Arc Length Homework Solutions

Related Topics: More Geometry Lessons



 


In these lessons, we will learn
  • the arc of a circle
  • the central angle
  • the arc measure
  • the arc length formula when the arc measure is given in degrees
  • the arc length formula when the arc measure is given in radians
  • how to calculate the arc length using the arc length formula

Arc of a Circle

An arc is any connected part of the circumference of a circle.

In the diagram above, the part of the circle from M to N forms an arc. It is called arc MN.

An arc could be a minor arc, a semicircle or a major arc.

  • A semicircle is an arc that is half a circle.
  • A minor arc is an arc that is smaller than a semicircle.
  • A major arc is an arc that is larger than a semicircle.

Central Angle

A central angle is an angle whose vertex is at the center of a circle.

In the diagram above, the central angle for arc MN is 45°.

The sum of the central angles in any circle is 360°.

Arc Measure

The measure of a semicircle is 180°.

The measure of a minor arc is equal to the measure of the central angle that intercepts the arc. We can also say that the measure of a minor arc is equal to the measure of the central angle that is subtended by the arc. In the diagram below, the measure of arc MN is 45°.

The measure of the major arc is equal to 360° minus the measure of the associated minor arc.

The following video shows how to identify semicircle, minor arc and major arc and their measures.

Arc Length Formula

The arc length is the distance along the part of the circumference that makes up the arc.


Arc Measure given in Degrees

Since the arc length is a fraction of the circumference of the circle, we can calculated it in the following way. Find the circumference of the circle and then multiply by the measure of the arc divided by 360°. Remember that the measure of the arc is equal to the measure of the central angle.

The formula for the arc length of a circle is

where r is the radius of the circle and m is the measure of the arc (or central angle) in degrees.

Worksheet to calculate arc length and area of a sector (degrees).

Arc Measure given in Radians

If the measure of the arc (or central angle) is given in radians, then the formula for the arc length of a circle is the product of the radius and the arc measure.

Arc Length = r × m

where r is the radius of the circle and m is the measure of the arc (or central angle) in radians

The above formulas allow us to calculate any one of the values given the other two values.

Worksheet to calculate arc length and area of sector (radians).

Calculate Arc Length given Measure of Arc in degrees

From the formula, we can calculate the length of the arc.

Example:

If the circumference of the following circle is 54 cm, what is the length of the arc ABC?

Solution:

Circumference = 2πr = 54

Example:

If the radius of a circle is 5 cm and the measure of the arc is 110˚, what is the length of the arc?

Solution:



Arc Length in Degrees
These video shows how to define arc length and how it is different from arc measure. They will also show how to calculate the length of an arc when the arc measure is given in degrees.

Arc Length Theorem - How to Find the Length of an Arc?
How to use the Arc Length Formula when the arc measure is given in degrees?
This video lesson discusses how to find the length of an arc. First, the arc length theorem is reviewed and explained. An example of find the length of a major arc is modeled. The given information is the measure of the related minor arc and the radius of the circle. How to find the arc length on a circle when the central angle is given in degrees?

Calculate Arc Length given Measure of Arc in radians

If the measure of the arc (or central angle) is given in radians, then the formula for the arc length of a circle is

Arc Length = r × m

where r is the radius of the circle and m is the measure of the arc (or central angle) in radians

This video shows how to use the Arc Length Formula when the measure of the arc is given in radians. Definition of Arc Length and Finding the Arc Length when the central angle is given in radians.

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Solution:

Diameter of the cart wheel is 4.3 m
[Given.]

Circumference of the wheel = 4.3
[Circumference = .]

Distance covered for 10 revolutions = 10 × 4.3 = 43 m.
[Distance covered = circumference × no. of revolutions.]


Correct answer : (3)



Solution:

Radius of the wheel is 50 cm
[Given.]

Diameter = 100 cm = 1 m

Circumference of the wheel = 1
[Circumference = .]

No. of revolutions made in traveling a distance of 270 m = = 90
[.]


Correct answer : (3)



Solution:

Base of the right triangle = 13.5 cm.

Height of the right triangle = 18 cm.

Radius of the circle = = 11.25 cm.
[The radius of the circumcircle of a right triangle is half of the hypotenuse.]

Circumference of the circle = 2r = 2 × × 11.25 = 67.5 cm.


Correct answer : (1)



Solution:


Perimeter of the figure = Length of the line segment AB + Length of arc BC + Length of line segment CD + length of arc DE + length of line segment EF + length of line segment FA

AB = 2 cm
[Given.]

Length of arc BC = × × 2 = 1.57 cm
[Formula for the length of the arc = × × radius.]

CD = 5 cm
[Given.]

Length of arc DE = × = 6.28 cm

Δ AEF is equilateral.
[AF = EF, AFE = 60.]

AF = EF = AE = 1 + 5 + 4 = 10 cm
[AE = AC + CD + DE.]

