Area, Surface Area and Volume Lesson Test

Answer: B

Correct answer: 54 square feet. Use

• A: This adds the length and width instead of multiplying them.
• B (correct): Area of a rectangle is length times width: 9 times 6 is 54.
• C: This doubles the area, as if a perimeter-style factor of 2 belonged in the formula.
• D: This is the perimeter, 2 times (9 + 6), which measures the edge, not the covered floor.

Answer: D

Correct answer: 42 square meters. Use

• A: This halves twice, taking one fourth of 12 times 7 instead of one half.
• B: This multiplies base by height but forgets to take half.
• C: This adds the base and height instead of multiplying them.
• D (correct): Half of base times height: one half of 12 times 7 is 42.

Answer: C

Correct answer: 36π square inches. Change the diameter to a radius: 12 divided by 2 is 6.

• A: This is the circumference, pi times the diameter, which is a length, not an area.
• B: This squares the diameter instead of the radius.
• C (correct): The radius is half of 12, or 6, and pi times 6 squared is 36 pi.
• D: This multiplies pi by the radius without squaring it.

Answer: D

Correct answer: 120 cubic inches. Use

• A: This adds the three dimensions instead of multiplying them.
• B: This is only the base area, 6 times 4, without multiplying by the height.
• C: This is the surface area, 2 times (24 + 30 + 20), which measures covering, not capacity.
• D (correct): Volume of a box is length times width times height: 6 times 4 times 5 is 120.

Answer: D

Correct answer: Area, in square feet. Covering a surface is an area question, so the answer is in square units such as square feet. Perimeter is for edges and volume is for capacity.

• A: Perimeter measures the distance around the edge, which suits trim or fencing, not paint coverage.
• B: A single height is only one length; covering a surface takes two lengths multiplied together.
• C: Volume measures how much fits inside the room, such as air, not how much surface is covered.
• D (correct): Paint covers a surface, so the painter needs the area of the walls, measured in square units.

Answer: B

Correct answer: 96 square centimeters. A cube has six equal square faces, so

square centimeters.

• A: This multiplies 6 by the edge length without squaring the edge.
• B (correct): Six square faces of 16 each give 6 times 16, or 96.
• C: This is the area of only one of the six faces.
• D: This is the volume, 4 cubed, which is in cubic units, not square units.

Answer: A

Correct answer: 55 square feet. Use

The average base is (8 + 14) divided by 2, or 11, so the area is 11 times 5, or 55 square feet.

• A (correct): The average of the bases is 11, and 11 times 5 is 55.
• B: This multiplies the sum of the bases by the height but forgets the one half.
• C: This uses only the longer base, 14 times 5, as if the shape were a rectangle.
• D: This uses only the shorter base, 8 times 5, as if the shape were a rectangle.

Answer: C

Correct answer: 72π cubic centimeters. The radius is half of 6, or 3.

cubic centimeters.

• A: This uses the diameter 6 as the radius, giving pi times 36 times 8.
• B: This is the curved side area, 2 pi times 3 times 8, not the volume.
• C (correct): With radius 3, pi times 9 times 8 is 72 pi.
• D: This multiplies pi by the radius and height but never squares the radius.

Answer: A

Correct answer: 48π cubic inches. Use

The base area is 16 pi, and one third of 9 is 3, so

• A (correct): One third of pi times 16 times 9 is 48 pi.
• B: This takes one third of pi times 4 times 9 without squaring the radius.
• C: This is the volume of a cylinder with the same base and height; the one third is missing.
• D: This is only the area of the circular base, not multiplied by the height.

Answer: C

Correct answer: 972π cubic feet. Use

and 729 divided by 3 is 243, times 4 is 972, so

• A: This is the surface area, 4 pi times 81, which is in square units.
• B: This squares the radius instead of cubing it before taking four thirds.
• C (correct): 9 cubed is 729, and four thirds of 729 is 972, so the volume is 972 pi.
• D: This multiplies 729 by 4 but forgets to divide by 3.

Answer: B

Correct answer: 225 square feet. Use the subtract method.

Whole room: 20 times 12 is 240. Closet: 5 times 3 is 15. Carpet: 240 minus 15 is 225 square feet.

• A: This is the whole rectangle; the closet corner was never subtracted.
• B (correct): The full rectangle is 240 and the closet is 15, so 240 minus 15 is 225.
• C: This is the distance around the room, 2 times (20 + 12), a perimeter in feet, not an area.
• D: This adds the closet area instead of removing it.

