Area & Volume - PSAT Math

1. State the formula for the volume of a cone with radius $r$ and height $h$.

   $V = \frac{1}{3}\pi r^2h$. One-third of cylinder volume with same base and height.

2. Identify the volume of a cylinder with radius $3$ and height $5$.

   $45\pi$. Using $V=\pi r^2h$ with $r=3$ and $h=5$.

3. Find the volume of a cylinder with radius $r=2$ and height $h=5$ in terms of $\pi$.

   $20\pi$. Apply $V = \pi r^2h$: $\pi(2)^2(5) = 20\pi$.

4. What is the area of a rectangle with $l=9$ and $w=4$?

   $36$. Apply $A = lw$: $9 \times 4 = 36$.

5. Find the area of a rectangle with $l=9$ and $w=4$.

   $36$. Apply $A = lw$: $9 \times 4 = 36$.

6. What is the surface area formula for a rectangular prism with dimensions $l,w,h$?

   $SA=2(lw+lh+wh)$. Sum of all six rectangular face areas.

7. What is the volume formula for a rectangular prism with dimensions $l,w,h$?

   $V=lwh$. Multiply all three dimensions for box volume.

8. State the formula for the area of a triangle with base $b$ and height $h$.

   $A = \frac{1}{2}bh$. Half the product of base and height gives triangular area.

9. What is the surface area formula for a sphere with radius $r$?

   $SA=4\pi r^2$. Four times pi times radius squared.

10. What is the area of a square with side length $s$?

    $A = s^2$. Square area equals side length squared.

11. State the formula for the area of a trapezoid with bases $b_1, b_2$ and height $h$.

    $A = \frac{1}{2}(b_1+b_2)h$. Average of parallel bases times height.

12. Find the area of a circle with radius $r=3$ in terms of $\pi$.

    $9\pi$. Apply $A = \pi r^2$: $\pi(3)^2 = 9\pi$.

13. What is the area of a circle with diameter $10$?

    $25\pi$. Radius is $5$, so area is $\pi(5)^2 = 25\pi$.

14. State the formula for the surface area of a sphere with radius $r$.

    $\text{SA} = 4\pi r^2$. Sphere surface area equals four times circle area with same radius.

15. State the formula for the area of a sector with radius $r$ and central angle $\theta$ degrees.

    $A = \frac{\theta}{360}\pi r^2$. Fraction of circle's area based on angle ratio.

16. What is the volume of a sphere with $r=3$? (Leave in terms of $\pi$.)

    $36\pi$. Apply $V = \frac{4}{3}\pi r^3$: $\frac{4}{3} \times \pi \times 3^3 = 36\pi$.

17. State the formula for the arc length of a circle with radius $r$ and central angle $\theta$ (in radians).

    $s = r\theta$. Arc length equals radius times angle in radians.

18. What is the surface area of a cube with side length $s=4$?

    $SA = 96$. Cube has 6 faces: $6 \times 4^2 = 96$.

19. State the formula for the area of a sector with central angle $\theta$ (degrees) and radius $r$.

    $A = \frac{\theta}{360}\pi r^2$. Fraction of circle's area based on angle ratio.

20. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

    $V = lwh$. Multiply all three dimensions for box volume.

21. What is the area of a sector with $r=6$ and $\theta=90^\circ$? Express in terms of $\pi$.

    $9\pi$. Sector is $\frac{90}{360} = \frac{1}{4}$ of circle: $\frac{1}{4} \times \pi \times 6^2 = 9\pi$.

22. What is the area of a rectangle with $l=9$ and $w=4$?

    $36$. Apply $A = lw$: $9 \times 4 = 36$.

23. Identify the area of a trapezoid with $b_1=4$, $b_2=10$, and $h=6$.

    $42$. Apply trapezoid formula: $\frac{1}{2}(4+10) \times 6 = 42$.

24. What is the surface area formula for a cylinder with radius $r$ and height $h$?

    $SA=2\pi r^2+2\pi rh$. Two circular bases plus curved lateral surface.

25. Identify the volume of a rectangular prism with $l=4$, $w=3$, and $h=5$.

    $60$. Apply $V = lwh$: $4 \times 3 \times 5 = 60$.

