Area & Volume - PSAT Math

1. In the figure, trapezoid $ABCD$ has $AB\parallel CD$, with $AB=10$, $CD=22$, and legs $AD=BC=10$. Point $E$ lies on $CD$ such that $BE\perp CD$. What is the area of triangle $BEC$?
   Answer: $24$ Drop perpendiculars from $A$ and $B$ to $CD$ at points $F$ and $E$. Then $DF=EC=(22-10)/2=6$. In right triangle $BEC$: $BC=10$, $EC=6$, so $BE=\sqrt{100-36}=8$. Area $=\frac{1}{2}(6)(8)=24$. (B) uses $EC=5$ (half of $10$). (C) uses $BE=12$. (D) computes $\frac{1}{2}(6)(16)$ using total base length.
2. Refer to the figure. A right circular cone has a base radius of $6\text{ ft}$ and a height of $8\text{ ft}$. What is the volume of the cone, in cubic feet?
   Answer: $96\pi$ Volume $=\dfrac13\pi r^{2}h=\dfrac13\pi(6)^{2}(8)=\dfrac13\pi(36)(8)=96\pi$. B uses $h=12$ instead of 8. C omits the 1/3 factor's effect incorrectly. D multiplies without the 1/3 and doubles the height.
3. A rectangular prism has dimensions $0.5\text{ m}$ by $40\text{ cm}$ by $30\text{ cm}$. What is the volume of the prism in cubic centimeters?
   Answer: $60{,}000\text{ cm}^3$ The question asks for the volume in cubic centimeters of a rectangular prism with dimensions 0.5 m by 40 cm by 30 cm. The formula for the volume of a rectangular prism is length × width × height, requiring consistent units. Convert 0.5 m to 50 cm for consistency in centimeters. Then, volume = 50 × 40 × 30 = 2000 × 30 = 60,000 cm³, noting cubic units for volume. A key error is neglecting unit conversion, leading to incorrect multiplication. Always convert to the requested units before applying the formula to ensure accuracy.
4. A rectangular prism has dimensions $2.5\text{ m}$ by $1.2\text{ m}$ by $0.8\text{ m}$. What is the volume of the prism?
   Answer: $2.4\text{ m}^3$ The question asks for the volume of a rectangular prism with dimensions 2.5 m by 1.2 m by 0.8 m, with the answer in cubic meters. The formula for the volume of a rectangular prism is length × width × height, with all dimensions in meters for consistent units. Multiply the dimensions: first, 2.5 × 1.2 = 3.0, then 3.0 × 0.8 = 2.4 m³. This direct multiplication ensures accuracy with decimal values. A key error is misaligning decimals, such as 2.5 × 1.2 = 3.0 correctly but then 3.0 × 0.8 = 2.0 incorrectly. Another mistake might be adding instead of multiplying, giving sums like 4.5 m, confusing with perimeter. As a test-taking strategy, break down multiplications step-by-step and verify with approximations, like 2.5 × 1 × 1 ≈ 2.5, close to 2.4.
5. A cylinder has radius $4\text{ cm}$ and height $15\text{ cm}$. What is the volume of the cylinder? Use $\pi$ in your answer.
   Answer: $240\pi\text{ cm}^3$ The question asks for the volume in cubic centimeters of a cylinder with radius $4 \text{ cm}$ and height $15 \text{ cm}$, including $\pi$ in the answer. The formula for the volume of a cylinder is $V = \pi r^2 h$, where $r$ is radius and $h$ is height. Substitute $r = 4 \text{ cm}$ and $h = 15 \text{ cm}$ into the formula: $V = \pi (4)^2 \times 15 = \pi \times 16 \times 15$. Calculate $16 \times 15 = 240$, so $V = 240\pi \text{ cm}^3$, ensuring units are consistent in centimeters for all measurements. A key error might be using diameter instead of radius or forgetting to square the radius. When dealing with volumes, always confirm the formula and unit consistency to avoid calculation mistakes.
6. Refer to the figure. Parallelogram $ABCD$ has side lengths $AB=10\text{ cm}$ and $AD=7\text{ cm}$ with an included angle of $30^{\circ}$ at vertex $A$. What is the area of the parallelogram, in square centimeters?
   Answer: 35 Area $=ab\sin\theta=10\times7\times\sin30^{\circ}=70\times0.5=35$. B multiplies the sides but omits the sine factor. C halves only one side before multiplying. D multiplies the shorter side by itself.
7. A circle is inscribed in a square so that it touches all four sides of the square. The diameter of the circle is $10\text{ cm}$. What is the area of the region inside the square but outside the circle? Use $\pi=3.14$.