Answer: B

Correct answer: 16π square centimeters. Subtract the hole from the whole:

square centimeters.

• A: This subtracts the radii first and squares the difference, 2 squared, which is not the same as subtracting the areas.
• B (correct): Pi times (25 minus 9) gives 16 pi.
• C: This adds the hole's area to the outer circle instead of subtracting it.
• D: This is the whole outer disk; the hole in the middle was not removed.

Answer: A

Correct answer: 188 square inches. Bottom: 10 times 6 is 60. Long sides: 2 times 10 times 4 is 80. Short sides: 2 times 6 times 4 is 48.

Total:

The open top is not counted.

• A (correct): Bottom 60 plus long sides 80 plus short sides 48 gives 188.
• B: This counts only the four sides and leaves out the bottom.
• C: This is the surface area of a closed box; the missing top of 60 was counted.
• D: This is the volume, 10 times 6 times 4, which measures capacity in cubic inches.

Answer: B

Correct answer: 432 square inches. A square foot is 12 inches by 12 inches, or 144 square inches.

• A: This converts only one square foot and does not multiply by 3.
• B (correct): One square foot is 144 square inches, and 3 times 144 is 432.
• C: This multiplies by 1728, the cubic factor, which is for volume, not area.
• D: This multiplies by 12, the linear factor, instead of 12 squared.

Answer: B

Correct answer: 7 inches. Write the formula and substitute.

So 40h equals 280, and h is 7 inches.

• A: This is the base area, 8 times 5, mistaken for the height.
• B (correct): The base area is 40, and 280 divided by 40 is 7.
• C: This divides the volume by the length only, leaving out the width.
• D: This divides the volume by the width only, leaving out the length.

Answer: D

Correct answer: 320 square centimeters. When lengths are multiplied by k, area is multiplied by $k^2$.

and 20 times 16 is 320 square centimeters.

• A: This multiplies by 64, 4 cubed, which is the factor for volume, not flat area.
• B: This adds 4 to the area instead of multiplying by a scale factor.
• C: This multiplies the area by 4, the length factor, instead of 4 squared.
• D (correct): Area grows by the square of the scale factor: 20 times 16 is 320.

Answer: A

Correct answer: 324π cubic centimeters. Volume needs the vertical height. The radius, height and slant height form a right triangle, so

cubic centimeters.

• A (correct): The vertical height is 12, and one third of pi times 81 times 12 is 324 pi.
• B: This uses the correct height but forgets the one third, giving a cylinder's volume.
• C: This is the curved surface area, pi times 9 times 15, not the volume.
• D: This uses the slant height 15 in place of the vertical height.

Answer: A

Correct answer: 800 square meters. The slant height spans half the base, 8, and the height, 15:

• A (correct): The slant height is 17, so the surface is 256 plus 2 times 16 times 17, or 800.
• B: This treats each face as 16 times 17 without taking half of base times slant height.
• C: This counts the four triangular faces but leaves out the square base.
• D: This uses the vertical height 15 for the triangular faces instead of the slant height.

Answer: C

Correct answer: 44π cubic inches. The radius is 2, so each base area is 4 pi.

Cylinder: 4 pi times 9 is 36 pi. Cone: one third of 4 pi times 6 is 8 pi. Total: 36 pi + 8 pi = 44 pi cubic inches.

• A: This uses the diameter 4 as the radius for both pieces.
• B: This is only the cylinder; the cone on top was never added.
• C (correct): The cylinder holds 36 pi and the cone holds 8 pi, for 44 pi in all.
• D: This leaves out the one third for the cone, using 24 pi instead of 8 pi.

Answer: C

Correct answer: 4 cubic yards. Four inches is 1/3 of a foot, so the volume is 27 times 12 times 1/3, or 108 cubic feet.

One cubic yard is 3 times 3 times 3, or 27 cubic feet, so 108 divided by 27 is 4 cubic yards.

• A: This divides 108 cubic feet by 3, the linear factor, instead of 27.
• B: This treats the 4 inches as 4 feet before dividing by 27.
• C (correct): The volume is 108 cubic feet, and 108 divided by 27 is 4.
• D: This divides 108 cubic feet by 9, the square-yard factor, instead of 27.

Answer: A

Correct answer: 4.8 liters. The stone pushes the water up by its own volume. The rise is 32 minus 30, or 2 cm.

cubic centimeters, and since 1 liter is 1000 cubic centimeters, that is 4.8 liters.

