PSAT 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 PSAT 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: State the chord-chord power theorem: chords intersect at $E$ with segments $AE,EB$ and $CE,ED$.

Answer: $AE\cdot EB=CE\cdot ED$. Products of chord segments are equal at intersection.

Flashcard 2: Find the arc length when $r=6$ and $\theta=60^\circ$.

Answer: $2\pi$. $s=\frac{60}{360}\cdot 2\pi(6)=\frac{1}{6}\cdot 12\pi=2\pi$.

Flashcard 3: State the equation of a circle with center $(h,k)$ and radius $r$ in standard form.

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

Flashcard 4: What is the relationship between diameter $d$ and radius $r$ in a circle?

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

Flashcard 5: Find the distance between the centers of two circles with centers $(1,2)$ and $(4,6)$.

Answer: $5$. Distance formula: $\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=5$.

Flashcard 6: What is the measure of an inscribed angle that intercepts an arc of measure $m$ degrees?

Answer: $\frac{m}{2}$. Inscribed angle theorem: half the intercepted arc.

Flashcard 7: State the area formula for a circle with radius $r$.

Answer: $A=\pi r^2$. Area equals $\pi$ times radius squared.

Flashcard 8: Identify the radius of the circle $(x+1)^2+(y-4)^2=49$.

Answer: $7$. Radius equals the square root of the constant term.

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

Answer: Center $(4,-3)$, radius $5$. Complete the square: $(x-4)^2+(y+3)^2=25$.

Flashcard 10: Find the arc length for $r=6$ and $\theta=60^\circ$.

Answer: $2\pi$. $s=\frac{60}{360}\cdot 2\pi(6)=\frac{1}{6}\cdot 12\pi=2\pi$.

Flashcard 11: What is the length of a semicircle arc with radius $r$ (arc only, not including the diameter)?

Answer: $\pi r$. Half the circumference gives the semicircle arc.

Flashcard 12: Find the measure of an inscribed angle that intercepts a $110^\circ$ arc.

Answer: $55^\circ$. Inscribed angle is half the intercepted arc: $\frac{110°}{2}=55°$.

Flashcard 13: Find the radius if the center is $(2,3)$ and a point on the circle is $(2,11)$.

Answer: $8$. Distance from $(2,3)$ to $(2,11)$ is $|11-3|=8$.

Flashcard 14: State the standard form equation of a circle with center $(h,k)$ and radius $r$.

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

Flashcard 15: Find the radius of the circle $(x+5)^2+(y-7)^2=49$.

Answer: $7$. $r^2=49$, so $r=\sqrt{49}=7$.

Flashcard 16: Use $PT^2=PA\cdot PB$: if $PA=4$ and $PB=16$, find $PT$.

Answer: $8$. $PT^2=4\cdot 16=64$, so $PT=\sqrt{64}=8$.

Flashcard 17: What is the diameter of a circle in terms of radius $r$?

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

Flashcard 18: State the tangent-secant power theorem from external point $P$: tangent $PT$, secant meets circle at $A$ and $B$.

Answer: $PT^2=PA\cdot PB$. Tangent squared equals product of secant segments from P.

Flashcard 19: What is the circumference formula of a circle with radius $r$?

Answer: $C=2\pi r$. Circumference equals $2\pi$ times the radius.

Flashcard 20: State the standard form of a circle with center $(h,k)$ and radius $r$.

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

Flashcard 21: State the circumference formula for a circle with radius $r$.

Answer: $C=2\pi r$. Circumference equals $2\pi$ times the radius.

Flashcard 22: What are the center and radius of $(x-3)^2+(y+2)^2=25$?

Answer: Center $(3,-2)$, radius $5$. Read $(h,k)$ from $(x-h)^2+(y-k)^2$ and $r$ from $r^2=25$.

Flashcard 23: What is the arc length formula for central angle $\theta$ in degrees and radius $r$?

Answer: $s=\frac{\theta}{360}\cdot 2\pi r$. Arc length is the fraction of circumference.

Flashcard 24: Identify the center and radius from $(x-3)^2+(y+4)^2=25$.

Answer: Center $(3,-4)$, radius $5$. Read $(h,k)$ from $(x-h)^2+(y-k)^2$ and $r=\sqrt{25}=5$.

Flashcard 25: Find the center of the circle $(x+5)^2+(y-7)^2=49$.

Answer: $(-5,7)$. Center is $(h,k)$ where equation is $(x-h)^2+(y-k)^2$.

Flashcard 26: State the relationship between diameter $d$ and radius $r$.

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

Flashcard 27: What is the area formula of a circle with radius $r$?

Answer: $A=\pi r^2$. Area equals $\pi$ times radius squared.

Flashcard 28: What is the radius of a circle in terms of diameter $d$?

Answer: $r=\frac{d}{2}$. Radius is half the diameter.

Flashcard 29: What is the distance formula used to find a radius from center $(h,k)$ to point $(x,y)$?

