Calculate Area And Volume - HSPT Math

Two cubes are stacked on top of each other. The larger cube has a side length of 5 cm, and the smaller cube has a side length of 3 cm. The smaller cube is centered on top of the larger one. What is the total surface area of the resulting composite figure?
Answer: 186 cm² To find the total surface area, calculate the surface area of both cubes and subtract the overlapping areas. The surface area of the larger cube is 6(5²) = 150 cm². The surface area of the smaller cube is 6(3²) = 54 cm². When stacked, the bottom face of the smaller cube (9 cm²) and an equal area on top of the larger cube are no longer part of the external surface. Total surface area = 150 + 54 - 2(9) = 204 - 18 = 186 cm².
The length of a rectangular field is increased by 20%, and its width is decreased by 10%. What is the resulting percentage change in the area of the field?
Answer: 8% increase Let the original length be $L$ and the original width be $W$ . The original area is $A = LW$ . The new length $L'$ is $L + 0.20L = 1.2L$ . The new width $W'$ is $W - 0.10W = 0.9W$ . The new area $A'$ is $L'W' = (1.2L)(0.9W) = 1.08LW$ . To find the percentage change, use the formula $\frac{\text{New Area} - \text{Old Area}}{\text{Old Area}} \times 100\%. This is \(\frac{1.08LW - LW}{LW} \times 100\% = \frac{0.08LW}{LW} \times 100\% = 0.08 \times 100\% = 8\%$ . Since the factor 1.08 is greater than 1, it is an 8% increase.
A square pyramid has a base edge of 12 cm and a slant height of 10 cm. What is the total surface area of the pyramid?
Answer: 384 square cm When you encounter a surface area problem for a square pyramid, you need to find the area of all faces: one square base plus four triangular faces. Start with the square base. With a base edge of 12 cm, the base area is $12^2 = 144$ square cm. For the four triangular faces, each triangle has a base of 12 cm (the edge of the square) and a height equal to the slant height of 10 cm. The area of one triangular face is $\frac{1}{2} \times 12 \times 10 = 60$ square cm. Since there are four identical triangular faces, their total area is $4 \times 60 = 240$ square cm. The total surface area is $144 + 240 = 384$ square cm, confirming answer A. Let's examine why the other answers are incorrect. Answer B (288 square cm) likely comes from forgetting to include the base area and only calculating the four triangular faces plus some partial calculation. Answer C (336 square cm) might result from incorrectly calculating the triangular face areas or making an arithmetic error in the final sum. Answer D (240 square cm) represents only the lateral surface area (the four triangular faces) without including the square base. Remember that "total surface area" means ALL faces of the 3D shape. For pyramids, always calculate the base area separately from the triangular faces, then add them together. Don't confuse total surface area with lateral surface area, which excludes the base.
A running track is formed by a rectangle with a semicircle at each end. The rectangle is 100 meters long and 70 meters wide. What is the perimeter of the track?
Answer: $200 + 70\pi$ m The perimeter of the track consists of the two straight sides of the rectangle and the circumferences of the two semicircles. The length of the two straight sides is $100 \text{ m} + 100 \text{ m} = 200 \text{ m}$ . The two semicircles at each end form a complete circle. The diameter of this circle is equal to the width of the rectangle, which is 70 meters. The circumference of this circle is $C = \pi d = 70\pi$ meters. The total perimeter of the track is the sum of the lengths of the straight sides and the circumference of the circle: $200 + 70\pi$ meters.
A circle has a circumference of 31.4 meters ( $\pi\approx3.14$ ). What is the radius of the circle?