• A (correct): The water rises 2 cm, so the stone is 60 times 40 times 2, or 4800 cubic centimeters, which is 4.8 liters.
• B: This finds 4800 cubic centimeters but divides by 100 instead of 1000 to get liters.
• C: This is the volume of the water before the stone was added, not the stone itself.
• D: This is the volume of everything below the new 32 cm level, water plus stone.

Answer: D

Correct answer: 216 cubic inches. When every length is multiplied by k, volume is multiplied by $k^3$.

As a check, a 4 inch cube becomes a 6 inch cube, and

• A: This multiplies the volume by 1.5, the length factor, instead of cubing it.
• B: This multiplies by 1.5 squared, which is the factor for surface area, not volume.
• C: This multiplies by 3 times 1.5, or 4.5, instead of raising 1.5 to the third power.
• D (correct): Volume scales by 1.5 cubed, which is 3.375, and 64 times 3.375 is 216.

Answer: C

Correct answer: 96π square feet. The radius is 4. An open-top cylinder has one base plus the curved side.

The answer is 96 pi square feet.

• A: This counts only the curved side and leaves out the bottom.
• B: This uses the diameter 8 as the radius in both the bottom and the side.
• C (correct): With radius 4, the bottom is 16 pi and the side is 80 pi, for 96 pi.
• D: This adds a top of 16 pi, but the barrel has no lid.

Answer: D

Correct answer: 56 square inches. Each cube has 6 faces of 4 square inches, or 24, so three cubes total 72.

Two joints each hide 2 faces, so 4 faces of 4 are removed: 72 minus 16 is 56 square inches. Check: the row is a 6 by 2 by 2 box, and

• A: This is the three cubes' surface area added separately; no hidden faces were removed.
• B: This removes only one face per joint, but each joint hides a face on both cubes.
• C: This is the volume of the row, 6 by 2 by 2, which is in cubic inches.
• D (correct): The cubes total 72, and the two glued joints hide 4 faces of 4 each, so 72 minus 16 is 56.

Answer: D

Correct answer: 256 - 9π square feet. Lawn: 20 times 14 is 280. Shed: 6 times 4 is 24. The bed has radius 3, so its area is 9 pi.

• A: This squares the diameter 6 instead of the radius 3, so the flower bed is counted four times too large.
• B: This removes the flower bed but forgets to subtract the 24 square foot shed.
• C: This subtracts the circle's circumference, 2 pi r, which is a length, not the area pi r squared.
• D (correct): Subtract the shed (24) and the circle with radius 3 (9 pi) from the 280 square foot lawn.

Answer: C

Correct answer: 105 square inches. A parallelogram's area is base times perpendicular height. The slanted side is not used.

• A: This multiplies the base by the slanted side, 9, which is not the height.
• B: This takes half of base times height, the triangle formula, but a parallelogram needs no one half.
• C (correct): Base 15 times perpendicular height 7 gives 105.
• D: This multiplies the slanted side by the height instead of using the base.

Answer: C

Correct answer: 96 square inches. In a right triangle the two legs serve as base and height.

• A: This multiplies the legs but forgets the one half in the triangle formula.
• B: This uses the 20 inch longest side as if it were a leg; the legs are the base and the height.
• C (correct): The two legs are the base and height: half of 12 times 16 is 96.
• D: This adds the three sides, giving the perimeter in inches, not the area.

Answer: B

Correct answer: 100π square centimeters. A sphere's surface area is four times the area of its largest circle.

• A: This is pi r squared, the area of one flat circle, not four times it.
• B (correct): 4 pi r squared with r of 5 gives 4 times 25 pi, or 100 pi.
• C: This is the volume formula, four thirds pi r cubed, which measures space inside, not the surface.
• D: This multiplies 4 pi by the radius without squaring it.

Answer: A

Correct answer: 20 square yards. Floor area: 15 times 12 is 180 square feet. One square yard is 3 ft by 3 ft, or 9 square feet, so 180 divided by 9 is 20 square yards.

• A (correct): 180 square feet divided by 9 square feet per square yard gives 20.
• B: This divides by 3, the linear factor; a square yard is 3 by 3, or 9 square feet.
• C: This is the area in square feet with no conversion to square yards.
• D: This multiplies by 3 instead of dividing; square yards are larger units, so the number must get smaller.

