Circles - SAT Math

The equation of a circle is $(x-3)^2 + (y+4)^2 = 25$ . What is the radius of the circle?
Answer: 5 This question asks for the radius of a circle given its equation in standard form. The standard form of a circle equation is $(x-h)^2 + (y-k)^2 = r^2$ , where $(h,k)$ is the center and r is the radius. In the equation $(x-3)^2 + (y+4)^2 = 25$ , we can see that $r^2 = 25$ . Taking the square root of both sides: $r = \sqrt{25} = 5$ .
A circle has circumference $30\pi$ cm. What is the radius of the circle, in centimeters?
Answer: $15$ Given a circumference of 30π cm, we need to find the radius. The circumference formula is C = 2πr. Setting up the equation: 30π = 2πr. Dividing both sides by 2π: r = 30π/(2π) = 15 cm. The π cancels out, leaving just 15. A common error is forgetting to divide by 2 or not canceling the π properly. When working backwards from circumference to radius, divide by 2π.
A circle has radius $10$ . A central angle $\angle AOB$ measures $72^\circ$ . What is the length of arc $AB$ , in terms of $\pi$ ?
Answer: $4\pi$ This question asks for the arc length of a circle with radius 10 and central angle $72^\circ$ . The formula for arc length is $s = \frac{\theta}{360^\circ} \times 2\pi r$ , where $\theta$ is the central angle in degrees. Substituting the values: $s = \frac{72^\circ}{360^\circ} \times 2\pi(10) = \frac{1}{5} \times 20\pi = 4\pi$ . A common error is forgetting to include the factor of 2 in the circumference formula or using the wrong fraction for the angle ratio. Remember that arc length is a fraction of the circumference determined by the central angle.
In a circle, the measure of an inscribed angle that intercepts arc $AB$ is $35^\circ$ . What is the measure of arc $AB$ ?
Answer: $70^\circ$ This question involves the inscribed angle theorem, which states that an inscribed angle is half the measure of its intercepted arc. Given that the inscribed angle is 35°, the intercepted arc AB has measure $2 \times 35° = 70°$ . This is a fundamental theorem in circle geometry: inscribed angle = $\frac{1}{2}$ × intercepted arc. A common error is confusing this with central angles, which equal their intercepted arcs. For test-taking, remember: inscribed angles are always half their intercepted arcs.
In the coordinate plane, the circle with center $(4,1)$ is tangent to the $x$ -axis. What is the equation of the circle?
Answer: $(x-4)^2+(y-1)^2=1$ This question involves a circle tangent to the $x$ -axis. When a circle with center $(4, 1)$ is tangent to the $x$ -axis, the radius equals the y-coordinate of the center (the perpendicular distance to the $x$ -axis). Therefore, $r = 1$ . The equation is $(x-4)^2 + (y-1)^2 = 1^2 = 1$ . A common mistake is using the x-coordinate as the radius when tangent to the $x$ -axis. Remember: tangent to the $x$ -axis means $r = | \text{y-coordinate of center} |$ .
A semicircle has diameter 14 ft. What is the area of the semicircle, in square feet, in terms of $\pi$ ?
Answer: $\dfrac{49\pi}{2}$ The question asks for the area of a semicircle with diameter 14 ft, in terms of $\pi$ . A semicircle's area is $\frac{1}{2} \pi r^2$ , where r is the radius; distinguish that diameter $d=14$ means $r=d/2=7$ ft. Substitute r=7: area = $\frac{1}{2} \times \pi \times 7^2 = \frac{1}{2} \times 49\pi = \frac{49\pi}{2}$ ft $^2$ . This is half the full circle area of $49\pi$ . A key error is using diameter in place of radius, like $\frac{1}{2} \pi (14)^2=98\pi$ , or forgetting the half. Emphasize radius as half the diameter to avoid miscalculation. For semicircles, confirm if it includes the diameter base or just the curved part, but area is always $\frac{1}{2} \pi r^2$ .
The circumference of a circle is $30\pi$ cm. What is the area of the circle, in square centimeters? (Use $\pi$ in your answer.)
