Circular Measure | A Level Maths Revision Notes

Why are we interested in this topic?

Circle is an amazing shape. The sun, the moon and the planets look circular (yes they are actually spherical in shape). The bicycle you probably love to ride has circular tyres. The can of icy cool carbonated soda you cherish on a hot day has a circular base. Even the pair of glasses of world-famous Harry Potter has circular glass lenses.

There are some astonishing features this shape has to offer. In civil engineering, the arc shape has been used to construct strong bridges that still stand today. You will not see a soap bubble that has any other shape except spherical (loosely speaking, 3D form of a circle). There would have been no fun in playing football if it were not round.

This does not stop here. Mathematicians have defined a constant number called “pi” (written as “π”) that basically relates to the boundary length and the diameter of the circle (we will discuss it in detail later). This constant appears at several and unexpected places in maths, physics and engineering where there will be no circles and arcs. The concept of periodicity, frequency, wavelength etc. all stem from the idea of circles. Thus, it is not possible to assert enough the significance of circles.

Circle and Polygon

Polygon is a two-dimensional figure with connected straight-line segments that form a close figure. For example, a square is a polygon with four equal sides. A pentagon is a five-sided polygon. A hexagon is six-sided polygon. You will find it interesting to note that a football surface is made of pentagon and hexagon shapes.

Now here is the interesting thing. Imagine what happens as you keep increasing the number of sides of a polygon (lengths of sides should be equal) i.e., 10 sides, 20 sides, 100 sides, 1000 sides and so on. Yes, that is right. It will approach the shape of a circle! In other words, you can think of a circle as a polygon with unlimited number of sides.

Concept of angle and its units

When two lines intersect, an angle is formed between them. The angle is a measure of the amount of rotation required (with the intersection point as the pivot) to align one line with the other. You can think of two line segments as two hands of a clock. The angle between the hands is the measure of ‘rotational difference’ between the hands.

Angle is measured in the units of degrees and radians. A circle is defined to be constituted of 360 degrees, written as 360⁰. Therefore, a semi-circle (half a circle) consists of 180 degrees or 180⁰. Consider the clock example with one hand pointing to 12:00 and other pointing to 06:00. The angle between the two hands is said to be 180⁰ in that case. What will be the angle (in degrees) between the two hands when one hand is at 12:00 and other at 03:00?

In the clock example you may have noted that we can define angle between two hands in two ways. We may end up with two different angles depending on the direction of rotation traversed to align the two hands. For example, when on hand is at 6:00 and other is at 5:00, then we can either rotate the hand through the shorter path from 5:00 increasing towards 6:00 or through the longer opposite path (all the way traversing 4:00, 3:00…12:00, 11:00,…7:00 and then 6:00). To solve the problem, we define a reference and a standard direction of rotation to measure the angle. Or depending on the application, we may choose the shorter angle among the two. As a matter of fact, this is not something to worry about.

There is another unit of measurement of angle. It is called radian. One radian is approximately 57.3⁰. How do we get this unique value? It will be discussed later.

Exercise: Keeping in mind the above relation between radian and degrees, find how many radians are there is 1⁰, 90⁰ and 180⁰.

In physics, the radian is considered a dimensionless quantity because of the fact that it is actually a ratio of two quantities with same units. Most scientific calculators support both the units of degree and the radian. Care should be taken while dealing with angle values. When the calculator is set to the “degree mode”, you should enter the angle in degrees and if the calculator is in “radian mode” then correspondingly the angle in radians should be entered. Trigonometric functions make use of angles.

Arc length

Let us consider the example of the clock once again. At the start of a minute, the “Second hand” points towards 12:00. It can be realized that as the second hand traverses an angle, a corresponding arc is also traversed. As the angle increases, the arc length also increases. We can also take another example. How do we make a circle? We make use of a technical instrument called pair of compasses. It helps make accurate circles of variable radii. While using the compass, it is observed that greater the angle we rotate it through, greater is the arc length that is sketched. A full rotation yields a full circle (when an angle of 360⁰ is traversed). In fact, the arc length is directly proportional to the angle. It is represented as:

S ∝ θ (1)

The above relation says that “S (arc length) is directly proportional to θ (Greek letter known as theta. It is usually used to represent the angle. There are other symbols that can be used to represent an angle for example, α, β, ϕ, γ)”.