26. What is the volume of a cylinder with $r=2$ and $h=5$ (leave in terms of $\pi$)?

    $20\pi$. Apply $V=pi r^2h$: $pi \times 2^2 \times 5 = 20pi$.

27. What is the area of a triangle with $b=10$ and $h=6$?

    $30$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 10 \times 6 = 30$.

28. Identify the volume of a cone with radius $6$ and height $4$.

    $48\pi$. Using $V=\frac{1}{3}\pi r^2h$ with $r=6$ and $h=4$.

29. Identify the scale factor effect: if all lengths scale by $k$, how does volume scale?

    Volume scales by $k^3$. Volume grows with cube of linear scale factor.

30. What is the volume of a cylinder with $r=2$ and $h=5$? Express in terms of $\pi$.

    $20\pi$. Apply $V = \pi r^2h$: $\pi \times 2^2 \times 5 = 20\pi$.

31. Identify the scale factor for area if all linear dimensions are multiplied by $k$.

    Area scales by $k^2$. Area increases by the square of the scale factor.

32. Identify the volume of a cone with $r=6$ and $h=4$ in terms of $\pi$.

    $48\pi$. Apply $V = \frac{1}{3}\pi r^2h$: $\frac{1}{3} \times \pi \times 6^2 \times 4 = 48\pi$.

33. State the formula for the arc length of a circle with radius $r$ and central angle $\theta$ degrees.

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

34. What is the total surface area of a right circular cylinder with radius $r$ and height $h$?

    $S=2\pi r^2+2\pi rh$. Total surface includes lateral area plus two circular bases.

35. Identify the area of a triangle with $b=10$ and $h=6$.

    $30$. Apply $A=\frac{1}{2}bh=\frac{1}{2}(10)(6)=30$.

36. Identify the scale factor effect: if all lengths scale by $k$, how does area scale?

    Area scales by $k^2$. Area grows with square of linear scale factor.

37. What is the volume of a rectangular prism with $l=5$, $w=3$, and $h=2$?

    $30$. Apply $V = lwh$: $5 \times 3 \times 2 = 30$.

38. State the formula for the area of a sector with central angle $\theta$ (in radians) and radius $r$.

    $A = \frac{1}{2}r^2\theta$. Fraction of circle's area proportional to angle.

39. What is the area of a square with side length $s$?

    $A=s^2$. Square the side length for square area.

40. Find the area of a circle with $r=5$ in terms of $\pi$.

    $25\pi$. Apply $A = \pi r^2 = \pi(5)^2 = 25\pi$.

41. State the formula for the length of an arc with radius $r$ and central angle $\theta$ (degrees).

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

42. Identify the area of a trapezoid with bases $8$ and $14$ and height $5$.

    $55$. Using $A=\frac{1}{2}(8+14)(5)=\frac{1}{2}(22)(5)$.

43. What is the volume formula for a prism with base area $B$ and height $h$?

    $V=Bh$. Base area times height for any prism.

44. State the formula for the area of a sector with radius $r$ and central angle $\theta$ (degrees).

    $A = \frac{\theta}{360}\pi r^2$. Fraction of circle's area based on angle ratio.

45. State the formula for the area of a semicircle with radius $r$.

    $A = \frac{1}{2}\pi r^2$. Half a circle's area is $\frac{1}{2}\pi r^2$.

46. State the formula for the surface area of a rectangular prism with dimensions $l,w,h$.

    $SA = 2(lw+lh+wh)$. Sum of all six rectangular face areas.

47. What is the volume formula for a cylinder with radius $r$ and height $h$?

    $V=\pi r^2h$. Circular base area times height.

48. State the formula for the area of a sector with central angle $\theta$ degrees and radius $r$.

    $A = \frac{\theta}{360}\pi r^2$. Sector area is the fraction of circle area based on angle ratio.

49. State the formula for the surface area of a rectangular prism with dimensions $l$, $w$, and $h$.

    $SA = 2(lw+lh+wh)$. Sum of all six rectangular face areas.

50. Identify the area of a rectangle with $l=8$ and $w=3$.

    $24$. Apply $A = lw$: $8 \times 3 = 24$.

