ACT Math Flashcards: Circles

What this deck covers

This deck focuses on Circles, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: If the circumference is 31.4, estimate the radius.

Answer: Radius $\approx 5$. Using $C = 2\pi r$, solve for $r$.

Flashcard 2: State the radius of a circle with equation $x^2 + y^2 = 81$.

Answer: Radius = 9. Radius is square root of constant term.

Flashcard 3: What is the relationship between radius $r$ and diameter $d$ of a circle?

Answer: $d=2r$. Diameter is twice the radius.

Flashcard 4: State the conversion from degrees to radians for an angle of $\theta^\circ$.

Answer: $\theta\cdot\frac{\pi}{180}$. Multiply by $\frac{\pi}{180}$ to convert degrees.

Flashcard 5: What is the relationship between a radius and a tangent line at the point of tangency?

Answer: They are perpendicular. Forms a $90°$ angle at point of contact.

Flashcard 6: Identify the distance between the center and any point on the circle.

Answer: The radius. Definition of radius in a circle.

Flashcard 7: Find the diameter of a circle with circumference $18\pi$.

Answer: $18$. From $C=\pi d$, solve for $d$.

Flashcard 8: What is the center of the circle $(x+1)^2 + (y-1)^2 = 9$?

Answer: $(-1, 1)$. Center is $(-(-1), -(-1)) = (-1, 1)$.

Flashcard 9: What is the measure of an inscribed angle that intercepts a semicircle?

Answer: $90^\circ$. Inscribed angle in semicircle is always right.

Flashcard 10: Find the area of a circle with radius $\frac{1}{2}$ in terms of $\pi$.

Answer: $\frac{\pi}{4}$. Use $A=\pi r^2$ with $r=\frac{1}{2}$.

Flashcard 11: What is the arc length of a circle with radius $4$ cm and angle $90^\circ$?

Answer: $2\pi$ cm. Using $L = \frac{90}{360} \times 2\pi(4) = 2\pi$.

Flashcard 12: Identify the center and radius of $(x-1)^2 + (y+2)^2 = 16$.

Answer: Center: $(1, -2)$, Radius: 4. Read values from standard form equation.

Flashcard 13: Identify the term for a section of a circle bounded by two radii.

Answer: A sector. A sector is a pie-slice shaped region.

Flashcard 14: Find the area of a circle with diameter $10$ in terms of $\pi$.

Answer: $25\pi$. Use $A=\pi r^2$ where $r=5$ (half of diameter).

Flashcard 15: What is the circumference of a circle with diameter $14$?

Answer: $14\pi$. Using $C = \pi d$ with $d = 14$.

Flashcard 16: Find the arc length when $r=4$ and $\theta=\frac{\pi}{3}$ (radians).

Answer: $\frac{4\pi}{3}$. Apply $s=r\theta$ with given values.

Flashcard 17: Identify the definition of a chord in a circle.

Answer: A segment with endpoints on the circle. A line segment inside the circle.

Flashcard 18: State the equation of a circle centered at the origin with radius $r$.

Answer: $x^2+y^2=r^2$. Special case where center is at $(0,0)$.

Flashcard 19: What is the length of a radius if the circle's diameter is $10$?

Answer: $5$. Radius is half the diameter.

Flashcard 20: State the circumference formula for a circle with radius $r$.

Answer: $C=2\pi r$. Perimeter of a circle using radius.

Flashcard 21: Find the radius of a circle with area $49\pi$.

Answer: $7$. From $A=\pi r^2$, solve for $r$.

Flashcard 22: Identify the center of a circle with equation $(x-3)^2 + (y+2)^2 = 25$.

Answer: Center: $(3, -2)$. Center is $(h, k)$ from standard form.

Flashcard 23: Determine the area of a circle with radius $3$.

Answer: $9\pi$. Using $A = \pi r^2$ with $r = 3$.

Flashcard 24: If the radius is $7$, what is the area of the circle?

Answer: $49\pi$. Using $A = \pi r^2$ with $r = 7$.

Flashcard 25: Define a sector of a circle.

Answer: Sector: A region bounded by two radii and an arc. Part of circle enclosed by two radii.

