anyone looking to learn about circles.