Answer: C

Correct answer: 120 cubic inches. A pyramid holds one third of the matching prism. The base area is 6 times 6, or 36.

• A: This is the volume of a box with the same base and height; a pyramid holds only one third of it.
• B: This uses one half, as for a triangle's area, instead of the pyramid's one third.
• C (correct): Base area 36 times height 10 is 360, and one third of that is 120.
• D: This uses the side length 6 instead of the base area 36 before taking one third.

Answer: C

Correct answer: 170π square feet. The radius is half the diameter, 5 ft. Ends: 2 times 25 pi is 50 pi. Side: 2 pi times 5 times 12 is 120 pi.

• A: This counts only one circular end; a closed drum has a top and a bottom.
• B: This uses the diameter 10 as the radius in both parts of the formula.
• C (correct): Two ends of 25 pi each give 50 pi, the side gives 120 pi, for 170 pi.
• D: This is the curved side alone, leaving out both circular ends.

Answer: B

Correct answer: 204 square centimeters. Block:

Cube: 6 times 9 = 54. The cube's bottom and the block patch it covers are each 9 square centimeters and are hidden.

• A: This adds both surface areas but does not remove the two hidden faces where they touch.
• B (correct): The block has 168, the cube 54; remove the cube's bottom and the covered patch, 9 each, to get 204.
• C: This removes only one hidden 9 square centimeter face; the cube's bottom and the patch it covers are both hidden.
• D: This also removes the cube's top face, but the top is still exposed to the air.

Answer: D

Correct answer: 8 feet. Write the volume formula and substitute what is known.

Divide both sides by 36 pi: h = 8 feet.

• A: This divides by the radius 6 instead of the radius squared, 36.
• B: This divides by 2 times the radius, 12, mixing in the circumference idea.
• C: This squares the diameter, 12, giving 144, instead of squaring the radius.
• D (correct): Dividing 288 pi by pi times 36 gives 8.

Answer: C

Correct answer: 1080 cubic inches. Area scales by k squared, so k squared = 9 and k = 3. Volume scales by k cubed = 27.

• A: This multiplies the volume by 9, the area factor, but volume scales by k cubed, not k squared.
• B: This multiplies by the length factor 3 only; volume grows by 3 cubed.
• C (correct): Area grew by 9, so k is 3; volume grows by 3 cubed, 27, and 40 times 27 is 1080.
• D: This multiplies by 9 squared, 81, treating 9 as the length factor instead of the area factor.

Answer: B

Correct answer: 20 liters. Volume: 40 times 25 times 30 is 30,000 cubic centimeters. Since 1 liter is 1000 cubic centimeters, that is 30 liters. Two thirds of 30 is 20 liters.

• A: This is the full capacity; the cooler is filled only two thirds of the way.
• B (correct): The cooler holds 30,000 cubic centimeters, or 30 liters, and two thirds of that is 20 liters.
• C: This divides 20,000 cubic centimeters by 100 instead of 1000.
• D: This divides 20,000 cubic centimeters by 10,000; one liter is 1000 cubic centimeters.

Answer: B

Correct answer: 36π square feet. The outer radius is

Subtract the pool from the whole circle.

• A: This subtracts the radii first and squares the 2; the formula needs R squared minus r squared.
• B (correct): The outer radius is 10, so the ring is pi times 100 minus 64, or 36 pi.
• C: This is the whole outer circle; the pool in the middle must be subtracted.
• D: This adds 2 feet to each side of the radius, making R 12; the radius grows by only 2.

Answer: A

Correct answer: Perimeter, in feet. Ask what the fence does: it goes around the edge. A boundary is a perimeter, measured in plain length units such as feet.

• A (correct): Fencing runs along the edge of the run, so it is a length around the boundary.
• B: Area measures the ground inside the run, which would be needed for sod, not fencing.
• C: Volume measures how much fits inside a solid, not the length of the boundary.
• D: Surface area covers the faces of a solid; a fence follows only the edge.

Answer: B

Correct answer: 49π square meters. Use the circle area formula with the radius.

• A: This is 2 pi r, the circumference, which is a length, not an area.
• B (correct): Pi times 7 squared is 49 pi.
• C: This squares the diameter, 14, instead of the radius.
• D: This squares the radius but leaves out pi.

Answer: D

Correct answer: 8 meters. Write the formula and substitute.

So h = 104 divided by 13 = 8 meters.