51. What is the volume of a cone with $r=3$ and $h=4$? (Leave in terms of $\pi$.)

    $12\pi$. Apply $V = \frac{1}{3}\pi r^2h$: $\frac{1}{3} \times \pi \times 3^2 \times 4 = 12\pi$.

52. State the formula for the area of a sector with radius $r$ and central angle $\theta$ (degrees).

    $A = \frac{\theta}{360}\pi r^2$. Sector area is fraction of circle based on central angle.

53. State the formula for the surface area of a cylinder with radius $r$ and height $h$.

    $SA = 2\pi r^2 + 2\pi rh$. Two circular bases plus curved lateral surface.

54. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

    $V = lwh$. Product of all three dimensions gives box volume.

55. What is the area of a circle with radius $r=3$?

    $A = 9\pi$. Apply $A = \pi r^2$: $\pi \times 3^2 = 9\pi$.

56. What is the area of a rectangle with $l=8$ and $w=3$?

    $24$. Apply $A=lw$: $8 \times 3 = 24$.

57. State the formula for the volume of a rectangular prism with length $l$, width $w$, and height $h$.

    $V=lwh$. Box volume equals length times width times height.

58. What is the volume formula for a pyramid with base area $B$ and height $h$?

    $V=\frac{1}{3}Bh$. One-third of prism volume with same base and height.

59. What is the area of a circle with $r=3$? (Leave in terms of $\pi$.)

    $9\pi$. Apply $A = \pi r^2$: $\pi \times 3^2 = 9\pi$.

60. State the formula for the volume of a sphere with radius $r$.

    $V = \frac{4}{3}\pi r^3$. Four-thirds pi times radius cubed for spherical volume.

61. State the formula for the volume of a rectangular prism with length $l$, width $w$, height $h$.

    $V = lwh$. Box volume equals length times width times height.

62. State the formula for the surface area of a rectangular prism with dimensions $l,w,h$.

    $SA = 2(lw+lh+wh)$. Sum of all six rectangular face areas.

63. What is the area of a semicircle with radius $r$?

    $A=\frac{1}{2}\pi r^2$. Half of a full circle's area.

64. State the formula for the arc length of a sector with radius $r$ and central angle $\theta$ (degrees).

    $s = \frac{\theta}{360}2\pi r$. Arc length is fraction of circumference based on angle.

65. State the formula for the surface area of a rectangular prism with edges $l$, $w$, and $h$.

    $SA = 2(lw+lh+wh)$. Sum of areas of all six rectangular faces.

66. State the formula for the volume of a right circular cylinder with radius $r$ and height $h$.

    $V=\pi r^2h$. Cylinder volume is base area times height.

67. Find the area of a triangle with base $b=10$ and height $h=7$.

    $35$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 10 \times 7 = 35$.

68. Identify the area of a circle with diameter $10$.

    $25\pi$. Diameter $10$ gives radius $5$, so area is $\pi(5)^2$.

69. Identify the scale factor for volume if all linear dimensions are multiplied by $k$.

    Volume scales by $k^3$. Volume increases by the cube of the scale factor.

70. State the formula for the volume of a cylinder with radius $r$ and height $h$.

    $V = \pi r^2h$. Base area times height for cylindrical volume.

71. State the formula for the circumference of a circle with radius $r$.

    $C = 2\pi r$. Distance around circle is $2\pi$ times radius.

72. State the formula for the volume of a right circular cone with radius $r$ and height $h$.

    $V=\frac{1}{3}\pi r^2h$. Cone volume is one-third of cylinder with same base and height.

73. If a rectangle's length doubles and width stays the same, what happens to its area?

    $\text{The area doubles.}$. Since $A = lw$, doubling $l$ doubles the area.

74. State the formula for the area of a sector with radius $r$ and central angle $\theta$ (degrees).

    $A = \frac{\theta}{360}\pi r^2$. Sector area is the fraction of circle area based on angle.

75. What is the volume of a cone with $r=3$ and $h=4$? Express in terms of $\pi$.

    $12\pi$. Apply $V = \frac{1}{3}\pi r^2h$: $\frac{1}{3} \times \pi \times 3^2 \times 4 = 12\pi$.