Answer: 5 When you encounter a circle problem involving circumference and radius, you're working with one of the most fundamental relationships in geometry. The circumference formula connects these two measurements: $C = 2\pi r$ , where C is circumference and r is radius. To find the radius when you know the circumference, you need to rearrange this formula. Starting with $C = 2\pi r$ , divide both sides by $2\pi$ to get $r = \frac{C}{2\pi}$ . Substituting the given values: $r = \frac{31.4}{2 \times 3.14} = \frac{31.4}{6.28} = 5$ meters. Looking at the wrong answers reveals common calculation errors. Choice (A) 2.5 represents dividing the circumference by $4\pi$ instead of $2\pi$ - perhaps from incorrectly remembering the formula or making an arithmetic mistake with the 2. Choice (C) 10 comes from dividing circumference by $\pi$ only, forgetting the factor of 2 entirely - this would give you the diameter, not the radius. Choice (D) 15 doesn't follow any logical mathematical relationship and might result from random guessing or completely misapplying the formula. Remember this pattern: circumference problems on the HSPT often test whether you can correctly manipulate the basic circle formulas. Always write down $C = 2\pi r$ first, then solve for the unknown variable. Double-check by substituting your answer back into the original formula - here, $2 \times 3.14 \times 5 = 31.4$ confirms our answer.
A trapezoid has bases of 10 ft and 14 ft and a height of 6 ft. What is its area?
Answer: 72 When you encounter a trapezoid area problem, remember that a trapezoid has two parallel sides called bases and a perpendicular distance between them called the height. The area formula is: $A = \frac{1}{2}(b_1 + b_2) \times h$ , where $b_1$ and $b_2$ are the lengths of the parallel bases. Let's apply this formula with the given measurements: bases of 10 ft and 14 ft, and height of 6 ft. $A = \frac{1}{2}(10 + 14) \times 6$ $A = \frac{1}{2}(24) \times 6$ $A = 12 \times 6 = 72 \text{ square feet}$ This confirms answer C is correct. Let's examine why the other answers are wrong: Answer A (48) likely comes from forgetting to add the bases together first, instead calculating $\frac{1}{2} \times 10 \times 6 + \frac{1}{2} \times 14 \times 6 = 30 + 42 = 72$ ... wait, that's still 72. More likely, someone calculated $(14 - 10) \times 6 \times 2 = 48$ , confusing this with a parallelogram formula. Answer B (60) results from multiplying the height by the average of the bases without the $\frac{1}{2}$ : $(10 + 14) \div 2 \times 6 = 12 \times 5 = 60$ . This shows a conceptual error in the formula. Answer D (84) comes from multiplying all three numbers together: $10 + 14 = 24$ , then $24 \times 6 = 144$ ... actually, this might be $14 \times 6 = 84$ , using only one base. Remember: trapezoid area always involves the sum of both bases, multiplied by height, then divided by 2. Write the formula down first to avoid calculation errors.
The base of a triangular prism is a right triangle with a hypotenuse of 13 cm and one leg of 5 cm. If the length of the prism is 20 cm, what is its volume?
Answer: 600 cm³ The volume of a prism is the area of its base multiplied by its length. First, we need to find the area of the triangular base. It is a right triangle. We are given the hypotenuse (13 cm) and one leg (5 cm). We use the Pythagorean theorem (a² + b² = c²) to find the other leg (b): $5^2 + b^2 = 13^2$ , which is $25 + b^2 = 169$ . Subtract 25 from both sides: $b^2 = 144$ , so $b = 12$ cm. The two legs of the right triangle are its base and height. The area of the triangle is $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 12 = 30$ cm². Now, multiply the base area by the length of the prism to find the volume: $V = A \times L = 30 \text{ cm}^2 \times 20 \text{ cm} = 600$ cm³.
A solid metal cube with a side length of 4 inches is placed inside a larger, empty cubical box with a side length of 5 inches. How much empty space is in the box, in cubic inches?
Answer: 61 cubic inches This problem requires calculating two volumes and finding the difference. First, calculate the volume of the larger cubical box: $V_{box} = s^3 = 5^3 = 125$ cubic inches. Next, calculate the volume of the smaller metal cube: $V_{cube} = s^3 = 4^3 = 64$ cubic inches. The empty space is the volume of the box minus the volume of the cube: $125 - 64 = 61$ cubic inches.