The proportionality can be converted into an equality by the use of a constant k.

S = kθ (2)

Intuitively, this constant is actually equal to the length of the line-segment that is being rotated. It is represented by the symbol ‘r’ since it equates to the radius of the circle formed when the line segment is rotated through 360 degrees. Thus, we have,

S = rθ (3)

It should be kept in mind that this relation works when θ is in radians. As a matter of fact, you should realize that this is the reason we have the unit of radian. Moreover, observe that if S=r i.e., if the arc length traversed by a line segment is equal to the length of the line segment, then θ evaluates to 1 radian. This is the definition of a radian. The angle through which a line segment of length ‘r’ rotates to traverse an arc of length also equal to ‘r’.

Figure 1 Arc length when angle is 1 radian

When the angle becomes equal to 2π, the arc takes the shape of a complete circle. Using the arc length formula, we can find the circumference or arc-length of one complete circle. Plugging in the angle equal to 2π, we get the circumference equal to 2πr. From here you can also realize that a 2π radian angle is traversed when one complete circle is formed, therefore,

2π radians = 360⁰ (4)

This gives the accurate conversion factor between radian and degrees.

1 radian = 180⁰/π (5)

There is another important result that is worth noting. We can write the following expression.

$\pi = \frac{2\pi\,r}{2r}$ (6)

Now, realize what the numerator and the denominator of the above expression are. We note that the numerator is actually the circumference of the circle, and the denominator is the diameter of the circle. Thus,

$\pi = \frac{Circumference\, of\, the\, circle}{Diameter\, of\, the\, circle}$ (7)

Wow! We have just proved that π is actually the ratio of circumference and the diameter of a circle. All circles follow this rule.

Exercise: Find the angle subtended by a line segment of length 15 cm when it traverses an arc of length 20 cm. What is the angle subtended if another line segment of length 40 cm traverses an arc of identical length.

The unit of degree is subdivided into minutes and seconds.

1 degree = 60 arc minutes (8)

1 arc minute = 60 arc seconds (9)

The arc minutes and arc seconds are represented by ‘ and “ respectively. For example, an angle of 12.14⁰ can be represented as 12⁰ and (0.14)(60 arc minutes) = 12⁰8.4’ = 12⁰8’ and (0.4)(60 arc seconds) = 12⁰8’24”

It is similar to how we represent length in feet and inches. Similarly, an angle given in the form of arc minutes and arc seconds can be converted to the angle in degrees in form of decimal number by reversing the process.

There is one interesting thing to note here. As the angle subtended by an arc becomes very small, the arc becomes more like a straight line-segment. Conversely, as the length ‘r’ increases, a given arc length subtends less and less angle.

Exercise: Angle subtended in the eye of a person by a building 1.5 km away is 40 arc minutes. Using this information find the height of the building. (Hint: Convert arc minutes to degrees and then to radians. The height of the building is analogue to the arc length. The symbol ‘r’ is the distance between the building and the person. Note that since the angle subtended is very small therefore, the arc-length is practically equal to a straight line segment i.e., the building).

The number ‘π’ is really a mysterious number. It is basically an irrational number, meaning that it cannot be written as a ratio of two integer numbers. Its value is 3.1415926…. (the digits continue). A commonly used approximation is 22/7. But remember that the fraction 22/7 is just an approximation. The exact value of π is not completely known till date. Currently, 31.4 trillion digits have been evaluated by computers! You will find it astonishing that you may even find your birthday in the digits of π. You can google “mypiday” and find at which position your birthday appears.

Perimeter: In textbooks, you may come across questions related to the perimeter of a shape. The perimeter is the length of the total boundary of any shape. The perimeter of a square is four times its side length. The perimeter of a circle is the circumference of that circle equal to 2πr.

What if you are asked to find the perimeter of the shape as in Figure 1? You may be tempted to answer that its perimeter is ‘S’ i.e., the arc length but that would be wrong. As you can see that the complete boundary of the shape also involves the two side lengths ‘r’. Therefore, as a whole, the perimeter of the shape given in Figure 1 is ‘S+r+r’ = ‘S+2r’. When the angle subtended by the shape is 1 radian then you may observe that the perimeter reduces to ‘r+2r’ = ‘3r’ (because S = r when θ = 1 radian). Moreover, given the angle subtended by the arc, we can find the arc-length by using the formula S=rθ.

