Circles: arcs, sectors and angles — Digital SAT Math practice
Arc length, sector area, inscribed and central angles.
Key terms
- Central Angle: Vertex at the center; measure equals its intercepted arc.
- Inscribed Angle: Vertex on the circle; measure is half its intercepted arc.
- Arc: A portion of the circumference defined by two endpoints.
- Sector: The region bounded by two radii and their intercepted arc.
- Tangent Line: A line touching the circle at exactly one point, perpendicular to the radius there.
Common traps
- Assuming an inscribed angle equals its intercepted arc instead of being exactly half.
- Using the diameter value where the radius is required for area or circumference calculations.
- Treating a tangent line as passing through the center rather than meeting the radius at 90 degrees.
Key ideas to know
Central vs inscribed angles
In circle O, angle AOC is a central angle intercepting arc AC, and angle ABC is an inscribed angle (B is on the circle) intercepting the same arc. If arc AC measures 80 degrees, what are the measures of angles AOC and ABC?
Angle AOC = 80 degrees and angle ABC = 40 degrees. A central angle equals its intercepted arc, while an inscribed angle is half of it.
Watch out: Applying the half rule to the central angle, or forgetting that the vertex of a central angle is the center of the circle.
Calculating Sector Area
A circle has a radius of 6 units. What is the area of the sector formed by a central angle of 90 degrees? Use pi for calculations.
The sector area is 9 pi square units. Multiply the total circle area (pi r2) by the fraction of the full rotation (central angle / 360). Here, (90/360) * pi * 62 = 9 pi.
Watch out: Using the arc length formula instead of the area formula, or forgetting to square the radius when calculating the full circle area first.
Tangent and Radius Relationship
Line l is tangent to circle P at point T, and segment PT connects the center to the point of tangency. If the angle between line l and segment PT measures x degrees, what value must x equal?
x must equal 90 degrees. A tangent line is always perpendicular to the radius drawn to the point of tangency, forming a right triangle you can solve with the Pythagorean theorem or trigonometry.
Watch out: Assuming the tangent and radius form an acute or obtuse angle, which breaks right triangle properties needed for most circle geometry problems.
Semicircle Inscribed Angle Error
Triangle ABC is inscribed in a circle with diameter AC. A student writes that angle B measures 70 degrees because the other angles are 55 and 55. What is wrong with this solution?
The error is assuming angle B can be any value. An angle inscribed in a semicircle must always be a right angle (90 degrees). Angles A and C would then need to sum to 90 degrees, not 110.
Watch out: Treating the semicircle theorem as optional or miscalculating the remaining angles without enforcing the fixed 90-degree vertex first.
Arc Length Calculation
A circle has a radius of 8 units. What is the length of the arc intercepted by a central angle of 45 degrees? Express your answer in terms of pi.
The arc length is 2 pi units. Multiply the circumference (2 pi r) by the fraction of the full rotation (central angle / 360). Here, (45/360) * 2 * pi * 8 = 2 pi.
Watch out: Using the area formula fraction instead of the circumference for arc length, or forgetting to multiply by 2 when calculating the full circumference.
Sample questions
Question 1
A circle has area 196π. What is its circumference?
- 28π
- 196π
- 56π
- 14π
Show answer and explanation
Answer: 28π
πr² = 196π → r = 14. Circumference = 2πr = 28π.
Question 2
A circle has a circumference of 10pi. What is its area?
- 10pi
- 100pi
- 5pi
- 25pi
Show answer and explanation
Answer: 25pi
Circumference 2*pi*r = 10pi gives r = 5, so area = pi*r^2 = 25pi.