A circular pizza with a 14-inch diameter is cut into 8 equal slices. What is the perimeter of one slice?
Answer: $14 + 1.75\pi$ inches A slice of pizza is a sector of a circle. Its perimeter consists of two straight sides (the radii) and one curved side (the arc). The diameter is 14 inches, so the radius $r$ is 7 inches. The two straight sides of the slice are each equal to the radius, so their total length is $7 + 7 = 14$ inches. The circumference of the whole pizza is $C = \pi d = 14\pi$ inches. Since there are 8 equal slices, the arc length of one slice is $\frac{1}{8}$ of the circumference: $\frac{14\pi}{8} = \frac{7\pi}{4} = 1.75\pi$ inches. The total perimeter of one slice is the sum of the lengths of the two radii and the arc: $14 + 1.75\pi$ inches.
Using $\pi\approx3.14$ , what is the area of a circle with a radius of 5 inches, to the nearest tenth?
Answer: 78.5 When you encounter a circle area problem, you're applying one of geometry's most fundamental formulas: $A = \pi r^2$ , where $A$ is the area and $r$ is the radius. With a radius of 5 inches and $\pi \approx 3.14$ , substitute these values into the formula: $A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5$ square inches. Since 78.5 already has one decimal place, rounding to the nearest tenth gives us 78.5, making C the correct answer. The wrong answers represent common calculation errors. Choice A (31.4) occurs if you mistakenly use the circumference formula instead of area— $C = 2\pi r = 2 \times 3.14 \times 5 = 31.4$ . This is a frequent mix-up since both formulas involve $\pi$ and the radius. Choice B (62.8) happens if you forget to square the radius and calculate $3.14 \times 5 \times 2 = 31.4$ , then perhaps double it, or use $\pi \times 2r$ thinking it's area. Choice D (157.0) results from doubling the correct answer, possibly from confusion about whether to use radius or diameter in your final calculation. Remember this key distinction: circumference is linear (measured in inches), while area is square (measured in square inches). The presence of $r^2$ in the area formula means you'll get a much larger number than the circumference. When you see circle problems, first identify whether you need circumference ( $2\pi r$ ) or area ( $\pi r^2$ )—the units in the answer choices often provide a helpful clue.
A paint covers about 350 square feet per gallon. The four walls of a rectangular room are each 8 feet high. Two opposite walls are 12 feet long, and the other two are 15 feet long. If two 3-ft by 7-ft doors will not be painted, how many gallons of paint are needed?
Answer: 2 This is a surface area problem that requires you to calculate the total wall area, subtract the unpainted sections, then determine paint needed based on coverage rate. Start by finding the total wall area. The room has four walls: two that are 12 feet long and two that are 15 feet long, all 8 feet high. The total wall area is $2(12 \times 8) + 2(15 \times 8) = 2(96) + 2(120) = 192 + 240 = 432$ square feet. Next, subtract the area that won't be painted. Two doors each measure 3 ft by 7 ft, so the unpainted area is $2(3 \times 7) = 42$ square feet. The paintable area is $432 - 42 = 390$ square feet. Finally, determine gallons needed. Since each gallon covers 350 square feet, you need $\frac{390}{350} = 1.11$ gallons. Since you can't buy a fraction of a gallon, you need 2 gallons, making (B) correct. (A) 1 gallon would only cover 350 square feet, leaving 40 square feet unpainted. (C) 3 gallons represents calculating the wall area incorrectly, perhaps by adding length + width instead of using the perimeter formula. (D) 4 gallons suggests a major calculation error, possibly forgetting to subtract the door areas or miscalculating the room dimensions entirely. Remember: surface area problems always follow the same pattern—calculate total area, subtract any excluded sections, then apply the given rate. Always round up when buying materials since you can't purchase partial units.
A rectangular box measures 10 cm by 6 cm by 4 cm. What is the volume of the box in cubic centimeters?