76. A rectangle's length doubles and width triples; what is the area scale factor?

    $6$. Area scales by product of dimension changes: $2 \times 3 = 6$.

77. State the formula for the volume of a pyramid with base area $B$ and height $h$.

    $V = \frac{1}{3}Bh$. One-third base area times height for any pyramid.

78. A cube has side length $6$. What is its volume?

    $216$. Cube volume is $s^3$: $6^3 = 216$.

79. State the formula for the arc length of a sector with central angle $\theta$ (degrees) and radius $r$.

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

80. What is the area of a triangle with base $b=12$ and height $h=5$?

    $A = 30$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 12 \times 5 = 30$.

81. State the formula for the area of a trapezoid with bases $b_1,b_2$ and height $h$.

    $A = \frac{1}{2}(b_1+b_2)h$. Average of parallel bases times height.

82. State the formula for the area of a rhombus (or kite) with diagonals $d_1$ and $d_2$.

    $A = \frac{1}{2}d_1d_2$. Half the product of perpendicular diagonals.

83. State the formula for the volume of a right circular cone with radius $r$ and height $h$.

    $V = \frac{1}{3}\pi r^2h$. Cone volume is one-third of cylinder with same base and height.

84. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

    $V = lwh$. Box volume equals length times width times height.

85. What is the volume of a cylinder with $r=2$ and $h=7$? (Leave in terms of $\pi$.)

    $28\pi$. Apply $V = \pi r^2h$: $\pi \times 2^2 \times 7 = 28\pi$.

86. State the formula for the area of a circle with radius $r$.

    $A = \pi r^2$. Pi times radius squared gives circular area.

87. State the formula for the volume of a rectangular prism with dimensions $l$, $w$, and $h$.

    $V = lwh$. Product of all three dimensions gives box volume.

88. What is the surface area of a sphere with radius $r$?

    $S=4\pi r^2$. Sphere surface area equals four pi times radius squared.

89. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

    $V = lwh$. Product of all three dimensions gives box volume.

90. State the formula for the area of a rhombus with diagonals $d_1$ and $d_2$.

    $A = \frac{1}{2}d_1d_2$. Half the product of perpendicular diagonals.

91. Identify the volume of a cylinder with $r=3$ and $h=7$ in terms of $\pi$.

    $63\pi$. Apply $V = \pi r^2h$: $\pi \times 3^2 \times 7 = 63\pi$.

92. If all linear dimensions scale by $k$, what is the volume scale factor?

    $k^3$. Volume scales as cube of linear scale factor.

93. State the formula for the area of a parallelogram with base $b$ and height $h$.

    $A = bh$. Base times perpendicular height for parallelogram area.

94. State the formula for the lateral surface area of a right circular cylinder with radius $r$ and height $h$.

    $\text{LSA} = 2\pi rh$. Lateral area wraps around cylinder: circumference times height.

95. What is the area of a triangle with $b=12$ and $h=5$?

    $30$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 12 \times 5 = 30$.

96. Identify the area of a triangle with $b=10$ and $h=6$.

    $30$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 10 \times 6 = 30$.

97. State the formula for the area of a sector with central angle $\theta$ (in degrees) and radius $r$.

    $A=\frac{\theta}{360}\pi r^2$. Fraction of circle's area based on angle ratio.

98. What is the volume of a rectangular prism with $l=5$, $w=2$, and $h=3$?

    $V = 30$. Apply $V = lwh$: $5 \times 2 \times 3 = 30$.

99. State the formula for the area of a rectangle with length $l$ and width $w$.

    $A = lw$. Multiply length by width for rectangular area.

100. Identify the area of a circle with radius $r=5$ in terms of $\pi$.

     $25\pi$. Apply $A = \pi r^2$: $\pi \times 5^2 = 25\pi$.

101. Find the area of a triangle with $b=10$ and $h=6$.

     $30$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 10 \times 6 = 30$.

102. State the formula for the surface area of a rectangular prism with dimensions $l,w,h$.

     $SA = 2(lw+lh+wh)$. Sum of all six face areas, paired opposites.

103. What is the volume formula for a sphere with radius $r$?

     $V=\frac{4}{3}\pi r^3$. Four-thirds pi times radius cubed.