Answer: $225\pi$ This question requires finding area given circumference. From C = 2πr = 30π, we get r = 15 cm. The area is A = πr² = π(15)² = 225π square centimeters. A common error is using the circumference value directly without first finding the radius. Always work through the radius: circumference → radius → area.
A circular garden has area $196\pi$ square feet. What is the circumference of the garden, in feet? (Use $\pi$ in your answer.)
Answer: $28\pi$ This question requires finding circumference given the area. From A = πr² = 196π, we get r² = 196, so r = 14 feet. The circumference is C = 2πr = 2π(14) = 28π feet. A common mistake is taking the square root of 196π instead of just 196. Remember to isolate r² by dividing out π first, then take the square root to find r.
A circle has area $81\pi$ square inches. What is the diameter of the circle, in inches?
Answer: $18$ The question asks for the diameter of a circle with area $81\pi$ square inches. The area formula is $A = \pi r^2$ , where $r$ is the radius. Setting $\pi r^2 = 81\pi$ and dividing both sides by $\pi$ : $r^2 = 81$ , so $r = 9$ inches. Since diameter $d = 2r$ , we have $d = 2(9) = 18$ inches. A common error is confusing radius and diameter—the area formula uses radius, but the question asks for diameter. Always read carefully to determine whether radius or diameter is requested, and remember that diameter is twice the radius.
In the coordinate plane, a circle has equation $(x-4)^2+(y+1)^2=49$ . What are the center and radius of the circle?
Answer: Center $(4,-1)$ , $r=7$ This question asks us to identify the center and radius from the standard form equation of a circle. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius. From (x-4)² + (y+1)² = 49, we can identify: h = 4, k = -1 (note that y+1 = y-(-1)), and r² = 49, so r = 7. The center is (4,-1) and the radius is 7. A common mistake is confusing the signs in the center coordinates or thinking r² is the radius instead of taking the square root.
A tangent line touches a circle at point $T$ . The radius to $T$ has length $9$ and the tangent segment from $T$ to point $P$ has length $12$ . What is the distance from the center of the circle to point $P$ ?
Answer: $15$ This question involves a tangent line to a circle, where the radius to the tangent point T has length 9 and the tangent segment TP has length 12. Since a tangent line is perpendicular to the radius at the point of tangency, we have a right triangle with the radius (9) and tangent segment (12) as legs. The distance from the center O to point P is the hypotenuse: $OP = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ . A key property to remember is that tangent lines are always perpendicular to the radius at the point of tangency.
Two radii of a circle form a central angle of $150^\circ$ . If the radius is $8$ , what is the length of the intercepted arc, in terms of $\pi$ ?
Answer: $\tfrac{20\pi}{3}$ We need to find the arc length for a central angle of 150° in a circle with radius 8. The arc length formula is $s = \frac{\theta}{360°} \times 2\pi r$ when angle is in degrees. Substituting: $s = \frac{150°}{360°} \times 2\pi(8) = \frac{5}{12} \times 16\pi = \frac{80\pi}{12} = \frac{20\pi}{3}$ . A common error is forgetting to multiply by $2\pi r$ (the full circumference) or using the wrong angle fraction. Always check that your fraction simplifies correctly.
A circle in the coordinate plane has center $(2,-1)$ and passes through the point $(8,3)$ . Which equation represents the circle?
Answer: $(x-2)^2+(y+1)^2=52$ The question asks for the equation of a circle with center $(2,-1)$ passing through $(8,3)$ . The standard form is $(x-h)^2+(y-k)^2=r^2$ , where we know $(h,k)=(2,-1)$ . To find $r^2$ , calculate the distance from center to the given point: $r^2 = (8-2)^2 + (3-(-1))^2 = 6^2 + 4^2 = 36 + 16 = 52$ . Therefore, the equation is $(x-2)^2+(y+1)^2=52$ . A common error is calculating $r$ instead of $r^2$ for the equation, or making sign errors when substituting the center coordinates. When writing circle equations, double-check that $(x-h)$ means subtracting the x-coordinate of the center.
A circle has equation $(x+4)^2+(y-5)^2=9$ . Which point lies on the circle?