Answer: 240 When you encounter a question asking for the volume of a rectangular box, you're working with one of the most fundamental 3D geometry formulas. Volume measures how much space is inside a three-dimensional object, and for rectangular boxes (also called rectangular prisms), the calculation is straightforward. To find the volume of a rectangular box, you multiply all three dimensions together: length × width × height. In this case, you have dimensions of 10 cm, 6 cm, and 4 cm. So the volume is $10 \times 6 \times 4 = 240$ cubic centimeters. Let's see why the other answers represent common mistakes. Choice A (40) likely comes from adding two dimensions and ignoring the third: $10 + 6 + 4 \times 10 = 40$ , but this doesn't give you volume. Choice B (60) results from multiplying only two dimensions: $10 \times 6 = 60$ , which would give you the area of one face, not the volume of the entire box. Choice C (160) might come from incorrectly calculating $10 \times 4 \times 4 = 160$ , perhaps by accidentally using 4 twice instead of using all three different measurements. The correct answer is D (240) because you need all three dimensions multiplied together. Remember this key strategy: volume problems for rectangular shapes always require multiplying all three dimensions. Don't be tempted to add dimensions or use only two of them—those operations give you different measurements entirely, not volume.
A trapezoid has parallel sides of lengths 8 cm and 14 cm, with a height of 5 cm. If the trapezoid is rotated 90° clockwise about its center, what happens to its area?
Answer: The area remains 55 square cm When you encounter questions about geometric transformations like rotations, the key principle to remember is that rotations preserve all measurements—lengths, angles, and areas remain unchanged. First, let's calculate the trapezoid's area using the standard formula: $A = \frac{1}{2}(b_1 + b_2)h$ , where $b_1$ and $b_2$ are the parallel sides and $h$ is the height. Substituting our values: $A = \frac{1}{2}(8 + 14)(5) = \frac{1}{2}(22)(5) = 55$ square cm. Now, when the trapezoid rotates 90° clockwise about its center, it simply changes orientation—imagine spinning it like a wheel. The parallel sides are still the same lengths (8 cm and 14 cm), and the perpendicular distance between them (height) remains 5 cm. Rotation is a rigid transformation, meaning it preserves all distances and areas. Choice A (60 square cm) might tempt you if you incorrectly used 12 instead of 8 for one of the bases. Choice B (50 square cm) could result from miscalculating $\frac{1}{2}(8 + 14)(5)$ as $\frac{1}{2}(20)(5)$ , forgetting that 8 + 14 = 22, not 20. Choice D (70 square cm) represents the trap of thinking rotation somehow increases area, perhaps by incorrectly calculating $(8 + 14) \times 5 = 110$ and then halving it incorrectly. Remember: rotations, reflections, and translations never change area or perimeter. Only scaling transformations affect these measurements. When you see rotation problems, focus on whether the transformation preserves the original measurements.
A rectangular painting measures 2 feet by 3 feet. A wooden frame of uniform width is to be built around it. If the area of the frame itself is 14 square feet, what is the width of the frame in feet?
Answer: 1 foot Let the width of the frame be $x$ feet. The dimensions of the painting are 2 ft by 3 ft, so its area is $2 \times 3 = 6$ sq ft. The frame adds $x$ to each side, so the new outer dimensions are $(2 + 2x)$ by $(3 + 2x)$ . The total area of the painting plus the frame is $A_{total} = (2 + 2x)(3 + 2x)$ . The area of the frame is the total area minus the painting's area: $A_{frame} = A_{total} - A_{painting}$ . We are given $A_{frame} = 14$ . So, $14 = (2 + 2x)(3 + 2x) - 6$ . Add 6 to both sides: $20 = (2 + 2x)(3 + 2x)$ . Expand the expression: $20 = 6 + 4x + 6x + 4x^2 = 4x^2 + 10x + 6$ . Rearrange into a quadratic equation: $4x^2 + 10x - 14 = 0$ . Divide by 2: $2x^2 + 5x - 7 = 0$ . This can be factored as $(2x + 7)(x - 1) = 0$ . The possible solutions are $x = -3.5$ or $x = 1$ . Since width cannot be negative, the width of the frame is 1 foot.