104. State the formula for the arc length of a sector with central angle $\theta$ (degrees) and radius $r$.

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

105. State the formula for the lateral area of a cylinder with radius $r$ and height $h$.

     $LA = 2\pi rh$. Curved surface area excluding circular bases.

106. What is the area of a triangle with $b=12$ and $h=5$?

     $30$. Apply $A = \frac{1}{2}bh$: $\frac{1}{2} \times 12 \times 5 = 30$.

107. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

     $V = lwh$. Product of all three dimensions gives box volume.

108. State the formula for the area of a sector with central angle $\theta$ (degrees) and radius $r$.

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

109. State the formula for the volume of a right circular cylinder with radius $r$ and height $h$.

     $V = \pi r^2h$. Cylinder volume equals base area times height.

110. State the formula for the area of a sector with central angle $\theta$ (degrees) and radius $r$.

     $A=\frac{\theta}{360}\pi r^2$. Fraction of circle's area based on angle ratio.

111. What is the volume of a cylinder with $r=2$ and $h=10$? (Leave in terms of $\pi$.)

     $40\pi$. Apply $V = \pi r^2h$: $\pi \times 2^2 \times 10 = 40\pi$.

112. Find the volume of a cylinder with $r=3$ and $h=7$ in terms of $\pi$.

     $63\pi$. Apply $V = \pi r^2h = \pi(3)^2(7) = 63\pi$.

113. Find the volume of a cylinder with $r=3$ and $h=5$ in terms of $\pi$.

     $45\pi$. Apply $V = \pi r^2h$: $\pi \times 3^2 \times 5 = 45\pi$.

114. State the formula for the arc length of a sector with central angle $\theta$ (degrees) and radius $r$.

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

115. What is the circumference of a circle with diameter $d=12$? (Leave in terms of $\pi$.)

     $12\pi$. Diameter 12 means $r = 6$, so $C = 2\pi(6) = 12\pi$.

116. What is the lateral surface area of a right circular cylinder with radius $r$ and height $h$?

     $L=2\pi rh$. Lateral area wraps around cylinder without top/bottom.

117. What is the area of a rectangle with $l=9$ and $w=4$?

     $36$. Apply $A = lw$: $9 \times 4 = 36$.

118. State the formula for the surface area of a rectangular prism with dimensions $l,w,h$.

     $SA = 2(lw+lh+wh)$. Add areas of all six rectangular faces.

119. Find the volume of a rectangular prism with $l=3$, $w=5$, and $h=2$.

     $30$. Apply $V = lwh$: $3 \times 5 \times 2 = 30$.

120. What is the area of a circle with radius $r=3$? (Leave in terms of $\pi$.)

     $9\pi$. Apply $A = \pi r^2$: $\pi \times 3^2 = 9\pi$.

121. State the formula for the total surface area of a right circular cylinder with radius $r$ and height $h$.

     $\text{SA} = 2\pi rh + 2\pi r^2$. Total surface includes lateral area plus two circular bases.

122. What is the area of a circle with radius $r=3$? Express in terms of $\pi$.

     $9\pi$. Apply $A = \pi r^2$: $\pi \times 3^2 = 9\pi$.

123. State the formula for the volume of a prism with base area $B$ and height $h$.

     $V = Bh$. Any prism's volume equals base area times height.

124. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

     $V=lwh$. Product of length, width, and height.

125. What is the volume formula for a cone with radius $r$ and height $h$?

     $V=\frac{1}{3}\pi r^2h$. One-third of cylinder volume with same dimensions.

126. Find the area of a rectangle with $l=8$ and $w=3$.

     $24$. Apply $A = lw$: $8 \times 3 = 24$.

127. State the formula for the area of a sector with radius $r$ and central angle $\theta$ degrees.

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

128. Identify the volume of a cylinder with $r=3$ and $h=5$ in terms of $\pi$.

     $45\pi$. Apply $V=\pi r^2h=\pi(3)^2(5)=45\pi$.

129. What is the area of a trapezoid with $b_1=8$, $b_2=14$, and $h=5$?

     $55$. Apply trapezoid formula: $\frac{1}{2}(8+14) \times 5 = 55$.

Study these flashcards