Flashcard 26: Find the circumference of a circle with diameter $14$ in terms of $\pi$.

Answer: $14\pi$. Use $C=\pi d$ with $d=14$.

Flashcard 27: What is the formula for the area of a sector?

Answer: Area = $\frac{1}{2}r^2\theta$ (in radians). Half radius squared times central angle.

Flashcard 28: State the sector area formula for radius $r$ and central angle $\theta$ in radians.

Answer: $A=\frac{1}{2}r^2\theta$. Half of radius squared times angle.

Flashcard 29: What is the meaning of 'tangent' in relation to circles?

Answer: A line touching the circle at exactly one point. A tangent line intersects the circle at one point.

Flashcard 30: What is the formula for the area of a circle segment?

Answer: Area = $\frac{1}{2}r^2(\theta - \sin\theta)$ (in radians). Area between chord and arc.

Flashcard 31: Define the term 'radius' in the context of a circle.

Answer: Distance from the center to any point on the circle. The radius is a fundamental measure of a circle's size.

Flashcard 32: What is the distance between parallel tangents to a circle if diameter is 10?

Answer: Distance = 10. Distance between parallel tangents equals diameter.

Flashcard 33: Identify the definition of a diameter in a circle.

Answer: A chord passing through the center. The longest chord in a circle.

Flashcard 34: Find the diameter of a circle with radius $9$ cm.

Answer: $18$ cm. Diameter is twice the radius.

Flashcard 35: Find the diameter of a circle with radius $9$ cm.

Answer: $18$ cm. Diameter is twice the radius.

Flashcard 36: What is the relationship between radius and circumference?

Answer: Circumference = $2\pi r$. Circumference is proportional to radius.

Flashcard 37: Find the diameter of a circle with circumference $18\pi$.

Answer: $18$. From $C=\pi d$, solve for $d$.

Flashcard 38: How do you calculate the chord length in a circle?

Answer: Chord Length $= 2r\sin(\frac{\theta}{2})$. Uses central angle and radius.

Flashcard 39: Find the sector area when $r=3$ and $\theta=120^\circ$ in terms of $\pi$.

Answer: $3\pi$. Use degrees formula: $\frac{120}{360}\cdot \pi\cdot 9$.

Flashcard 40: What is the formula for the length of an arc?

Answer: $L = \frac{\theta}{360} \times 2\pi r$. Proportion of the full circumference based on angle.

Flashcard 41: Find the circumference of a circle with radius $5$ cm.

Answer: $10\pi$ cm. Using $C = 2\pi r$ with $r = 5$.

Flashcard 42: What is the standard form equation of a circle?

Answer: $(x-h)^2 + (y-k)^2 = r^2$. General equation with center $(h, k)$.

Flashcard 43: If a circle's equation is $(x-2)^2 + (y-3)^2 = 16$, what is the center?

Answer: $(2, 3)$. Center is $(h,k) = (2,3)$ from the equation.

Flashcard 44: Calculate the radius of a circle with circumference $12\pi$.

Answer: $6$. From $C = 2\pi r$, solve for $r$.

Flashcard 45: Find the area of a circle with diameter $10$ cm.

Answer: $25\pi$ cm$^2$. Using $A = \pi r^2$ with $r = 5$ (since $d = 10$).

Flashcard 46: State the area formula for a circle with radius $r$.

Answer: $A=\pi r^2$. Interior space enclosed by a circle.

Flashcard 47: What is the arc length of a circle with radius $4$ cm and angle $90^\circ$?

Answer: $2\pi$ cm. Using $L = \frac{90}{360} \times 2\pi(4) = 2\pi$.

Flashcard 48: Find the measure of an inscribed angle intercepting an arc of $110^\circ$.

Answer: $55^\circ$. Inscribed angle is half the intercepted arc.

Flashcard 49: What is the relationship between radius and circumference?

Answer: Circumference = $2\pi r$. Circumference is proportional to radius.

Flashcard 50: What is the formula for the area of a circle?

Answer: Area = $\pi r^2$. Standard formula for circular area.

