Geometry: Measuring Circles
This is a free lesson from our course in Geometry
 
   
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...)
<h2> Geometry - Measuring Circles</h2> <p> center, angle, basics, video, circles, point, tangent, secants, chord, relationship, geometry, find, angle, diameter, degrees, radians, arc length example, radius, measure, solution, math help, circumference of a circle, formulas for working with angles, practice questions, quizzes</p> <p> Geometry - Measuring Circles</p>
Other useful lessons:
Equation of a circle in standard form
Circles- arcs, chords, tangents, sector, segment, secant
Inscribed and Circumscribed Polygons
(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.
Circle-Tangent
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. Degree Measure of an Arc
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 re-written 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 m2.
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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