Circles - SAT Math

1. Identify the longest chord in a circle.

   The diameter. Diameter passes through center, maximizing length.

2. What is the term for the distance from the center to any point on the circle?

   Radius. Fundamental distance measurement in circles.

3. Calculate the area when the radius is 4.

   $A = 16\pi$. Apply $A = \pi r^2$ with $r = 4$.

4. Identify the radius if the diameter of a circle is 10.

   $r = 5$. Radius equals half the diameter.

5. What is the formula for the angle of an inscribed angle?

   Half the measure of the intercepted arc. Inscribed angle theorem relates to intercepted arc.

6. Find the circumference if the radius of a circle is 7.

   $C = 14\pi$. Apply $C = 2\pi r$ with $r = 7$.

7. Find the center of the circle: $(x-5)^2 + (y+3)^2 = 25$.

   Center = $(5, -3)$. The center coordinates are $(h,k)$ from $(x-h)^2 + (y-k)^2 = r^2$.

8. What is the formula for the equation of a circle in general form?

   $Ax^2 + Ay^2 + Dx + Ey + F = 0$. The expanded form where $A$, $D$, $E$, and $F$ are constants.

9. Calculate the area of a sector with radius 3 and angle 30°.

   $A = \frac{3\pi}{2}$. Use $A = \frac{1}{2}r^2\theta$ with $\theta = \frac{\pi}{6}$.

10. Find the radius if the circumference is $20\pi$.

    $r = 10$. Solve $20\pi = 2\pi r$ to get $r = 10$.

11. What is the formula for the equation of a circle in standard form?

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

12. What defines a concentric circle?

    Circles with the same center but different radii. Circles sharing center point with different sizes.

13. Identify the center and radius of $x^2 + y^2 = 49$.

    Center is $(0, 0)$, $r = 7$. Origin center with $r = \sqrt{49} = 7$.

14. Determine the diameter given the circle's circumference is $31.4$ cm.

    Diameter = $10$ cm. Using $C = 2\pi r$: $31.4 = 2\pi r$, so $r = 5$, diameter = $10$.

15. Identify the center of the circle $(x - 3)^2 + (y + 2)^2 = 16$.

    Center is $(3, -2)$. Center coordinates are $(h,k) = (3,-2)$.

16. Identify the circumference of a circle with radius 12.

    $C = 24\pi$. Apply $C = 2\pi r$ with $r = 12$.

17. What is the term for a circle's distance across through the center?

    Diameter. Standard term for longest chord through center.

18. What is the term for a circle's boundary?

    Circumference. Standard term for circle's perimeter.

19. What is the formula for the area of a circle?

    $A = \pi r^2$. Standard formula for area using radius squared.

20. State the definition of a tangent line to a circle.

    A line that touches a circle at exactly one point. Key property distinguishing tangent from secant lines.

21. What is the formula for the circumference of a circle?

    $C = 2\pi r$. Standard formula relating circumference to radius.

22. Find the diameter if the circumference is $18\pi$.

    $d = 18$. Solve $18\pi = 2\pi r$ to get $r = 9$, so $d = 18$.

23. State the formula to find the arc length of a circle.

    $L = \frac{\theta}{360} \times 2\pi r$. Arc length equals the fraction of the circle times the circumference.

24. What distinguishes a minor arc from a major arc?

    A minor arc is less than 180°; a major arc is more. Classification based on arc's angular measure.

25. What is the length of an arc with radius 5 and angle 60°?

    $\text{Arc Length} = \frac{5\pi}{3}$. Use $s = r\theta$ where $\theta = \frac{\pi}{3}$ radians.

26. State the formula for the length of an arc.

    $\text{Arc Length} = r\theta$. Formula with angle $\theta$ in radians.

27. State the definition of a secant line in a circle.

    A line that intersects a circle at two points. Distinguishes secant from tangent lines.

28. What is the radius of the circle $(x + 1)^2 + (y - 4)^2 = 25$?

    $r = 5$. Radius equals $\sqrt{25} = 5$.

29. Identify the area of a circle with diameter 8.

    $A = 16\pi$. Use $A = \pi r^2$ with $r = 4$.

30. What is the formula for the equation of a circle centered at the origin?

    $x^2 + y^2 = r^2$. Special case of standard form with center at origin.

31. What defines a chord in a circle?

    A line segment with both endpoints on the circle. Distinguishes chord from other circle segments.

32. Identify the radius given the circle's equation: $(x-3)^2 + (y+2)^2 = 16$.

    Radius = $4$. In standard form $(x-h)^2 + (y-k)^2 = r^2$, the radius is $\sqrt{16} = 4$.

33. Find the circumference of a circle with diameter 14.

    $C = 14\pi$. Apply $C = \pi d$ with $d = 14$.

34. What is the formula for the area of a sector with radius $r$ and angle $\theta$?

    $A = \frac{1}{2}r^2\theta$. Formula for sector area with angle in radians.

35. Determine the radius of the circle $(x - 5)^2 + (y + 6)^2 = 64$.

    $r = 8$. Radius equals $\sqrt{64} = 8$.

36. What is the relationship between a radius and a tangent?

    They are perpendicular at the point of tangency. Radius and tangent form $90°$ angle at contact point.

37. What is the term for the line that divides a chord into two equal parts?

    Perpendicular bisector. Property of line from center to chord midpoint.

38. Determine the length of an arc with radius 4 and angle 90°.

    $\text{Arc Length} = 2\pi$. Use $s = r\theta$ with $\theta = \frac{\pi}{2}$.

39. Determine the radius if the diameter is 20.

    $r = 10$. Radius equals half the diameter.

40. Calculate the diameter if the radius is $7$ cm.

    Diameter = $14$ cm. Diameter equals twice the radius: $2 \times 7 = 14$ cm.

41. State the formula for the sector area of a circle.

    $A = \frac{\theta}{360} \times \pi r^2$. Sector area equals the fraction of the circle times the total area.

42. What is the standard form equation of a circle?

    $(x-h)^2 + (y-k)^2 = r^2$. Where $(h,k)$ is the center and $r$ is the radius.

43. Identify the equation of a circle with center $(0, 0)$ and radius 9.

    $x^2 + y^2 = 81$. Standard form equation with $r^2 = 81$.

44. What is the name for the part of a circle bounded by a chord and the arc?

    Segment. Region between chord and its corresponding arc.

45. What is the formula to find the diameter of a circle given the radius?

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

46. Calculate the radius if the area is $36\pi$.

    $r = 6$. Solve $36\pi = \pi r^2$ to get $r = 6$.

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