Flashcard 51: Find the circumference of a circle with radius $6$ in terms of $\pi$.

Answer: $12\pi$. Use $C=2\pi r$ with $r=6$.

Flashcard 52: What is the measure of an inscribed angle for a semicircle?

Answer: Inscribed Angle = 90 degrees. Inscribed angle in semicircle is always 90°.

Flashcard 53: Define the term 'diameter' in a circle.

Answer: Distance across the circle through the center, $2r$. The diameter is the longest chord in a circle.

Flashcard 54: What is the relationship between diameter and circumference?

Answer: Circumference = $\pi$ times Diameter. Circumference equals $\pi$ times diameter.

Flashcard 55: What is the angle measure of a full circle in degrees?

Answer: $360^\circ$. Complete rotation around a circle.

Flashcard 56: Determine the area of a circle with radius $3$.

Answer: $9\pi$. Using $A = \pi r^2$ with $r = 3$.

Flashcard 57: Determine the diameter of a circle with area $16\pi$.

Answer: $8$. From $A = \pi r^2$, find $r = 4$, so $d = 8$.

Flashcard 58: Identify the definition of a diameter in a circle.

Answer: A chord passing through the center. The longest chord in a circle.

Flashcard 59: What is the circumference of a circle with diameter $14$?

Answer: $14\pi$. Using $C = \pi d$ with $d = 14$.

Flashcard 60: What is the arc length of a $60^\circ$ sector in a circle with radius $5$?

Answer: $\frac{5\pi}{3}$. Using $L = \frac{60}{360} \times 2\pi(5) = \frac{5\pi}{3}$.

Flashcard 61: State the sector area formula for radius $r$ and central angle $\theta$ in radians.

Answer: $A=\frac{1}{2}r^2\theta$. Half of radius squared times angle.

Flashcard 62: State the standard equation of a circle with center $(h,k)$ and radius $r$.

Answer: $(x-h)^2+(y-k)^2=r^2$. Standard form with center $(h,k)$ and radius $r$.

Flashcard 63: What is the length of chord $AB$ if it is a diameter in a circle of radius $9$?

Answer: $18$. Diameter equals twice the radius.

Flashcard 64: State the radius of a circle with equation $x^2 + y^2 = 81$.

Answer: $\text{Radius} = 9$. Radius is square root of constant term.

Flashcard 65: Identify the center and radius of $(x-1)^2 + (y+2)^2 = 16$.

Answer: Center: $(1, -2)$, Radius: 4. Read values from standard form equation.

Flashcard 66: Find the radius of a circle with area $49\pi$.

Answer: $7$. From $A=\pi r^2$, solve for $r$.

Flashcard 67: Identify the center and radius of $x^2+y^2-6x+8y+9=0$.

Answer: Center $(3,-4)$, radius $4$. Complete the square to find center and radius.

Flashcard 68: What is the formula for the circumference of a circle?

Answer: $C = 2\pi r$. Standard formula where $r$ is the radius.

Flashcard 69: Find the area of a circle segment with radius 6 and angle 90 degrees.

Answer: Area $\approx 9\pi - 18$. Apply segment area formula.

Flashcard 70: Find the sector area when $r=6$ and $\theta=\frac{\pi}{2}$ (radians).

Answer: $9\pi$. Apply $A=\frac{1}{2}r^2\theta$ with given values.

Flashcard 71: Find the measure of an arc intercepted by a central angle of $140^\circ$.

Answer: $140^\circ$. Central angle equals its intercepted arc measure.

Flashcard 72: Identify the length of a chord if radius is 5 and angle is 60 degrees.

Answer: Chord Length $\approx 5$. Apply chord formula with given values.

Flashcard 73: What is the relationship between a radius and a tangent line at the point of tangency?

Answer: They are perpendicular. Forms a $90°$ angle at point of contact.

Flashcard 74: Find the circumference of a circle with radius $6$ in terms of $\pi$.

Answer: $12\pi$. Use $C=2\pi r$ with $r=6$.

Flashcard 75: What is the equation of a circle with center at origin and radius 4?

Answer: $x^2 + y^2 = 16$. Standard form with center at origin.