   Answer: $21.5\text{ cm}^2$ The question asks for the area inside a square but outside an inscribed circle with diameter 10 cm, in square centimeters, using π=3.14. The formulas needed are square area (side²) and circle area (πr²), with side equal to diameter, 10 cm. Square area: 10 × 10 = 100 cm². Circle radius is 5 cm, area: 3.14 × 25 = 78.5 cm². Remaining area: 100 - 78.5 = 21.5 cm². A common error is using radius for square side, leading to 25 cm² square minus smaller area. When finding regions between shapes, calculate each area separately and subtract, verifying units like cm².
8. A metal sheet is cut into a circle with diameter $20\text{ cm}$. What is the area of the circle? (Use $\pi$ in your answer.)
   Answer: $100\pi\text{ cm}^2$ To find the area of a circle with diameter 20 cm, we first need the radius: r = d/2 = 20/2 = 10 cm. The area formula for a circle is A = πr². Substituting: A = π(10)² = π(100) = 100π cm². A very common mistake is using the diameter directly in the formula instead of converting to radius first, which would incorrectly give 400π cm².
9. A rectangular prism has dimensions $2.5\text{ ft}$ by $4\text{ ft}$ by $6\text{ ft}$. What is its volume?
   Answer: $60\text{ ft}^3$ To find the volume of a rectangular prism with dimensions 2.5 ft × 4 ft × 6 ft, we use the formula V = length × width × height. Substituting the values: V = 2.5 × 4 × 6 = 10 × 6 = 60 ft³. When multiplying decimals, it's helpful to first multiply 2.5 × 4 = 10, then multiply by 6. Common errors include miscalculating with the decimal or confusing surface area formulas with volume formulas.
10. The figure shows a regular hexagon with side length $s$ inscribed in a circle. The area of the shaded region (inside the circle but outside the hexagon) is $12\pi - 18\sqrt{3}$. What is the value of $s$?
    Answer: $2\sqrt{3}$ For a regular hexagon inscribed in a circle of radius $r$, the side length equals $r$. Area of circle $=\pi r^2$; area of hexagon $=\frac{3\sqrt{3}}{2}r^2$. Shaded area $=\pi r^2-\frac{3\sqrt{3}}{2}r^2$. Setting this equal to $12\pi-18\sqrt{3}$: $r^2=12$, so $r=2\sqrt{3}$ and $s=r=2\sqrt{3}$. (B) comes from miscomputing hexagon area as $\frac{3\sqrt{3}}{4}s^2$. (C) results from $r^2=36$ via dividing wrong. (D) results from confusing $s$ with apothem.
11. In the figure, a rectangle with dimensions $12 \times 18$ has a semicircle removed from one of its longer sides (diameter along the side). The diameter of the semicircle equals the shorter dimension of the rectangle. What is the perimeter of the resulting shape?
    Answer: $48+6\pi$ Rectangle perimeter would be $2(12+18)=60$. The semicircle is removed from an $18$-length side; its diameter is $12$, so it takes up $12$ units of that side. Remaining flat portion on that side $=18-12=6$. The two other long sides contributions become: $18$ (opposite side) $+ 6$ (remaining on cut side) $+ 12 + 12$ (short sides) $= 48$. Add the semicircle arc $=\frac{1}{2}(2\pi\cdot 6)=6\pi$. Total $=48+6\pi$. (B) forgets to remove the diameter. (C) uses full circumference. (D) uses arc of wrong radius.
12. A trapezoid has bases of lengths $14\text{ cm}$ and $8\text{ cm}$. Its height is $6\text{ cm}$. What is the area of the trapezoid?
    Answer: $66\text{ cm}^2$ The question asks for the area in square centimeters of a trapezoid with bases 14 cm and 8 cm, and height 6 cm. The formula for the area of a trapezoid is (1/2) × (base1 + base2) × height. Substitute the values: (1/2) × (14 + 8) × 6 = (1/2) × 22 × 6. Calculate 22 × 6 = 132, then (1/2) × 132 = 66 cm², with squared units for area. A common error is using only one base or forgetting the 1/2 factor. When working with trapezoids, emphasize selecting the correct formula and verifying unit consistency.
13. A right triangle has legs of length $9\text{ cm}$ and $12\text{ cm}$. A rectangle is formed using the triangle's hypotenuse as the rectangle's length and using a width of $5\text{ cm}$. What is the area of the rectangle?
    Answer: $75\text{ cm}^2$ First, we need to find the hypotenuse of the right triangle with legs 9 cm and 12 cm using the Pythagorean theorem: c² = a² + b². So c² = 9² + 12² = 81 + 144 = 225, giving c = 15 cm. The rectangle uses this hypotenuse (15 cm) as its length and has width 5 cm. Rectangle area = length × width = 15 × 5 = 75 cm². A common mistake is using one of the legs instead of the hypotenuse, or forgetting to apply the Pythagorean theorem.
