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Lesson Explainer: Areas and Circumferences of Circles Mathematics • 11th Grade

In this explainer, we will learn how to find the area and the circumference of a circle given its radius or diameter and how to relate both the area and circumference to solve various problems.

Let us begin by recapping the form that a circle takes mathematically and some of its key features.

Definition: Circle

A circle is a set of points that are a constant distance from a point in the center, which we usually denote by 𝑀.

A radius of the circle is a line segment that goes from the center to the edge, which we have labeled here by π‘Ÿ.

A diameter of the circle is a line segment that starts and ends at two points on the edge of the circle and passes through the center, which we have labeled by 𝑑. Diameters are twice the length of radii.

Finally, the circumference is the distance going around the circle, which we have labeled 𝐢.

Circles are special shapes in that their features are related to each other by the mathematical constant πœ‹β‰ˆ3.14159…. In particular, the circumference is related to the diameter (and the radius) by the following formula.

Formula: Circumference of a Circle

For a circle with diameter 𝑑 and circumference 𝐢, their lengths are related by 𝐢=πœ‹π‘‘.

Alternatively, the circumference can be written in terms of a radius π‘Ÿ as 𝐢=2πœ‹π‘Ÿ.

In any situation in which we are given either the circumference or the diameter (or radius) of a circle, we can work out the other measurement using the above formulas. Let us see an example of this.

Example 1: Finding the Circumference of a Circle Using a Diagram

Work out the circumference of the circle, giving your answer accurate to two decimal places.

Answer

From the diagram, we have been given that the length of the diameter is 6 units. Since we need to work out the circumference of the circle, we can use the formula relating the circumference to the diameter: 𝐢=πœ‹π‘‘, where 𝐢 is the circumference and 𝑑 is the diameter. Substituting 𝑑=6, this is 𝐢=πœ‹Γ—6.

We can calculate this value using a calculator, giving us 𝐢=18.8496…=18.85(2).d.p.

We can use the formula for the circumference of a circle in many different settings. Suppose, for instance, we needed to calculate the perimeter of a semicircle or quarter-circle.

We can see that the circular part of the perimeter will be a fraction of the overall circumference 𝐢. Since a semicircle is half of a circle, the length of the circular portion will be 𝐢2, while for a quarter-circle, it will be 𝐢4.

The total perimeter can be obtained in each case by adding the remaining length to the circular part. In particular, we have perimeterofasemicircleperimeterofaquarter-circle=𝑑+𝐢2,=𝑑+𝐢4, where 𝑑 is the diameter (which is equal to two times the radius).

Let us consider an example where we calculate the perimeter of a shape composed of quarter-circles.

Example 2: Finding the Perimeter of a Shaded Area

Using 3.14 to approximate πœ‹ and the fact that 𝐴𝐡𝐢𝐷 is a square, calculate the perimeter of the shaded part.

Answer

At first, it appears as though the perimeter of this area will be tricky to calculate since it has an unorthodox shape. On closer inspection, however, we can see that it is composed of four quarter-circles. We show this for the bottom-left part below:

Here, we have highlighted that the lower-left part of the shape is one-quarter of a larger circle with center 𝐴 and radius π‘Ÿ. In addition, because it is a quarter-circle, the length of the circular part will be a quarter of the overall circumference.

Recall that the circumference of a circle is related to its radius by the formula 𝐢=2πœ‹π‘Ÿ.

Here, we can see that π‘Ÿ is half of the given length 68 cm, so π‘Ÿ=34cm. Thus, the circumference of the larger circle is 𝐢=2πœ‹Γ—34=68πœ‹.cm

Now, we want to calculate the perimeter of the entire shaded part. Since it is composed of four identical quarter-circles, the overall perimeter 𝑃 will be 𝑃=4×𝐢4=𝐢.

Thus, the perimeter is the same as the circumference of the larger circle. That is, the perimeter is 68πœ‹. Using πœ‹=3.14 to approximate this, we get 𝑃=3.14Γ—68=213.52.cm

So far, we have only considered the circumferences of circles, so, let us now approach the topic of the areas of circles. Recall that the area of a shape is the amount of space it takes up on a 2D plane. We denote this by 𝐴 in the diagram below.

Much like the circumference, the area is related to the radius (or diameter) by πœ‹ as given by the following formula.

Formula: Area of a Circle

For a circle with radius π‘Ÿ, its area 𝐴 is given by 𝐴=πœ‹π‘Ÿ.

Note that in situations where we are given the diameter rather than the radius, we can calculate the radius by dividing by two, after which we can substitute it into the above formula.

Much like for the circumference, the application of this formula is quite straightforward. Nevertheless, let us demonstrate it with an example.

Example 3: Finding the Area of a Circle given a Diagram

Using 3.14 as an approximation for πœ‹, find the area of the circle.

Answer

Recall that the area of a circle is given by 𝐴=πœ‹π‘Ÿ.

