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Central Angles and Arcs

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In Geometry, each and every shape makes an arcs and Central Angles; we will discuss how we evaluate central angles and arcs–
First of all, we discuss evaluation of central angles of different geometric shapes–
For central angle of different geometric shapes, we use following steps

Step 1: First of all, we calculate circumference of given figure by using their respective formulas, like circumference of Circle (C) = 2 * π* r, here π= 3.14 and ‘r’ is radius.
Step 2: After evaluation of circumference, now we calculate central angle of given geometric shape by using following formula–
Central angle of circle (θ) = (arc length * 360) / (2 * π* r).

Suppose we have to find out central angle of circle, whose Arc Length is 7.33 inch and radius is 7 inch, then we apply above central angle formula–
Central angle of circle (θ) = (arc length * 360) / (2 * π* r),
= (7.33 * 360) / 43.96,
= 60 degree.
So, central angle of circle is 60 degree, whose radius is 7 inch and arc length is 7.33 inch.
arcs and central angles are correlated with each other means for evaluation of arcs depend on central angles and evaluation of central angles are depend on arcs like when we evaluate arcs, we use following steps–

Step 1: First of all, we calculate length, like length of radius in circle.
Step 2: After length, now we calculate central angle of given geometric shape, which is generally given in degree form and we have to convert this central angle into radian form by multiplying π/180 to the angle, here value of ‘π’ is equal to 3.14.
Step 3: after first two steps, now we calculate arc of given geometric shape by using the following formula–
Arc of shape (s) = l * θ, here s is use for representing arc of a shape.
Therefore, we can say that central angles and arcs are related with each other.

Central Angles

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Each and every geometric shape like Circle has one central angle and for calculating the central angle of circle, we use following steps-
Step 1: First we calculate circumference of given Geometry by using their geometry following formula like we have circle as a geometry, then -
Circumference (C) of circle = 2 * π* r, here π= 3.14 and ‘r’ is radius, which is generally given in question, but if Diameter of circle is given, then we convert diameter into radius by dividing diameter by 2.
Step 2: After evaluation of circumference of given geometry, we calculate arc length of given geometry by using their following formula like we have circle as geometry, then–
Arc length (s) = r * θ,
Step 3: After circumference and arc length, now we calculate central angle of given geometry by using following formula–
Central angle of geometry (θ) = (arc length * 360) / circumference of geometry.
Suppose, we have a circle, whose arc length is 6.28 inch and diameter is 12 inch, then we use following steps for evaluation of central angle of circle-

Step 1: First we calculate circumference of circle by following formula–
Circumference (C) = 2 * π * r,
= 2 * 3.14 * (12 / 2), here 12 is diameter, so radius is (diameter / 2),
= 2 * 3.14 * 6,
= 37.68 inch.
Step 2: Here Arc Length of Circle is given–
Arc length (s) = 6.28 inches,
Step 3: Now we calculate central angle of circle by using the following formula–
Central angle of circle (θ) = (arc length * 360) / (2 * π* r),
= (6.28 * 360) / 37.68,
= 60 degree,
So, central angle of circle is '60 degrees'.
Therefore, we use these 3 steps for evaluation of central angles in geometry.

Arc In Geometry

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When we take any 2 connecting part of circumference of any Geometry is called as arcs of geometry and we can calculate arc length of Circle geometry. For calculating this arc length, we use following steps-

Step 1: First we calculate length of geometry like length of circle is equal to Radius of Circle ‘r’ = 11 inch.
Step 2: After length of geometry, now we calculate central angle of given circle, which is generally given in degree form and if angle is given in degree form, then we convert center angle into radian form by using following formula.
Center angle of geometry (θ) = π/180 * θ,
Step 3: After length of geometry and center angle of geometry, now we calculate length of arc of geometry by using the following formula.
Length of an arcs = length of geometry (l) * center angle of circle (θ).

Here l is use as a length of geometry variable and θ is use as a central angle of circle variable.
Suppose, we have a circle, whose Diameter length is equal to 36 inch and central angle is 150 degree, then for Arc Length of Circle, we use following steps -

Step 1: First we calculate length of radius of circle, whose diameter is equal to 36 inch.
Length of radius (r) = diameter / 2 = 36 / 2 = 18 inch.
So, radius length of given circle is '18 inch'.
Step 2: Now we calculate central angle of circle;
Central angle = π/180 * θ = π/180 * 150 = 5π/6 = 2.61 radian,
So, central angle of circle is equal to '2.61 radian'.
Arc length of circle (S) = l * θ,
= 18 * 2.61,
= 46.98 inch,
So, length of arc of circle is '46.98 inch'.

Arc Length Geometry

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Whenever we talk about Circle then some shapes come in our mind like Ring, tyre etc. and these round shapes Geometry is known as a circle and whole length of circle is known as circumference. When we take two points of this circumference length, it produces an Arc Length and for calculating this arc length, we use, we use following steps-

Step 1: We first calculate length of radius, generally length of radius is given in circle and if radius is given in Diameter form, then we divide diameter by 2 for calculating Radius of Circle.
Radius of circle = (diameter/2) like diameter is 36 inch, then radius of circle is 36/2 = 18 inch.
Step 2: After length of radius, now we calculate central angle of given circle, which is given in degree form and if angle is given in degree form, then we convert this central angle into radian form by multiplying π/180 to the angle, here value of ‘π’ is equal to 3.14.
Step 3: After length of radius and center angle of circle, now we calculate length of an arc of circle by using the following formula.
Length of an arc = length of radius (l) * center angle of circle (θ).
Here ‘l’ is use as a length of radius variable and ‘θ’ is used as a central angle of circle variable.
Suppose, we have a circle, whose diameter length is equal to 28 inch and central angle is 120 degree, then for Arc Length of Circle, we use following steps-

Step 1: We first calculate length of radius of circle, whose diameter is equal to
28 inch-
Length of radius (r) = diameter / 2 = 28 / 2 = 14 inch.
So, radius length of given circle is '11 inch'.
Step 2: Now we find out central angle of circle-
Central angle = π/180 * θ = π/180 * 120 = 2π/3 = 2.08 radian
So, central angle of circle is '2.08 radian'.
Now we use arc length of circle formula-
Arc length of circle (S) = l * θ,
= 14 * 2.08,
= 28 inch,
So, length of arc of circle is '28 inch'.