In this lesson you’ll explore the concepts of and different ways of Measuring
Circles in line with relationship of different circle parts
learnt earlier in context with Geometry terms and shapes. This
section also covers about measuring lengths of segments and degree
of angles related to circles and further explains in more details
how to measure circles connecting special relationship between their
circumference and diameter. The contents will be explained by the
instructor in own handwriting and using video,
with the help of
several examples.
The measurement of circles has a close relationship with the measurement
of angles. It is by virtue of its rotational aspect i.e. a ray if turned
about the vertex, it coincides with a static location like a point.
In fact a circle is a useful tool for measuring ‘planes of angles’,
as any angle cuts off the same fraction part of the circle whose center
is at the vertex of the angle.
(More text below video...)
(Continued from above)
A quick review of few Geometric terms and definition of tangent and an arc (figures below):
PO = radius PQ = tangent
AB = chord and AB arc
The tangent here in Geometry is unlike the trigonometric ratio meaning.
It is a line or line segment that touches the perimeter of a circle at
one point only and is perpendicular to the radius that contains the point.
Degree Measure of an Arc: ‘arc of a circle is the
part of the circle between two points on the circle’. An arc of a circle is called an
intercepted arc, or an arc intercepted by an angle, if each end point of the arc is
on a different ray of the angle and the other points of the arc are in the interior
of the angle. The degree measure of an arc is equal to the measure of the central
angle that intercepts the arc. E.g.
Different Ways to Measure Circles
A degree is defined as 1/360 of a rotation of a radius about the center of the circle.
Simply put, a circle is divided into 360 equal degrees, and a right angle
(1/4th of the rotation) is 90. While degree measure divides a circle into 360
equal parts, gradient measure does it in 400 equal parts i.e. there are 100
gradients in a right angle making it a better fit with the decimal system.
However, commonly degree measure is used for measuring angles using fractional
parts of a circle.
Another method for measuring angles in case of the circle is radian
measure.
It is useful especially in applications of calculus involving trigonometric
functions—for example, sine, cosine, or tangent etc. In such cases the angle
of the trigonometric function be measured in radians. The geometric
definition of radians based on measuring distances, conceptually states
that the measure in radians is determined by the intersected arc length (s)
divided by radius (r) and it can be expressed as:
(radians) = arc length/radius = s/r
For example, 0.84 radians when converted to degrees, the result is
48.13.
Notice that it is important to understand that when using arc length, the same angle
cuts off a larger length arc on larger circles than on smaller circles. Therefore,
when using ‘arc length’ to measure an angle, this is an important aspect.
Next is how to determine the circumference (C). You may review from earlier learning that
the ratio of the circumference of a circle to the length of its diameter (d), is
; i.e.
multiplying both sides of this equation by d results in a formula: C = d.
As you know, d = 2r, it can be rewritten
C = ·2r
=
2r.
Remember now some equivalents:
Conversion factors: 2 radians
= 360 and
radian =
180.
An angle of one radian has the same measure as the central angle in a circle, whose arc
length and radius is same. One radian is about 57 17'45".
Example: Find the length of a 30
arc of a circle with 8 cm radius.
Steps to follow:
Length of arc =
2()r /
360
= 2 (8)(30) / 360
= The length of the arc is (4/3)() cm, and substituting for = 3.14
= 4.19 cm, as the final answer
How to Measure the Circumference of a Circle:
Circumference is the distance around a circle. The steps for calculating circumference
can be used to solve real life problems. For example, Steve is interested in knowing
the distance around a circular path, having estimated its diameter as 250m.
Step 1: Measure diameter.
Step 2: Find out radius i.e. half of the diameter of circle
Step 3: Pick up
i.e. the ratio of the circumference of a circle to its diameter.
This number is regardless of the size of the circle and its value is rounded to 3.14.
Step 4: Calculate circumference.
The estimated diameter of the circular path is 250m across. The circumference; therefore,
would be 250 times 3.14. It equals to 785m, as the final answer. Note that the units of
measurement should be same in both the cases.
Calculators can be used to determine the circumference and areas efficiently. E.g.
the question in example below can familiarize you with the process of using Casio
calculator and the respective formulas for circumference and area. For example,
using calculator find out the circumference and area of the circle, having radius 6.2m.
It is known that the he perimeter of a circle, usually called the circumference can be determined by the formulas:
Circumference = (d)
= 2r, where’d’
represents the diameter of the circle and 'r' the radius. Then, Area = r2
STEP 1: Choose MATH mode in SETUP menu, using
STEP 2: Enter 2
STEP 3: Multiply by radius
The use of key, followed by pressing the equals key, gives the result as a numerical
approx of 38.955 m.
Similarly for determining the area, follow the steps:
STEP 1: Enter
STEP 2: Multiply by radius squared
STEP 3: Evaluate area
It gives the area to the decimal approximation, 120.76 m^{2}.
To remember formulas for working with angles in circles:
• Central Angle A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.
• Inscribed Angle: An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.
• Tangent Chord Angle: An angle formed by an intersecting tangent and chord has its vertex "on" the circle.
• Angle Formed Inside of a Circle
by Two Intersecting Chords: When two chords intersect "inside" a circle, four angles are formed. At the point of intersection, two sets of vertical angles formed are equal.
•
Angle Formed Outside of a Circle by the Intersection of, "Two Tangents" or "Two Secants" or "a Tangent and a Secant": Angle formed outside is equal to half the difference of intercepted arcs.
The video above will explain more in detail about the "Measuring Circles
in line with relationship of different circle parts", with the help of several examples.
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