A circle has the following equation: $ x^2-8ax+y^2+10ay=-5a^2 $ Point O is its center and is in the second quadrant ($$ a\neq0 $$) Use the completing the square method to find the center of the circle and its radius in terms of $$ a $$.

$$ O(4a,-5a),\hspace{4pt}R=-6a $$

$$ O(4a,5a),\hspace{4pt}R=6a $$

$$ O(4a,10a),\hspace{4pt}R=6a $$

$$ O(4a,-5a),\hspace{4pt}R=6a $$

A circle has the equation: $$ x^2-6x+y^2-4y=7 $$ Use the completing the square method to find the center of the circle and its radius.

$$ (3,2),\hspace{6pt} R=\sqrt{20} $$

$$ (3,-2),\hspace{6pt} R=\sqrt{20} $$

Arcs in a Circle Practice Problems and Worksheets

Understanding Arcs in a Circle

Complete explanation with examples

Arcs in a Circle

An arc is a portion of the circumference of a circle, the part that is between $2$ points on the circle.
The arc is part of the circumference of the circle and does not pass inside the circle.
Arcs are categorized as either major (larger than half the circle) or minor (smaller than half the circle).

Diagram of a circle illustrating key components: a central angle labeled theta, a radius, an arc (green), and a sector (shaded blue). The diagram highlights the relationships between these elements in the context of circles. Featured in a guide on understanding arcs and sectors in a circle.

More relevant components of the circle:

Radius: The distance from the center of the circle to any point on the circumference.
Diameter: A straight line passing through the center that connects two points on the circumference, equal to twice the radius.
Arc: A portion of the circumference.
Chord: A line segment connecting two points on the circle.
Tangent: A line that touches the circle at exactly one point.

Arc Length - Advanced

The central angle formed by two radii connecting the center to the arc determines its size. The distance along the arc can be calculated using the formula

$\text{Length} = \frac{\theta}{360} \times 2\pi r$

$\theta$ is the central angle in degrees.

Sector: The area bounded by two radii and the arc, resembling a slice of pie.

Detailed explanation

Practice Arcs in a Circle

Test your knowledge with 6 quizzes

Calculate the length of the arc marked in red given that the circumference is 12.

Examples with solutions for Arcs in a Circle

Step-by-step solutions included

Exercise #1

In which of the circles is the point marked inside of the circle and not on the circumference?

Step-by-Step Solution

Let's remember that the circular line draws the shape of the circle, and the inner part is called a disk.

Therefore, in diagram B, the point is located in the inner part, meaning inside the disk.

Answer:

Video Solution

Exercise #2

Identify which diagram shows the radius of a circle:

Step-by-Step Solution

Remember that a radius is a line segment connecting the center of a circle to any point on the circle itself.

In drawing C we can see that the line coming from the center of the circle indeed connects to a point on the circle itself, while in the other drawings the lines don't touch any point on the circle.

Therefore, C is the correct drawing.

Answer:

Exercise #3

Identify which diagram shows the radius of a circle:

Step-by-Step Solution

Remember that a radius is a line segment connecting the center of the circle to a point that lies on the circle itself.

In drawing A, the line doesn't touch any point on the circle itself.

In drawing B, the line doesn't pass through the center of the circle.

We can see that in drawing C, the line that extends from the center of the circle is indeed connected to a point on the circle itself.

Answer:

Video Solution

Exercise #4

Where does a point need to be so that its distance from the center of the circle is the shortest?

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is less than the radius from the center of the circle will necessarily be inside the circle.

Answer:

Inside

Exercise #5

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is greater than the center of the circle will necessarily be outside the circle.

Answer:

greater

Frequently Asked Questions

Everything you need to know about Arcs in a Circle

What is the difference between a major arc and a minor arc?

+

A minor arc is smaller than half the circle (less than 180°), while a major arc is larger than half the circle (greater than 180°). The arc between two points creates both a major and minor arc, and we specify which one we mean by naming three points.

How do you calculate the length of an arc in a circle?

+

Use the formula: Arc Length = (θ/360°) × 2πr, where θ is the central angle in degrees and r is the radius. For angles in radians, use Arc Length = θ × r.

What's the difference between an arc and a chord in a circle?

+

An arc is the curved portion of the circumference between two points on the circle. A chord is a straight line segment connecting those same two points directly through the interior of the circle.

How do you name an arc in a circle?

+

For minor arcs, use two letters (like arc AB). For major arcs, use three letters with the middle letter on the arc (like arc ACB). This prevents confusion about which arc you're referring to.

What is a sector and how does it relate to an arc?

+

A sector is the pie-slice shaped region bounded by two radii and an arc. The arc forms the curved boundary of the sector, while the two radii form the straight boundaries meeting at the center.

Can an arc be the entire circumference of a circle?

+

Technically no, because an arc is defined as the portion between two distinct points on a circle. The entire circumference would require the two points to be the same point, which doesn't create a proper arc definition.

How do central angles relate to arc measurement?

+

The central angle and its corresponding arc have the same degree measurement. If a central angle measures 60°, then the arc it intercepts also measures 60°. This relationship is fundamental to all arc calculations.

What are some real-world examples of arcs?

+

Common examples include: rainbow shapes in the sky, hair ties or rubber bands when stretched, pizza slice crusts, windshield wiper paths, and portions of circular race tracks or roads.

Continue Your Math Journey

Master Arcs in a Circle first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Circle Distance from a chord to the center of a circle Chords of a Circle Central Angle in a Circle Perpendicular to a chord from the center of a circle Inscribed angle in a circle Tangent to a circle Circuit Components Elements of the circumference Area of a circle The Circumference of a Circle How is the radius calculated using its circumference?

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Calculate the ratio between radii of two circles Calculating arc length Calculating parts of the circle Identifying and defining elements