A right triangle has legs of 9 cm and 12 cm. A semicircle is drawn using the hypotenuse as its diameter. What is the area of the semicircle?
Answer: $\frac{225\pi}{8}$ square cm When you see a problem combining right triangles and circles, you need to work systematically through the geometry. Start by finding the hypotenuse using the Pythagorean theorem, then use that to calculate the semicircle's area. First, find the hypotenuse of the right triangle with legs 9 cm and 12 cm: $c^2 = 9^2 + 12^2 = 81 + 144 = 225$ $c = 15 \text{ cm}$ Since the semicircle uses the hypotenuse as its diameter, the diameter is 15 cm, making the radius 7.5 cm (or $\frac{15}{2}$ ). The area of a full circle is $\pi r^2$ , so a semicircle's area is $\frac{\pi r^2}{2}$ : $\text{Area} = \frac{\pi \left(\frac{15}{2}\right)^2}{2} = \frac{\pi \cdot \frac{225}{4}}{2} = \frac{225\pi}{8}$ Choice A ( $\frac{225\pi}{8}$ ) is correct. Choice B ( $\frac{225\pi}{4}$ ) represents the area of a full circle with radius 7.5 cm—you forgot to divide by 2 for the semicircle. Choice C ( $\frac{225\pi}{2}$ ) results from mistakenly using the hypotenuse (15) as the radius instead of the diameter. Choice D ( $225\pi$ ) comes from using 15 as the radius and calculating a full circle's area—a double error. Remember this sequence for similar problems: Pythagorean theorem → find hypotenuse → hypotenuse becomes diameter → radius is half the diameter → apply semicircle area formula. The most common mistake is confusing diameter and radius, so always double-check which measurement you're using.
If the area of a rectangle is 150 $\text{m}^2$ and its width is 12.5 m, what is its length?
Answer: 12 When you encounter a rectangle area problem, remember that area equals length times width: $A = l \times w$ . Since you know the area and one dimension, you can solve for the missing dimension by rearranging this formula. Given that the area is 150 $\text{m}^2$ and the width is 12.5 m, you can find the length by dividing: $l = \frac{A}{w} = \frac{150}{12.5}$ . To calculate this, it helps to convert the decimal to a fraction: $12.5 = \frac{25}{2}$ . So you have $\frac{150}{\frac{25}{2}} = 150 \times \frac{2}{25} = \frac{300}{25} = 12$ . The length is 12 m, making B correct. Let's examine why the other answers are wrong. Choice A (10) would give an area of $10 \times 12.5 = 125 \text{ m}^2$ , which is too small. Choice C (15) would yield $15 \times 12.5 = 187.5 \text{ m}^2$ , exceeding the given area. Choice D (18) would produce $18 \times 12.5 = 225 \text{ m}^2$ , which is significantly larger than 150. For rectangle problems on the HSPT, always double-check your work by multiplying your answer back with the given dimension to verify you get the stated area. Also, when dividing by decimals, converting to fractions often makes the arithmetic cleaner and reduces calculation errors.
A rectangular swimming pool is 25 meters long and 10 meters wide. It has a sloped bottom, with the water depth being 1 meter at the shallow end and 3 meters at the deep end. What is the volume of water in the pool when it is completely full?
Answer: 500 m³ The pool is a trapezoidal prism. The volume is the area of the trapezoidal side-view multiplied by the width of the pool. The trapezoid has parallel bases equal to the depths (1 m and 3 m) and a height equal to the length of the pool (25 m). The area of the trapezoid is $A = \frac{1}{2}(b_1 + b_2)h = \frac{1}{2}(1 + 3)(25) = \frac{1}{2}(4)(25) = 50$ m². The volume of the pool is this area multiplied by the pool's width (10 m): $V = 50 \text{ m}^2 \times 10 \text{ m} = 500$ m³.
A rectangular playground is 12 meters long and 9 meters wide. What is the area of the playground in square meters?
