Unit 10: Circles
Every angle a circle can make, from its center, from its edge, and from outside it entirely, plus the arcs and sectors those angles cut off.
11 topics · 39 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Circumference and π
- How it worksHow does unrolling a circle's edge onto a straight line reveal π as a fixed ratio?
- Worked exampleFind the circumference of a circle with a diameter of 14.
- Worked exampleFind the diameter of a circle with a circumference of 62.8.
- Common mistakeWhy isn't π exactly equal to 3.14, just closely approximated by it?
Your turn1 of 2
Find the circumference of a circle with a diameter of 35.Parts of a Circle
- How it worksHow do a radius, a chord, a diameter, and a tangent line each relate differently to a circle's center?
- Common mistakeWhy isn't a secant line the same thing as a tangent line to a circle?
- Going furtherWhat happens to a chord's length as it slides farther away from passing through the center?
Your turn1 of 2
Why isn't a secant line the same thing as a tangent line to a circle?Central Angles and Arcs
- How it worksHow does the sum of a circle's arcs always add up to 360 degrees no matter how it's divided?
- Worked exampleFind the measure of an arc cut off by a central angle of 82 degrees.
- Worked exampleFind the measure of a major arc if its corresponding minor arc measures 110 degrees.
- Common mistakeWhy isn't an arc's length the same thing as its degree measure?
Your turn1 of 2
Find the measure of an arc cut off by a central angle of 121 degrees.Inscribed Angles
- How it worksHow does an inscribed angle always end up exactly half the measure of the arc it intercepts?
- Worked exampleFind the measure of an inscribed angle that intercepts an arc of 96 degrees.
- Common mistakeWhy doesn't an inscribed angle equal its intercepted arc the way a central angle does?
- Going furtherWhat happens to an inscribed angle the moment its intercepted arc becomes a full semicircle?
Your turn1 of 2
Find the measure of an inscribed angle that intercepts an arc of 121 degrees.Angles Formed by Chords, Secants, and Tangents
- How it worksHow do two chords crossing inside a circle create an angle equal to the average of two arcs?
- How it worksHow do two secants meeting outside a circle create an angle equal to half the difference of two arcs?
- Worked exampleFind the angle formed by two chords intersecting inside a circle that cut off arcs of 70 and 40 degrees.
- Worked exampleFind the angle formed by two secants meeting outside a circle that cut off arcs of 100 and 30 degrees.
Your turn1 of 2
Two chords intersect inside a circle, cutting off arcs of 121 and 114 degrees. Find the measure of the angle formed at their intersection.Power of a Point: Segment Relationships in Circles
- How it worksHow do two intersecting chords split each other so that one segment's pieces multiply to match the other's?
- How it worksHow does a tangent segment's square end up equal to the product of a whole secant and its external part?
- Worked exampleFind the missing chord segment when two chords cross and their pieces are 4 and 9 on one chord and 6 and x on the other.
- Worked exampleFind the length of a tangent segment if an external secant measures 12 total with an external part of 3.
Your turn1 of 2
Two chords intersect inside a circle. One chord is split into pieces of 11 and 10. The other chord is split into a piece of 14 and a piece of unknown length x. Find x.Arc Length
- How it worksHow does an arc's central angle become a fraction of the circle's entire circumference?
- Worked exampleFind the arc length of a 90 degree arc on a circle with radius 8.
- Common mistakeWhy doesn't doubling a circle's radius also double the arc length of the same central angle?
Your turn1 of 2
Find the arc length of a 247 degree arc on a circle with radius 20.Area of a Sector
- How it worksHow does a sector's central angle become a fraction of the circle's entire area?
- Worked exampleFind the area of a sector with a central angle of 120 degrees on a circle with radius 9.
- Ties togetherHow does the sector area formula and the arc length formula both come from the exact same fraction of the circle?
Your turn1 of 2
Find the area of a sector with a central angle of 247 degrees on a circle with radius 20.Chord Relationships
- How it worksHow does the perpendicular from the center to a chord bisect it every single time?
- Worked exampleFind the distance from a circle's center to a chord of length 16 in a circle with radius 10.
- Common mistakeWhy don't two chords have to be equal in length just because they look close in a drawing?
Your turn1 of 2
Find the distance from a circle's center to a chord of length 11 in a circle with radius 17.Tangent Lines to a Circle
- How it worksHow does a tangent line always meet the radius drawn to the point of tangency at a right angle?
- Worked exampleFind the length of a tangent segment from an external point 13 units from the center of a circle with radius 5.
- Common mistakeWhy don't two tangent segments from the same external point have to be different lengths?
- Going furtherWhat happens to the two tangent segments drawn from an external point as that point moves farther from the circle?
Your turn1 of 2
Find the length of a tangent segment from an external point 25 units from the center of a circle with radius 10.Inscribed and Circumscribed Polygons
- How it worksHow does a cyclic quadrilateral force its opposite angles to always add up to 180 degrees?
- Worked exampleFind the missing angle of a cyclic quadrilateral if its opposite angle measures 108 degrees.
- Going furtherWhat happens to the circumscribed circle of a triangle as the triangle's angles approach 90 degrees?
Your turn1 of 2
A cyclic quadrilateral has one angle measuring 121 degrees. Find the measure of its opposite angle.