Carpentry Math Word Problems (With Worked Answers)
18 original carpentry math word problems with step-by-step answers: budgets, kerf leftovers, board feet, sheet goods, deck boards, stairs, miters and rafters.
These problems are written the way shop questions actually arrive: a budget, a board, a drawing in the wrong units. Each one is worked step by step and checked with the same functions the Woodshop Math calculators use, so you can rerun any of them with your own numbers. Try the problem first, then open the answer.
Put every length in one unit before doing anything else; most wrong answers start with mixed feet and inches.
Decide the rounding direction from the question: round down to stay inside a budget, round up to have enough to finish.
Cuts are not free. n parts cut in a row need n − 1 kerfs, and freeing the offcut takes one more.
Use actual dressed sizes for geometry and nominal sizes only where the trade convention says so, such as board-foot pricing.
Skills used
Warm-up 5 · Core 8 · Multi-step 5
Skill
Method
Problems
Calculator
Division and rounding
Round down when buying with a fixed budget; round up when buying enough to finish.
Every problem below is solved the same way. The method matters more than the arithmetic: once the quantities are named and in one unit, the calculation is usually a single line.
The drawing shows problem 3 laid out at scale, with the kerfs exaggerated. Sketching the board like this is the quickest way to see how many cuts there really are.
List what is given and what is asked, with units.
Convert everything to one unit (inches, feet or millimetres).
Write the relationship in words first — for example, parts plus kerfs must not exceed the board — then in numbers.
Check the answer against the question: does it need rounding up, rounding down, or a leftover?
Five 17-1/4 in. rails from a 96 in. board use 86-3/4 in. including four kerfs. The 9-1/4 in. that remains loses one more kerf when it is cut free, leaving a 9-1/8 in. offcut.
The problems
Problem 1Warm-up
How many studs fit the budget?
A builder has $183.00 for wall studs. Each 2x4x8 stud costs $4.27 (tax already included). How many studs can be bought, and how much money is left?
Show the worked answer
Divide the budget by the price of one stud: 183 ÷ 4.27 = 42.857.
You cannot buy part of a stud, so round down to 42. Rounding up would go over budget.
A fence repair needs 30 pickets. The plan allows 10% extra for splits and bad ends. Pickets cost $6.85 each. How many pickets should be bought, and what will they cost?
Show the worked answer
Add the allowance: 30 × 1.1 = 33 pickets.
That is already a whole number, so 33 pickets. Had it come out as, say, 32.4, you would round up to 33: rounding down would leave the job short.
Five rails, each 17-1/4 in long, are cut one after another from a 96 in board. The blade removes 1/8 in per cut. How long is the loose offcut once the fifth rail has been cut free?
Show the worked answer
Length of the parts: 5 × 17.25 in = 86.25 in.
Five parts in a row are separated by 4 kerfs: 4 × 0.125 in = 0.50 in.
Material used: 86.25 + 0.50 = 86.75 in, leaving 96 − 86.75 = 9.25 in.
Cutting the last rail free of the offcut is one more kerf, so the loose offcut measures 9.25 − 0.125 = 9.125 in.
Answer: The loose offcut is 9-1/8 in. Before the cut that frees the fifth rail, 9-1/4 in of board lies past it.
Shelves for a closet are 22-3/4 in long. How many can be cut from a 10 ft board if each cut removes 1/8 in, and how long is the loose offcut at the end?
Show the worked answer
Convert the board to inches: 10 ft × 12 = 120 in.
Each shelf after the first also costs one kerf, so n shelves need n × 22.75 + (n − 1) × 0.125 in.
Rearranged: n ≤ (120 + 0.125) ÷ (22.75 + 0.125) = 5.251, so n = 5.
Check: 5 × 22.75 + 4 × 0.125 = 114.25 in, leaving 5.75 in past the fifth shelf. A sixth shelf would need 137.125 in.
Cutting the fifth shelf free costs one more kerf, as in problem 3: 5.75 − 0.125 = 5.625 in.
Answer: 5 shelves. 5-3/4 in of board remains past the fifth shelf; after the cut that frees it, the loose offcut is 5-5/8 in.
A 14 ft board has a 21 in split at one end, which is cut off and thrown away. What percentage of the board's length was lost, and how long is the board now?
Show the worked answer
Put both lengths in the same unit: 14 ft × 12 = 168 in.
Percentage lost = part ÷ whole × 100 = 21 ÷ 168 × 100 = 12.5%.
Remaining length: 168 − 21 = 147 in, which is 12 ft 3 in.
Answer: 12.5% of the length was cut off; the board is now 147 in (12 ft 3 in) long.
A yard sells 1x8 boards by the board foot at $4.15 per board foot, measured on nominal size. How many board feet are in six 10 ft 1x8s, and what do they cost?
