Measurement Word Problems

Mathematics Knowledge asks you for the area of a trapezoid. Arithmetic Reasoning asks you how much sod to order for a yard. Same formulas, but here you have to work out which one applies from a description of a real job, and the formula is the easy part.

Which measurement does the situation want

The last two are the ones that catch people. Read the situation for what is physically being covered.

The formulas, in the form you need them

Perimeter is the distance around. For a rectangle, 2 times length plus 2 times width. For any other shape, add up the sides. There is no trick to it.

Circumference of a circle is pi times the diameter, or 2 pi r. Use 3.14 for pi unless the answer choices are written in terms of pi, in which case leave it.

Area

Volume

The last row is worth more than the rows above it. Volume of anything with a constant cross-section is the area of that cross-section times the length. That covers boxes, cylinders, triangular prisms and pipe, and it means you do not have to memorize a separate formula for each.

Composite shapes

A storage yard is L-shaped: a 40 by 30 foot rectangle with a 15 by 10 foot corner removed. How much gravel covers it, one inch deep?

Handle the area first.

• Full rectangle: 40 times 30 is 1,200 square feet.
• Removed corner: 15 times 10 is 150.
• Yard area: 1,050 square feet.

Now the depth. One inch is 1/12 of a foot, so the volume is 1,050 times 1/12, which is 87.5 cubic feet.

Two habits in there worth taking away. Subtract rather than dissect when a shape is a rectangle with a bite out of it - it is fewer operations and fewer chances to go wrong. And convert the odd dimension into the same unit as the others before multiplying, not after.

The other composite form adds rather than subtracts: a rectangle with a semicircle on one end, or two rectangles in an L. Split it at the obvious line, compute each piece, add.

Leftover and how-many-fit questions

Tiles are 8 inches square. How many are needed for a floor 12 feet by 10 feet?

Work in one unit. Twelve feet is 144 inches and 10 feet is 120 inches.

By area: floor is 144 times 120, which is 17,280 square inches; a tile is 64 square inches; 17,280 divided by 64 is 270 tiles.

By count along each edge, which is better: 144 divided by 8 is 18 tiles along one side, and 120 divided by 8 is 15 along the other, so 18 times 15 is 270.

The edge method is the honest one, because it tells you whether the tiles actually fit. If a side does not divide evenly, real tiles get cut and the area method quietly assumes the offcuts are reusable. A question that asks how many tiles to buy wants whole tiles, rounded up.

The same distinction governs every "how many fit" question - boxes in a truck, cans in a case. Divide along each dimension and round each one down to a whole number before multiplying, because half a box does not fit.

Scaling a measurement

If every dimension of a shape is multiplied by a factor:

• The perimeter is multiplied by that factor.
• The area is multiplied by the factor squared.
• The volume is multiplied by the factor cubed.

Double a room's dimensions and the floor takes four times the carpet. Double a tank's dimensions and it holds eight times the water. This is exactly the squared-unit rule from conversion, and it appears as a question in its own right.

Right triangles in a word problem

A ladder against a wall, a diagonal across a field, a ramp: each is a right triangle, and the long side is the hypotenuse. Leg squared plus leg squared equals hypotenuse squared, and the numbers are almost always a multiple of 3-4-5: legs of 9 and 12 give 15; 12 and 16 give 20. The Mathematics Knowledge lesson on triangles covers the theorem in full.

What you can skip

Across the 115 questions on this topic:

• Spheres and cones come up once each. Rectangular boxes and cylinders carry the volume questions.
• Exact pi arithmetic. When pi appears, the question says to use 3.14 or leaves pi in the answer; no question asks for more digits.

Where people lose points

Using area where perimeter was wanted. Fencing a field is not covering it. Check the unit of the answer choices: if they are in plain feet, you want a perimeter.

Forgetting the depth on a fill question. Gravel, concrete and soil questions give a depth in inches while the other dimensions are in feet, and the depth is easy to read past.

Using the diameter in a circle formula.

Mixing units inside one calculation. Feet with inches. Convert first.

Rounding a how-many-fit answer up when it should be down, or the reverse. Pieces that must fit round down; materials that must be bought round up.

Forgetting there are four walls on a painting question, or forgetting to subtract the door and windows when the question tells you their sizes.

Work one in under a minute

A circular tank is 6 feet across and 4 feet deep. How many cubic feet does it hold?

Diameter 6 means radius 3.

Base area: pi times 9, which is about 28.3 square feet.

Volume: 28.3 times 4 is about 113 cubic feet.

Check: a 6 by 6 by 4 box would hold 144, and a cylinder fills a bit under 79 percent of its bounding box, so something near 113 is right.

Where this leads

The same formulas appear on Mathematics Knowledge without the story around them, and the scaling rule is the conversion rule for squared and cubed units.