Answer: $r=\sqrt{(x-h)^2+(y-k)^2}$. Distance formula gives radius from center to any point.

Flashcard 30: What is the measure of a central angle that intercepts an arc of measure $m$ degrees?

Answer: $m$. Central angle equals its intercepted arc measure.

Flashcard 31: Find the area of a sector for $r=3$ and $\theta=120^\circ$.

Answer: $3\pi$. $A=\frac{120}{360}\cdot \pi(3)^2=\frac{1}{3}\cdot 9\pi=3\pi$.

Flashcard 32: State the sector area formula for central angle $\theta$ (in degrees) and radius $r$.

Answer: $A=\frac{\theta}{360}\cdot \pi r^2$. Fraction of full area based on angle ratio.

Flashcard 33: State the equation of a circle centered at the origin with radius $r$.

Answer: $x^2+y^2=r^2$. Center at origin means $(h,k)=(0,0)$.

Flashcard 34: State the arc length formula for central angle $\theta$ in degrees and radius $r$.

Answer: $s=\frac{\theta}{360}\cdot 2\pi r$. Arc length is the fraction of circumference for given angle.

Flashcard 35: Use $AE\cdot EB=CE\cdot ED$: if $AE=3$, $EB=12$, and $CE=6$, find $ED$.

Answer: $6$. $3\cdot 12=6\cdot ED$, so $ED=\frac{36}{6}=6$.

Flashcard 36: Find the arc length of a $90^\circ$ sector in a circle of radius $8$.

Answer: $4\pi$. Use $s=\frac{90}{360}\cdot 2\pi(8)=\frac{1}{4}\cdot 16\pi$.

Flashcard 37: Which statement is always true about a radius drawn to a point of tangency?

Answer: It is perpendicular to the tangent line. Radius meets tangent at $90°$ angle.

Flashcard 38: Identify the center and radius of $(x-3)^2+(y+2)^2=25$.

Answer: Center $(3,-2)$, radius $5$. Read $(h,k)$ from $(x-h)^2+(y-k)^2$; $r^2=25$ so $r=5$.

Flashcard 39: State the circumference of a circle with diameter $d$.

Answer: $C=\pi d$. Circumference equals $\pi$ times diameter.

Flashcard 40: Find the area of a circle with diameter $10$.

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

Flashcard 41: State the relationship between a tangent and the radius at the point of tangency.

Answer: Tangent is perpendicular to the radius. Forms a $90°$ angle at the point of tangency.

Flashcard 42: Find the area of a sector when $r=10$ and $\theta=90^\circ$.

Answer: $25\pi$. $A=\frac{90}{360}\cdot \pi(10)^2=\frac{1}{4}\cdot 100\pi=25\pi$.

Flashcard 43: What is the sector area formula for central angle $\theta$ in degrees and radius $r$?

Answer: $A=\frac{\theta}{360}\cdot \pi r^2$. Sector area is the fraction of circle area.

Flashcard 44: State the tangent property relating a radius to a tangent line at the point of tangency.

Answer: A radius is perpendicular to the tangent: $r\perp \text{tangent}$. Radius meets tangent at $90°$ angle.

Flashcard 45: Identify the radius of $(x+1)^2+(y-2)^2=\frac{49}{4}$.

Answer: $\frac{7}{2}$. $r^2=\frac{49}{4}$, so $r=\frac{7}{2}$.

Flashcard 46: State the circle equation with center $(2,-1)$ and radius $4$.

Answer: $(x-2)^2+(y+1)^2=16$. Substitute center $(h,k)=(2,-1)$ and $r^2=16$.

Flashcard 47: What is the length of a semicircle arc (not including the diameter) with radius $r$?

Answer: $\pi r$. Half the circumference, excluding the diameter.

Flashcard 48: State the arc length formula for central angle $\theta$ (in degrees) and radius $r$.

Answer: $s=\frac{\theta}{360}\cdot 2\pi r$. Fraction of full circumference based on angle ratio.

Flashcard 49: Find the circumference of a circle with diameter $12$.

Answer: $12\pi$. Use $C=2\pi r$ with $r=6$ (half of diameter).

Flashcard 50: Find the area of a $60^\circ$ sector of a circle with radius $6$.

Answer: $6\pi$. Use $A=\frac{60}{360}\cdot\pi(6)^2=\frac{1}{6}\cdot 36\pi$.

Flashcard 51: State the formula for the circumference of a circle with radius $r$.

Answer: $C=2\pi r$. Circumference equals $2\pi$ times the radius.

Flashcard 52: State the formula for the area of a circle with radius $r$.

Answer: $A=\pi r^2$. Area equals $\pi$ times radius squared.

Flashcard 53: State the sector area formula for central angle $\theta$ in degrees and radius $r$.

Answer: $A=\frac{\theta}{360}\cdot \pi r^2$. Sector area is the fraction of circle area for given angle.