Flashcard 76: What is the value of $\pi$ to two decimal places?

Answer: $\pi \approx 3.14$. Common approximation for pi.

Flashcard 77: State the equation of a circle centered at the origin with radius $r$.

Answer: $x^2+y^2=r^2$. Special case where center is at $(0,0)$.

Flashcard 78: What is the relationship between diameter and circumference?

Answer: Circumference = $\pi$ times Diameter. Circumference equals $\pi$ times diameter.

Flashcard 79: Identify $h$ and $k$ in the circle equation $(x-h)^2 + (y-k)^2 = r^2$.

Answer: Coordinates of the circle's center. $(h,k)$ represents the center point of the circle.

Flashcard 80: Determine the circumference for a circle with radius $2$.

Answer: $4\pi$. Using $C = 2\pi r$ with $r = 2$.

Flashcard 81: What is the term for a chord passing through the center of a circle?

Answer: A diameter. A chord through the center is the diameter.

Flashcard 82: Identify the length of a chord if radius is 5 and angle is 60 degrees.

Answer: Chord Length $\approx 5$. Apply chord formula with given values.

Flashcard 83: How do you calculate the diameter if given the circumference?

Answer: $d = \frac{C}{\pi}$. Rearranging $C = \pi d$ to solve for diameter.

Flashcard 84: What is the formula for the area of a circle?

Answer: Area = $\pi r^2$. Standard formula for circular area.

Flashcard 85: If a circle's equation is $(x-3)^2 + (y+4)^2 = 25$, what is the radius?

Answer: $5$. The radius is $\sqrt{25} = 5$.

Flashcard 86: State the circumference formula for a circle with diameter $d$.

Answer: $C=\pi d$. Perimeter of a circle using diameter.

Flashcard 87: State the area formula for a circle with radius $r$.

Answer: $A=\pi r^2$. Interior space enclosed by a circle.

Flashcard 88: What is the radian measure of a full circle?

Answer: $2\pi$. Complete rotation in radian measure.

Flashcard 89: Find the area of a semicircle with radius $4$ in terms of $\pi$.

Answer: $8\pi$. Half the area: $\frac{1}{2}\cdot \pi\cdot 16$.

Flashcard 90: Find the arc length when $r=5$ and $\theta=72^\circ$ in terms of $\pi$.

Answer: $2\pi$. Use degrees formula: $\frac{72}{360}\cdot 2\pi\cdot 5$.

Flashcard 91: Calculate the radius of a circle with circumference $12\pi$.

Answer: $6$. From $C = 2\pi r$, solve for $r$.

Flashcard 92: Identify the length of a semicircular arc with radius $8$ in terms of $\pi$.

Answer: $8\pi$. Half the circumference: $\frac{1}{2}\cdot 2\pi\cdot 8$.

Flashcard 93: What is the meaning of 'tangent' in relation to circles?

Answer: A line touching the circle at exactly one point. A tangent line intersects the circle at one point.

Flashcard 94: State the standard equation of a circle with center $(h,k)$ and radius $r$.

Answer: $(x-h)^2+(y-k)^2=r^2$. Standard form with center $(h,k)$ and radius $r$.

Flashcard 95: State the arc length formula for radius $r$ and central angle $\theta$ in radians.

Answer: $s=r\theta$. Arc length equals radius times angle.

Flashcard 96: Find the area of a sector with angle $120^\circ$ in a circle of radius $6$.

Answer: $12\pi$. Using $A = \frac{120}{360} \pi (6)^2 = 12\pi$.

Flashcard 97: What term describes the set of all points equidistant from a center point?

Answer: A circle. This is the geometric definition of a circle.

Flashcard 98: Find the radius if the area of a circle is $64\pi$.

Answer: Radius = 8. Use $A = \pi r^2$ and solve for $r$.

Flashcard 99: What is the radian measure of a full circle?

Answer: $2\pi$. Complete rotation in radian measure.

Flashcard 100: If the circumference is 31.4, estimate the radius.

Answer: Radius $\approx 5$. Using $C = 2\pi r$, solve for $r$.