Answer: $( -1, 5)$ This question asks which point lies on the circle $(x+4)^2+(y-5)^2=9$ . To check if a point lies on the circle, substitute its coordinates and see if the equation is satisfied. For point B $(-1,5)$ : $((-1)+4)^2+(5-5)^2=3^2+0^2=9+0=9$ ✓. This equals 9, so point B lies on the circle. A common error is making sign mistakes when substituting negative coordinates. Always be careful with signs when checking if points satisfy circle equations.
In a circle, an inscribed angle intercepts an arc that measures $110^\circ$ . What is the measure of the inscribed angle?
Answer: $55^\circ$ The question asks for the measure of an inscribed angle intercepting a 110° arc. The property is that an inscribed angle measures half the intercepted arc. Thus, the angle = 110° / 2 = 55°. This is a direct application of the inscribed angle theorem. A common error is doubling instead of halving, getting 220°, or confusing with central angle equaling the arc. Note that radius or diameter isn't directly needed here, as the relationship is angular. When seeing arc measures, recall inscribed angles are half, while central are equal; sketch the circle to visualize.
Point $P(6,1)$ lies on a circle centered at the origin. What is the equation of the circle?
Answer: $x^2+y^2=37$ This question asks for the equation of a circle centered at the origin that passes through point $P(6,1)$ . For a circle centered at the origin, the equation is $x^2 + y^2 = r^2$ , where $r$ is the radius. Since $P(6,1)$ lies on the circle, the radius equals the distance from the origin to $P$ : $r = \sqrt{6^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37}$ . Therefore, the equation is $x^2 + y^2 = 37$ . A common error is writing $r$ instead of $r^2$ on the right side of the equation, giving $x^2 + y^2 = \sqrt{37}$ .
Point $P$ lies on the circle $(x-1)^2 + (y-4)^2 = 25$ and has $x$ -coordinate $-2$ . If $P$ is in Quadrant II, what is the $y$ -coordinate of $P$ ?
Answer: 8 Plugging in $x=-2$ gives $(-3)^2+(y-4)^2=25\Rightarrow (y-4)^2=16\Rightarrow y=0$ or $8$ . Quadrant II requires $y>0$ , so $y=8$ ; 0 is the other intersection, and 16/4 are common errors.
A circular pizza has area $81\pi$ in $^2$ . What is the circumference of the pizza, in terms of $\pi$ ?
Answer: $18\pi$ in This question asks for the circumference of a pizza given its area is $81\pi$ square inches. First, find the radius using the area formula $A = \pi r^2$ : $81\pi = \pi r^2$ , so $r^2 = 81$ and $r = 9$ inches. Then use the circumference formula $C = 2\pi r$ : $C = 2\pi(9) = 18\pi$ inches. A common error is trying to find circumference directly from area without first finding the radius. Always work step-by-step: area → radius → circumference.
A circle has equation $x^2+y^2-4x+10y+13=0$ . What is the center of the circle?
Answer: $(2,-5)$ To find the center of the circle from $x^2 + y^2 - 4x + 10y + 13 = 0$ , we need to complete the square. Rearranging: $(x^2 - 4x) + (y^2 + 10y) = -13$ . For x: $(x-2)^2 - 4$ ; for y: $(y+5)^2 - 25$ . This gives us $(x-2)^2 + (y+5)^2 - 4 - 25 = -13$ , which simplifies to $(x-2)^2 + (y+5)^2 = 16$ . The center is $(2, -5)$ . A common error is mixing up the signs when identifying the center from the completed square form.
A circle is centered at $(2,-1)$ and passes through the point $(8,3)$ . What is the equation of the circle?
Answer: $(x-2)^2+(y+1)^2=52$ We need the equation of a circle centered at (2,-1) passing through (8,3). First, find the radius using the distance formula: $r = \sqrt{(8-2)^2 + (3-(-1))^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52}$ . The standard form equation is $(x-h)^2 + (y-k)^2 = r^2$ , so $(x-2)^2 + (y-(-1))^2 = 52$ , which becomes $(x-2)^2 + (y+1)^2 = 52$ . Common errors include forgetting to square the radius or making sign errors with the center coordinates.

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