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Area and Circumference of a Circle

A farmer is walking around his field, the shape of which is a circle. He starts walking on its boundary from a point, keeps himself on the border, and returns to the point where he began.

He wants to know how much distance he has covered. The only piece of information about the field he is aware of is the radius of the field (the shortest distance from the center of the field to its boundary).

He plans to grow some crops in the field and wants to buy the appropriate amount of pesticides and crops for it. But he needs to know the area of his circular field to buy the things mentioned above. Again, the only piece of information he has is the field's radius. How can he determine the area of the field?

Let us explore how we can help the farmer in this situation by developing the circumference and its area formula.

Area and Circumference of a Circle Definition

Circumference of a Circle Definition

Earlier, we saw the farmer going around his field and wanted to measure the distance he had covered. It is nothing but the circumference of a circle.

The circumference of a circle is the distance around a circle. It is just another word for the perimeter of a circle.

If you draw a circle and trace it from one point and stop at the same point after one round, the distance you have traced is the circumference of that circle.

Area of a Circle Definition

In order to help the farmer estimate how many pesticides and crops will he need for his field, we will be talking about the area of a circle.

The area of a circle is the region occupied by a circle in a two-dimensional plane.

Area and circumference of a circle equation

Circumference of a circle equation

In order to find the circumference of a circle, the concept of Pi is essential.

Every circle one can possibly draw, at its core, has one property in common. This property or characteristic is what gives rise to Pi, the ratio of the circumference of a circle to the diameter of the circle is known as pi. Translating this definition into a formula gives us,

where C denotes the circumference of the circle and r is its radius. We recall that diameter is twice the radius. This is how, from the definition of Pi, we get the formula for the circumference of a circle,

Pi is an irrational number, and is approximately given by 3.141592653589793… and it never ends. But for the convenience of calculations, it is approximated to 3.14 or as the fraction .

Area of a circle equation

The area of a circle can be derived by cutting the circle into small pieces as follows.

A circle broke up into pieces to form an approximate rectangle, StudySmarter Originals

If we break the circle into little triangular pieces (like that of a pizza slice) and put them together in such a way that a rectangle is formed, it may not look like an exact rectangle but if we cut the circle into thin enough slices, then we can approximate it to a rectangle.

Observe that we have divided the slices into two equal parts and colored them blue and yellow to differentiate them. Hence the length of the rectangle formed will be half of the circumference of the circle which will be . And the breadth will be the size of the slice, which is equal to the radius of the circle, r.

The reason why we did this, is that we have the formula to calculate the area of a rectangle: the length times the breadth. Thus, we have

Verbally, the area of a circle with radius r is equal to Pi times the radius squared. Hence the units of area are cm2, m2 or (unit)2 for appropriate units.

Calculating Area and Circumference of a Circle

In this section, we will work out some examples of the area and the circumference of a circle.

The radius of a circular pond is found to be 20 meters (assume it to be a perfect circle). Find the circumference of the pond in relevant units. Take .

Solution

The radius is given as , plugging it into the formula of circumference, we get

Therefore the circumference of the circular pond is 125.6 meters.

The perimeter of a circular bowl is measured using a measuring tape and is found to be 30 cm. But after some time the tape is lost but the radius of the bowl is yet to be measured. How can we determine the radius of the bowl without the tape? Take

Solution

We can use the formula of the circumference as it directly relates the radius to the circumference,

Thus we have

(rounded to two decimal places)

Hence, the radius of the bowl is 4.78 cm rounded to two decimal places.

The radius of a circular table is given by the manufacturer as 50 cm. A tablecloth has to be made for it and so its area is required. What is the area of the table?

Solution

The radius is 50 cm: .

Using the formula for the area of a circle, we have

Hence, the area of the circular table of radius 50 cm is 7850 cm2.

Difference between Circumference and Area of a circle

Using the formulae of the area and the circumference, we can deduce a formula that would explicitly relate to circumference and area. Recall that

and

Solving for r,

Substituting for r in the area formula we get,

Hence, if we are given the area of a circle and want to find its circumference then we don’t have to do the intermediary step of finding the radius and can directly calculate the circumference.

Given that the circumference of a trampoline is 10 meters, find its area without calculating its radius.

Solution

The circumference is given as .

Using the formula that relates circumference and the area of a circle,

Substituting for we get,

Therefore, the area of the trampoline with a circumference of 10 m is 7.962 m2.

Area and Circumference of a Circle - Key takeaways

• The circumference of a circle is the distance around it, it is just another term for the perimeter of a circle.
• The circumference of a circle with radius r is given by .
• The area of a circle is the region occupied by a circle in a two-dimensional plane. It can be calculated by reducing a circle to a triangle and calculating its area, and it is given by .
• The area and circumference are related explicitly to each other by the following equation .

The circumference of a circle with radius r is given by 2πr and its area is given by πr2.

The area of a circle can be found out using the circumference using the following formula: C2=4πA where C is the circumference, and A is the area.

Pi is an irrational number that is defined as the ratio of the circumference to the diameter of a circle.

Divide the circle into equal triangles, in the shape of pizza slices and arrange them to form a rectangle, whose length will be half the circumference of the circle and breadth will be the radius.

The area now will be length times breadth giving us the area of the rectangle, hence of the circle.

Final Area and Circumference of a Circle Quiz

Question

How to derive the formula for the area of a circle?

Dividing up the the circle into little little slices and rearranging them in a form of a rectangle. And now using the formula of area of the rectangle, the area of the circle can be determined.

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Question

What is the circumference of a circle?

The circumference of a circle is defined as the total length of the boundary of the circle.

Show question

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