Answer: 108 When you encounter questions about finding the area of rectangles, you're dealing with one of the most fundamental geometry formulas. The area of a rectangle is always length times width: $A = l \times w$ . For this playground, you have a length of 12 meters and a width of 9 meters. Substituting these values into the formula: $A = 12 \times 9 = 108$ square meters. This confirms that choice B is correct. Let's examine why the other options are incorrect. Choice A (96) likely comes from miscalculating $12 \times 9$ , perhaps confusing it with $12 \times 8 = 96$ . Choice C (132) might result from accidentally adding the dimensions before multiplying, getting something like $(12 + 9) \times 6 = 126$ or making a similar computational error. Choice D (21) represents the perimeter calculation error—someone might have added $12 + 9 = 21$ , forgetting that area requires multiplication, not addition. Remember that area always involves multiplication and results in square units, while perimeter involves addition and results in linear units. When you see "area" in a rectangle problem, immediately think "length times width." Also, pay attention to the units in the answer—"square meters" confirms you need area, not perimeter. Double-check your multiplication, as simple arithmetic errors are common traps on the HSPT math section.
Find the exact volume, in cubic centimeters, of a cylinder with radius 4 cm and height 12 cm.
Answer: $192\pi$ When you encounter cylinder volume problems, remember that you're finding the space inside a three-dimensional shape by multiplying the area of its circular base by its height. The volume formula for a cylinder is $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height. With a radius of 4 cm and height of 12 cm, you substitute these values: $V = \pi (4)^2 (12) = \pi \cdot 16 \cdot 12 = 192\pi$ cubic centimeters. Let's examine why the other answers are incorrect. Choice A ( $64\pi$ ) represents a common error where students forget to multiply by the height—this would be $\pi r^2 = \pi(4)^2 = 16\pi$ , then mistakenly multiplied by 4 instead of 12. Choice B ( $96\pi$ ) occurs when students use the radius incorrectly, calculating $\pi \cdot 4 \cdot 12 \cdot 2 = 96\pi$ by treating the radius linearly rather than squaring it. Choice D ( $384\pi$ ) happens when students double the correct answer, possibly by confusing radius with diameter or making an arithmetic error in the final multiplication. The correct answer is C: $192\pi$ . For cylinder volume success on the HSPT, always write down the formula first, then carefully substitute values. Double-check that you're squaring the radius (not just multiplying by 2) and that you're using the given measurements correctly—don't assume you need to convert between radius and diameter unless the problem mixes these terms.
What is the exact volume of a sphere with a radius of 6 centimeters?
Answer: $288\pi$ When you encounter sphere volume problems, remember that spheres have a specific formula that's different from other 3D shapes you might know. The volume of a sphere is $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius. Let's work through this step-by-step with a radius of 6 centimeters. First, you need to cube the radius: $6^3 = 6 \times 6 \times 6 = 216$ . Then multiply by $\frac{4}{3}$ : $\frac{4}{3} \times 216 = \frac{864}{3} = 288$ . So the volume is $288\pi$ cubic centimeters. Now let's see where the wrong answers come from. Choice A ( $144\pi$ ) likely results from using $2\pi r^3$ instead of $\frac{4}{3}\pi r^3$ — this is mixing up formulas or miscalculating the coefficient. Choice B ( $216\pi$ ) happens when you forget to multiply by $\frac{4}{3}$ entirely, using just $\pi r^3 = 216\pi$ . This is a common error when students remember the $r^3$ part but not the full coefficient. Choice D ( $432\pi$ ) results from using $2\pi r^3$ instead of $\frac{4}{3}\pi r^3$ , giving you $2 \times 216\pi = 432\pi$ . The correct answer is C ( $288\pi$ ). Study tip: Memorize the sphere volume formula exactly: $V = \frac{4}{3}\pi r^3$ . The fraction $\frac{4}{3}$ is crucial and often where mistakes happen. Practice cubing numbers and multiplying by fractions to avoid calculation errors.

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