• A: This divides 104 by the sum of the bases, 26, forgetting the one half.
• B: This divides 104 by only the longer base, 16.
• C: This divides 104 by only the shorter base, 10.
• D (correct): The average base is 13, and 104 divided by 13 is 8.

Answer: C

Correct answer: 30 cubic feet. Volume of a rectangular box is V = lwh = 5 times 2 times 3 = 30 cubic feet.

• A: This adds the three dimensions instead of multiplying them.
• B: This is the surface area of the box, the covering, not the space inside.
• C (correct): Length times width times height: 5 times 2 times 3 is 30.
• D: This adds the three face areas once, which is half the surface area, not the volume.

Answer: A

Correct answer: 200π square centimeters. First find the slant height from the right triangle of radius and height.

Then SA = pi r squared + pi r l = 64 pi + 136 pi = 200 pi square centimeters.

• A (correct): The slant height is 17, so the total is 64 pi plus 136 pi, or 200 pi.
• B: This uses the vertical height 15 in the curved area; the curved surface needs the slant height.
• C: This is the curved surface only; the circular base must be added.
• D: This is the cone's volume, one third pi r squared h, not its surface area.

Answer: B

Correct answer: 294 square inches. Find the edge first: s cubed = 343, so s = 7 inches. Each face is 7 times 7, or 49.

• A: This is the area of only one face; a cube has six.
• B (correct): The edge is 7 inches because 7 cubed is 343; six faces of 49 give 294.
• C: This counts only the four side faces and leaves out the top and bottom.
• D: This multiplies the volume by 6 instead of first finding the edge length.

Answer: A

Correct answer: 81 cubic feet. A cubic yard is a 3 ft by 3 ft by 3 ft block, which is 27 cubic feet. So 3 cubic yards is 3 times 27, or 81 cubic feet.

• A (correct): Each cubic yard is 27 cubic feet, and 3 times 27 is 81.
• B: This multiplies by 3, the linear factor; a cubic yard is 3 by 3 by 3 feet.
• C: This multiplies by 9, the square factor, which applies to area, not volume.
• D: This divides by 3 instead of multiplying; cubic feet are smaller, so the count must grow.

Answer: D

Correct answer: 80π cubic centimeters. The bolt's volume equals the volume of water pushed up, a cylinder of radius 4 cm and height 5 cm.

• A: This uses 2 pi r times the rise, which is curved side area, not volume.
• B: This uses the diameter 8 instead of the radius 4.
• C: This multiplies pi by the radius without squaring it.
• D (correct): The raised water is a cylinder of radius 4 and height 5: 16 pi times 5 is 80 pi.

Answer: D

Correct answer: 960 - 72π cubic inches. Find the whole block, then subtract the hollowed cylinder. Block: 12 times 10 times 8 is 960. Hole: pi times 3 squared times 8 is 72 pi.

• A: This subtracts 2 pi r h, the inside wall area of the hole, instead of its volume.
• B: This uses pi times r times h, forgetting to square the radius.
• C: This uses the diameter 6 in place of the radius when squaring.
• D (correct): The block is 960 and the hole is a cylinder of 9 pi times 8, or 72 pi, which is subtracted.

Answer: A

Correct answer: 169 square feet. For a square, A = s squared = 13 times 13 = 169 square feet.

• A (correct): Area of a square is side squared: 13 times 13 is 169.
• B: This is 4 times 13, the perimeter, which is a length.
• C: This doubles the side instead of squaring it.
• D: This doubles the correct area; a square's area is simply s squared.

Answer: D

Correct answer: 60π square feet. Leave out the missing faces. The radius is 2 ft, and only the curved side is metal.

• A: This adds two circular ends, but a duct open at both ends has no ends.
• B: This adds one circular end that the open duct does not have.
• C: This uses the diameter 4 as the radius in 2 pi r h.
• D (correct): Only the curved side exists: 2 pi times 2 times 15 is 60 pi.

Answer: A

Correct answer: 660 square feet. Lateral area is base perimeter times length: (5 + 12 + 13) times 20 = 600. Each triangular end is half of 5 times 12, or 30.

• A (correct): The side faces give perimeter 30 times 20, or 600, and the two triangle ends add 30 each, for 660.
• B: This is the lateral area only; the two triangular ends must be added.
• C: This adds only one triangular end; a prism has two matching bases.
• D: This treats each end as 5 times 12, forgetting the one half for a triangle.