Circles
59 flashcards covering Circles for the ACT Math section.
A circle is a basic geometric shape defined as the set of all points in a plane that are equidistant from a fixed center point. That distance is called the radius, and from it, we derive other key elements like the diameter (twice the radius), circumference (the perimeter around the circle), and area (the space inside). Understanding circles involves formulas for these properties, as well as working with their equations on a coordinate plane, such as (x - h)^2 + (y - k)^2 = r^2. These concepts are essential in math because they build foundational skills for more complex problems in geometry and trigonometry.
On the ACT Math section, circles show up in questions testing calculations of area and circumference, solving equations, or analyzing relationships like tangents and chords. Common traps include confusing radius with diameter, misapplying formulas under time pressure, or overlooking units in word problems. Focus on memorizing key formulas and practicing visualization skills, as questions often integrate circles with other shapes or require graphing. A concrete tip: Always sketch a diagram to clarify the problem.
Terms (59)
- 01
Circle
A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
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Radius
The radius of a circle is the distance from the center to any point on the circle.
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Diameter
The diameter of a circle is a line segment that passes through the center and connects two points on the circle, equal to twice the radius.
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Circumference
Circumference is the distance around a circle, calculated using the formula C = 2πr, where r is the radius.
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Area of a circle
The area of a circle is the space enclosed by its boundary, given by the formula A = πr², where r is the radius.
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Pi
Pi is a mathematical constant approximately equal to 3.14 or 22/7, used in formulas for circles to relate circumference and area to the radius.
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Chord
A chord is a straight line segment with endpoints on the circle.
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Tangent
A tangent is a straight line that touches the circle at exactly one point.
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Secant
A secant is a straight line that intersects the circle at two points.
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Arc
An arc is a portion of the circumference of a circle.
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Minor arc
A minor arc is the shorter path along the circumference between two points on a circle.
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Major arc
A major arc is the longer path along the circumference between two points on a circle.
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Central angle
A central angle is an angle formed by two radii at the center of the circle.
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Inscribed angle
An inscribed angle is an angle formed by two chords that share a common endpoint on the circle.
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Equation of a circle
The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r² in standard form.
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Center of a circle
The center of a circle is the fixed point equidistant from all points on the circle.
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Standard form of circle equation
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
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General form of circle equation
The general form of a circle's equation is x² + y² + Dx + Ey + F = 0, which can be rewritten to find the center and radius.
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Completing the square for circles
Completing the square is a method to rewrite the general form of a circle's equation into standard form by adding and subtracting constants to perfect squares.
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Distance from point to center
The distance from a point to the center of a circle determines if the point is inside, on, or outside the circle, using the distance formula.
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Intersecting chords theorem
The intersecting chords theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.
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Tangent-secant theorem
The tangent-secant theorem states that if a tangent and a secant are drawn from an external point to a circle, the square of the tangent segment equals the product of the secant segment and its external part.
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Length of an arc
The length of an arc is a fraction of the circumference, calculated as (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius.
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Area of a sector
The area of a sector is a fraction of the circle's area, calculated as (θ/360) × πr², where θ is the central angle in degrees and r is the radius.
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Area of a segment
The area of a segment is the region between a chord and the arc it subtends, found by subtracting the area of a triangular portion from the area of the sector.
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Inscribed quadrilateral
An inscribed quadrilateral is a four-sided polygon whose vertices all lie on the circle.
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Circumscribed circle
A circumscribed circle is one that passes through all vertices of a polygon, such as a triangle.
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Shaded region problems
Shaded region problems involve finding the area between parts of circles or between a circle and another shape, often requiring subtraction of areas.
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Pythagorean theorem in circles
The Pythagorean theorem applies in circles, such as finding the length of a radius when a tangent and a chord form a right triangle.
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Angle between tangent and chord
The angle between a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord.
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Common mistake: Radius vs. diameter
A common mistake is confusing radius and diameter, where using the wrong one in formulas like circumference or area leads to errors in calculations.
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Circumference in terms of diameter
Circumference can be expressed as C = πd, where d is the diameter, which is useful when diameter is given instead of radius.
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Finding radius from area
To find the radius from the area, solve the equation A = πr² for r by taking the square root after dividing by π.
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Finding diameter from circumference
To find the diameter from the circumference, divide the circumference by π, as C = πd.
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Circle equation with center at origin
For a circle centered at the origin (0,0), the equation simplifies to x² + y² = r².
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Translating circle equations
Translating circle equations involves shifting the center by changing h and k in (x - h)² + (y - k)² = r² to match the graph's position.
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Graphing a circle
Graphing a circle requires plotting the center point and drawing a curve r units away in all directions based on the equation.
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Midpoint and circles
The midpoint of a diameter is the center of the circle, which can be found using the midpoint formula for the endpoints.
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Diameter endpoints
The endpoints of a diameter are any two points on the circle that are separated by twice the radius and pass through the center.
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Perimeter of inscribed shapes
For shapes inscribed in a circle, the perimeter may involve chords, and calculating it requires knowing the lengths of those sides.
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Area between two circles
The area between two circles is the difference in their areas, useful for problems with concentric or overlapping circles.
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Overlap of circles
The overlap of two circles creates a lens-shaped region, whose area can be found by subtracting non-overlapping parts from the total.
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Tangent circles
Tangent circles touch at exactly one point, and the distance between their centers equals the sum or difference of their radii.
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Common external tangent
A common external tangent is a line that touches two circles without crossing between them.
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Arc length formula
The arc length formula is s = rθ, where θ is in radians, providing an alternative to the degree-based formula for calculations.
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Sector area formula
The sector area formula is (1/2)r²θ, where θ is in radians, for finding the area of a portion of the circle.
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Radian measure
Radian measure expresses angles as the ratio of arc length to radius, with 2π radians equal to 360 degrees.
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Degree to radian conversion
To convert degrees to radians, multiply the degree measure by π/180, which is helpful for arc and sector problems.
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Inscribed triangle
An inscribed triangle has all three vertices on the circle, and its angles relate to the arcs they subtend.
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Circumference ratio
The ratio of circumference to diameter is always π, a fundamental property used in deriving circle formulas.
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Circle and line intersection
A line can intersect a circle at two points, one point (tangent), or none, depending on the distance from the center to the line.
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Power of a point
The power of a point theorem relates the lengths of segments created by chords, secants, or tangents from a point outside the circle.
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Cyclic quadrilateral
A cyclic quadrilateral is one that can be inscribed in a circle, with opposite angles summing to 180 degrees.
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Radius of inscribed circle
The radius of a circle inscribed in a polygon touches all sides, and for a triangle, it is the area divided by the semi-perimeter.
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Concentric circles
Concentric circles share the same center but have different radii, often used in problems involving rings or annuli.
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Annulus area
The area of an annulus, the region between two concentric circles, is the difference between their individual areas.
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Tangent length from point
The length of a tangent from an external point to a circle is the square root of the power of that point with respect to the circle.
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Sector perimeter
The perimeter of a sector includes the two radii and the arc length, calculated based on the central angle.
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Circle symmetry
Circle symmetry means the shape looks the same after rotation by any angle around the center, affecting how equations are solved.