14. Refer to the figure. The right circular cylinder shown has radius $4\text{ in.}$ and height $9\text{ in.}$ What is the volume of the cylinder, in cubic inches?
    Answer: $144\pi$ Volume is $\pi r^{2}h=\pi(4)^{2}(9)=\pi\cdot16\cdot9=144\pi$. B halves the height. C uses $r=4$ but multiplies by 8 instead of 9. D doubles the correct height.
15. A trapezoid has bases of lengths $14\text{ m}$ and $8\text{ m}$. Its height is $6\text{ m}$. What is the area of the trapezoid?
    Answer: $66\text{ m}^2$ The question asks for the area of a trapezoid with bases 14 m and 8 m, and height 6 m, in square meters. The formula needed is the area of a trapezoid, (1/2) × (sum of bases) × height. Substitute the values: (1/2) × (14 + 8) × 6 = (1/2) × 22 × 6 = 11 × 6 = 66 m². This calculation directly gives the area without further steps. A common error is forgetting the (1/2) factor, resulting in 132 m², or using only one base. For trapezoid areas, always average the bases before multiplying by height, and confirm units like meters are squared for area.
16. A company is building a rectangular room that is $14\text{ ft}$ by $11\text{ ft}$. The floor will be covered with square tiles that are $6\text{ in}$ on each side. If tiles cannot be cut and must cover the floor exactly, how many tiles are needed?
    Answer: $616\text{ tiles}$ The question asks for the number of 6-inch square tiles needed to cover a 14 ft by 11 ft room exactly without cutting. First, convert feet to inches: 14 ft = 168 in, 11 ft = 132 in, noting no area formula is directly needed but division for tiling. Along the length, 168 / 6 = 28 tiles; along the width, 132 / 6 = 22 tiles. Total tiles = 28 × 22 = 616. A key error is forgetting to convert units, leading to incorrect divisions like 14 / 0.5 = 28 but mismatching width. For tiling problems, ensure dimensions are in the same units as tile size and verify exact fit by checking divisibility.
17. In the figure, $ABCD$ is a square with side length $8$. Points $E$ and $F$ are midpoints of sides $BC$ and $CD$, respectively. What is the area of triangle $AEF$?
    Answer: $24$ Place $A=(0,8)$, $B=(8,8)$, $C=(8,0)$, $D=(0,0)$. Then $E=(8,4)$, $F=(4,0)$. Using the shoelace formula: Area $=\frac{1}{2}|0(4-0)+8(0-8)+4(8-4)|=\frac{1}{2}|0-64+16|=\frac{1}{2}(48)=24$. (A) subtracts only two triangles. (B) is an arithmetic error. (D) omits one of the three corner triangles.
18. In the circle shown, $O$ is the center, the radius is $6$, and $\angle AOB=120°$. What is the area of the shaded region bounded by chord $AB$ and minor arc $AB$?
    Answer: $12\pi - 9\sqrt{3}$ Sector area $=\frac{120}{360}\pi(6)^2=12\pi$. Triangle $AOB$ area $=\frac{1}{2}(6)(6)\sin 120°=18\cdot\frac{\sqrt{3}}{2}=9\sqrt{3}$. Segment area $=12\pi-9\sqrt{3}$. (B) doubles the triangle area. (C) uses sector of $60°$. (D) uses full-circle area.
19. A cylindrical water tank has radius $3\text{ m}$ and height $10\text{ m}$. What is the volume of the tank? (Use $\pi$ in your answer.)
    Answer: $90\pi\text{ m}^3$ We need to find the volume of a cylinder with radius 3 m and height 10 m. The volume formula for a cylinder is V = πr²h, where r is the radius and h is the height. Substituting the given values: V = π(3)²(10) = π(9)(10) = 90π m³. Common mistakes include using diameter instead of radius or forgetting to square the radius. When working with π, leave it in symbolic form unless asked to approximate.
20. Refer to the figure. A sector of a circle with radius $10\text{ cm}$ and central angle $60^{\circ}$ is shaded. What is the area of the shaded sector, in square centimeters?
    Answer: $\dfrac{50\pi}{3}$ Sector area is $\dfrac{\theta}{360^{\circ}}\pi r^{2}=\dfrac{60^{\circ}}{360^{\circ}}\pi(10)^{2}=\dfrac16\cdot100\pi=\dfrac{50\pi}{3}$. B. $25\pi$ uses half the radius (5 cm) erroneously. C. $\dfrac{25\pi}{2}$ uses half the correct area of the full circle. D. $100\pi$ is the entire circle's area, not just the sector.

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