Perimeter of the figure = 1 + 1.57 + 5 + 6.28 + 10 + 10 = 33.85 cm
[Step 1.]


Correct answer : (3)



Solution:



When the circle is inscribed in a square (figure A), perimeter of the square = 4 × 2 = 8
[Side of the square = 2.]

Circumference of the circle = 2

2 < 8

So, in figure(A), the perimeter of the circle is always less than the perimeter of the square.

When the square is inscribed in the circle, side of the square = ×
[Diagonal of the square = 2, side = = × .]

Perimeter of the square = 4 ×

Perimeter of the circle = 2

4 < 2

So, in figure(B), the Perimeter of the square is less than perimeter of the circle.

Statements I and III are correct.


Correct answer : (4)



Solution:

Length of arc 1 = × 2 × 6 =
[Formula for the length of arc = × 2 × .]

Length of arc 2 = × 2 × 3 = ×

Length of arc 3 = × 2 = ×

The ascending order is arc 3, arc 1, arc 2.


Correct answer : (3)



Solution:

The radius of the first semicircle is 17.

Radius of the second semi circle =

Radius of the third semicircle =

Circumference of the first semicircle = × 17

Circumference of the second semicircle = ×

Circumference of the third semicircle = ×

Sum of the circumferences = × 17 + × + × + ------------------

= (17 + + + ---------------- )

= 2(17 + + + ---------------- ) = 7 m
[Given, the semicircles are drawn on a line of length 7 m.]

So, the sum of the arclengths of all the semi circular portions = × m = m.
[Given, line length = 7 m.]


Correct answer : (4)



Solution:

Perimeter of the outer curve = × 2 × 5 = 14.39 cm

Perimeter of the inner curve = × 2 × 6 = 8.06 cm

Perimeter = 14.39 - 8.06 = 6.33 cm


Correct answer : (4)



Solution:

The distance covered by the cyclists are the circumference of the semi circles.

Distance covered by the cyclist R = Circumference of the semi circle with AE as diameter.

=
[Circumference of semi circle with diameter is .]

= = 4
[Substitute AE = 8.]

Distance covered by the cyclist Q = Circumference of semicircle with diameter AC + circumference of semi circle with diameter CE.

=

=

Distance covered by the cyclist P = circumference of the semi circle with diameter AB + circumference of the semi circle with diameter BC + circumference of the semi circle with diameter CD + circumference of the semicircle with diameter DE.

= × × AB + × × BC + × × CD + × × DE

= × × 2 + × × 2 + × × 2 + × × 2 = 4

The distances covered by the cyclists P, Q and R are same.

Therefore, three cyclists P, Q and R reach the point E in same time.


Correct answer : (1)



Solution:

Radius of the cake = = 15 cm.
[Radius = × diameter.]

Length of the decorative band is 100 cm.
[Given.]

Perimeter of the cake = 2.
[Perimeter of the circle = 2 .]

2 × 3 × 15 = 90 cm
[Substitute the values and simplify.]

The required length of the band is 90 cm, therefore the given band is sufficient to decorate the cake.


Correct answer : (1)

 1.  

The diameter of a cart wheel is 4.3 m. How much distance will it cover if it makes 10 revolutions?

 2.  

The radius of a bicycle wheel is 50 cm. How many revolutions does the wheel make when the bicycle travels 270 m?[Take = 3.]

 3.  

Find the circumference of a circle in which a right triangle of base () 13.5 cm and height () 18 cm is inscribed. [Take = 3.]

 4.  

Find the perimeter of the figure if = 1 cm, = 5 cm, = 4 cm.

 5.  

Select the correct statement / statements.
I. When a square is inscribed in a circle, the perimeter of the square is always less than the perimeter of the circle.
II. When a circle is inscribed in a square, the perimeter of the circle will be less than the perimeter of the square if the radius of the circle is greater than 2.
III. When a circle is inscribed in a square, the perimeter of the circle will be always less than the perimeter of the square.

 6.  

Arrange the arcs in the figure in the ascending order of its lengths.

 7.  

Semi circles are drawn on a line of length 7 m as shown. The first semicircle has a radius of 17 cm. Radius of each consecutive semicircle is half the radius of the preceding semi circle. What is the sum of the arc lengths of all the semicircular portions?


 8.  

Find the perimeter of the figure.
[Take = 77o, = 165o, = 6 cm, = 5 cm.]

 9.  

Three cyclists P, Q and R with the same speed starts from A to reach E as shown in the figure. P will follow the circular route AB-BC-CD-DE; Q will follow the route AC-CE and R will follow the route AE. Who will reach point E first? [Take = 2.]


 10.  

Ebin bakes a christmas cake in a 30 cm diameter tin. He has a 100 cm length of decorative band to go around the outer line of the cake. Is this band long enough? [Take = 3.]



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