From the diagram, we can see that the radius is 12 cm. Thus, we can substitute it into the formula to find the area. We will also use πœ‹=3.14 as specified. This gives us 𝐴=3.14Γ—(12)=452.16.cmcm

It is important to keep in mind the units used for area; in the previous example, they were cm2, and in general, they will always be square units of length.

Now, just as we can relate the circumference of a full circle to semicircles and quarter-circles, a similar relationship exists for the area of a circle and its division into smaller parts. We can see this relationship below.

That is, if a circle has area 𝐴, then a semicircle has area half of that, 𝐴2, and a quarter-circle has area 𝐴4. In fact, for any shape that is a portion of a circle, we can relate its area to the full circle in a similar manner.

Let us see this in effect in the next example, where we relate the area of a semicircle to its perimeter.

Example 4: Finding the Perimeter of a Semicircle given the Area

The area of the given semicircle is 51.04 cm2. Find the perimeter of the semicircle to the nearest centimetre.

Answer

In this question, we have been given the area of a semicircle and need to calculate the perimeter.

Recall that the perimeter of a semicircle is related to the circumference of the entire circle and that the circumference 𝐢 of a circle is given by 𝐢=2πœ‹π‘Ÿ, where π‘Ÿ is the radius. Although we do not have π‘Ÿ here, it is possible to find it by using the information we have been given in the question.

In particular, we know that the area of a semicircle is half that of a full circle. Recall that the area of a circle is given by 𝐴=πœ‹π‘Ÿ.

We have been given that the area of the semicircle is 51.04 cm2, so 𝐴 is twice this, which is 102.08 cm2. Thus, we have 102.08=πœ‹π‘Ÿ.

We can rearrange this in terms of π‘Ÿ; first, we divide by πœ‹: 102.08πœ‹=π‘Ÿ.

Now, we can take the square root of both sides, giving us π‘Ÿ=ο„ž102.08πœ‹=5.70….cm

Finally, we need to calculate the perimeter of the semicircle. Consider the diagram below.

This shows us that the perimeter is given by 2π‘Ÿ+𝐢2. We know π‘Ÿ, and we can find 𝐢 using 𝐢=2πœ‹π‘Ÿ, so we substitute these values in to get 𝑃=2π‘Ÿ+2πœ‹π‘Ÿ2=2π‘Ÿ+πœ‹π‘Ÿ=2(5.70…)+πœ‹(5.70…)=29.308…=29().cmnearestcm

For our final example, let us consider a real-world problem where we must find the area of a field composed of sections of circles.

Example 5: Finding the Area that a Goat Can Cover in a Field

A goat is tethered by a 10-metre-long rope to the corner of a barn. What area of the field can the goat reach?

Answer

This example effectively has two parts. First, we must find what area is covered by the goat by keeping in mind the length of the rope and the dimensions of the barn. Then, we must calculate this area using the formula for the area of a circle, modified to find portions of a circle.

First, let us consider what space the goat can cover above and to the right of the barn. We can see that in this case, since the rope is 10 metres long, the area forms part of a circle with radius 10 metres, as shown below.

In addition to this area, it is also possible for the goat to move below the barn. In that case, 5 metres of the rope will be stuck against the right wall, leaving 5 metres of length left in the rope. Thus, a quarter of a circle with radius 5 metres is formed, which we show below.

Thus, the total area is the addition of both of these areas together. To find these areas, we can use the formula for the area of a circle, which is 𝐴=πœ‹π‘Ÿ.

For the larger area, we can see that the area forms three-quarters of a full circle. Another way to think of this is that we have a full circle with one quarter cut out. Thus, if we calculate the area of a circle with radius π‘Ÿ=10 and multiply it by 34, we will get this area. That is, areaoflargerpart=34πœ‹Γ—10.

For the smaller area, we have a quarter-circle with radius π‘Ÿ=5. Therefore, its area is the area of a circle with radius 5 multiplied by 14, which is areaofsmallerpart=14πœ‹Γ—5.

Adding these two areas together, we get totalaream=34πœ‹Γ—10+14πœ‹Γ—5=3004πœ‹+254πœ‹=3254πœ‹.

Let us summarize the key points we have learned in this explainer.

Key Points

  • The circumference of a circle is given by 𝐢=πœ‹π‘‘, where 𝑑 is a diameter. Alternatively, using 𝑑=2π‘Ÿ (where π‘Ÿ is a radius), we can write this as 𝐢=2πœ‹π‘Ÿ.
  • The area of a circle is given by 𝐴=πœ‹π‘Ÿ, where π‘Ÿ is the radius.
  • If we are given the radius, diameter, circumference, or area, we can use it to calculate the other values using the above formulas.
  • For semicircles and quarter-circles, the perimeter can be written in terms of the diameter 𝑑 and circumference 𝐢 of the full circle: perimeterofasemicircleperimeterofaquarter-circle=𝑑+𝐢2,=𝑑+𝐢4.
  • Similarly, the areas of semicircles and quarter-circles are just fractions of the overall area, as shown below.

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