Note that if θ is given in degrees then it should be converted in radians first before applying the arc length formula.

Let us solve an example to see how that works. Given an arc-length subtends an angle of 70 degrees. The arc-length is equal to 100 km. Find the perimeter of the shape (sector of a circle). First we convert the angle into radians. 70 degrees equal to $70^{\circ} \times \frac{\pi\, radians}{180^{\circ}}\approx 1.2217\, radians$ (Note that $\frac{\pi}{180^{\circ}}$ is used to convert degree to radian. To convert radian to degree, we use the conversion factor of $\frac{180^{\circ}}{\pi}$ ). Thus, θ=1.2217 rad. Next, arc -length i.e., S=100km. Using, S = rθ, we have r ≈ 81.85 km. Now, the final step. The perimeter of the sector of circular area is ‘S+2r’ = 100 km + 2(81.85 km) = 263.7 km.

Exercise: A biologist is working in a lab. She is using a microscope to view a sample taken from the Delamere Forest. As she looks into the eyepiece of the microscope, she finds out that some microbials are approximately of the shape of a semi-circle. With the help of the reticule/reticle (the microscopic scale etched in the glass of eyepiece), she figures out the length of the flat side to be 30 μm (micro-meter =10^-6 m). In order to study the diffusion rate of oxygen into the microbial cell, the biologist needs to figure out the total perimeter of the microbial cell. As a student of mathematics, you are asked to help her out. Find the boundary length of the cell.

Area of Circular Sector

We have observed the closed surface formed by the arc length and the two sides. That closed surface is called the Circular sector (because it forms part of a circle). Every surface has some area represented in square of the units of length. The area of square is square of the length of its side. The area of a circle is πr². Similarly, the circular sector has also a well-defined area, whose value is given by the formula:

$Area_{circular\, sector}=\frac{1}{2}r^{2}\theta$ (10)

The equation shows that the area of the sector is proportional to the square of the radius i.e., by doubling the radius, the area of the sector increases by a factor of 4. On the other hand, the area is proportional to the angle (not its square) i.e., doubling the angle of the sector will increase the area of the sector by only a factor of 2. The formula above can be rewritten in the form of the arc-length ‘S’ as:

$Area_{circular\, sector}=\frac{S^{2}}{2\theta }$ (11)

Now, you might be thinking that why on earth is the ‘θ’ in denominator. It would mean that the area of the sector is inversely related with the angle. This is evidently pure contradiction with what we get from the preceding equation. This “paradox” is resolved by considering that as θ is changed, the arc length ‘S’ also varies with the same factor (keeping ‘r’ constant). But since ‘S’ is squared, therefore, the net result is the same as pointed out by the previous equation.

Let us solve an example. We are required to find the area of a grass lawn in ‘m²‘ that has side length of 10 m and a subtended angle of π/2. Using the formula, we have Area_lawn = 0.5 x (10)² x π/2 = 78.54 m².

Exercise: Find the area of circular sector in ‘m²’, that has perimeter of 100 cm with arc length of 40 cm.

The proportional relation of the sector area with the subtended angle is exploited to solve certain numerical questions. For example, consider that one sector of a circle has an area A₁ with subtended angle of θ₁. Another sector of the same circle has an area A₂, with unknown subtended angle θ₂. We can find the unknown angle as:

$\theta _{2}=\frac{A_{2}\theta _{1}}{A_{1}}$ (11)

Exercise: An area of 1.2 km² is occupied by a sector of a large circular field in Edinburgh. The angle subtended by the sector is 50 degrees. Find the area occupied by another sector in the same field, that subtends an angle of 0.5 radians. Also, find the total area of the circular field.

Difference between sector and segment

You should be cautious of the difference between a sector of a circle and a segment of a circle. A segment is that part of a circle that is formed when a straight line, called secant, cuts off the circle at any two points. A sector, on the other hand, is that part of the circle that also possesses the subtended angle.

It is worth noting that in a special case, the sector and segment may equate. If the chord passes through the center then the region thus formed can be called a segment as well as a sector (with subtended angle of 180⁰).

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