Two shed walls are each 12 ft long and 8 ft high. They will be sheathed with 4 ft × 8 ft sheets, allowing 10% for offcuts. Sheets cost $38.50. How many sheets are needed and what is the cost? (Ignore door and window openings.)
Show the worked answer
Wall area: 2 × 12 × 8 = 192 sq ft.
Add 10%: 192 × 1.1 = 211.2 sq ft.
One sheet covers 4 × 8 = 32 sq ft, so 211.2 ÷ 32 = 6.6 sheets. Round up to 7.
Six cabinet sides, each 24 in × 30 in, are to be cut from 48 in × 96 in sheets with a 1/8 in kerf. Estimate the sheets by area with a 5% allowance, then check whether the parts actually fit that way.
Show the worked answer
Parts area: 6 × 24 × 30 = 4,320 sq in = 30 sq ft. With 5%: 31.5 sq ft ÷ 32 = 0.984, so 1 sheet by area.
Across the 48 in width, two 24 in parts plus one kerf need 48.125 in, which is 1/8 in too much. Only one part fits across.
Along the 96 in length, three 30 in parts plus two kerfs need 90.25 in, so three fit. Turning the parts 90° also gives three, so each sheet yields 3 sides.
Six sides ÷ 3 per sheet = 2 sheets.
Answer: 2 sheets. The area estimate says 1, but the kerf stops two 24 in parts fitting across a 48 in sheet, so each sheet holds only 3. Check the layout before trusting an area count.
A deck is 16 ft long and 12 ft wide. Boards run the 16 ft length, are 5-1/2 in wide, and are laid with a 3/16 in gap. Boards come in 16 ft lengths at $24.80 each, and the plan adds 10% for waste. How many boards should be bought?
Show the worked answer
The boards are laid side by side across the 12 ft width: 12 × 12 = 144 in to cover.
Each board plus its gap takes 5.5 + 0.1875 = 5.6875 in. The last board needs no gap after it, so rows = (144 + 0.1875) ÷ 5.6875 = 25.35, rounded up to 26.
Each row is one 16 ft board, so 26 boards (416 linear feet).
Add 10%: 26 × 1.1 = 28.6, rounded up to 29 boards, costing 29 × $24.80 = $719.20.
Answer: 29 boards (26 rows plus waste), about $719.20. Framing is not included.
Risers, treads and stringer length for porch steps
Porch steps climb a total rise of 43-1/2 in. The aim is risers close to 7-1/2 in and treads 10 in deep. How many risers and treads are there, how tall is each riser, and roughly how long is the stringer's sloped edge?
Show the worked answer
Risers: 43.5 ÷ 7.5 = 5.8, rounded to 6.
Actual riser height: 43.5 ÷ 6 = 7.25 in (7-1/4 in).
There is one fewer tread than risers: 5 treads, for a total run of 5 × 10 = 50 in.
Sloped length (Pythagoras): √(43.5² + 50²) = 66.27 in. The angle is arctan(43.5 ÷ 50) = 41.02°.
Answer: 6 risers of 7-1/4 in and 5 treads; the rise-and-run triangle has a 66.3 in hypotenuse at 41°. Check local stair rules before building.
A stair rises 2,650 mm from floor to floor. The target riser is 180 mm and each tread (going) is 250 mm. Find the number of risers, the actual riser height, the total going and the sloped length.
Show the worked answer
Risers: 2650 ÷ 180 = 14.72, rounded to 15.
Actual riser: 2650 ÷ 15 = 176.67 mm.
Treads: 14, so the total going is 14 × 250 = 3,500 mm.
Sloped length: √(2650² + 3,500²) = 4,390 mm, at arctan(2650 ÷ 3,500) = 37.13°.
Answer: 15 risers of 176.7 mm, 14 treads, 3,500 mm of going and a 4,390 mm sloped length at 37.1°.
A hexagonal planter box has 6 equal sides. What is each inside corner angle, and what angle should the miter saw be set to (measured from a square cut)?
Show the worked answer
The inside angles of any polygon add up to (sides − 2) × 180° = 720°.
Each corner: 720° ÷ 6 = 120°.
Each piece is cut so two ends meet at the corner. The cut from square is (180° − 120°) ÷ 2 = 30°. A shortcut: 180° ÷ 6 gives the same number.
Answer: Corners are 120°; set the saw to 30° on both ends of every piece.
An octagonal mirror frame has 8 equal sides. What is each inside corner angle, and what angle should the miter saw be set to (measured from a square cut)?
Show the worked answer
The inside angles of any polygon add up to (sides − 2) × 180° = 1080°.
Each corner: 1080° ÷ 8 = 135°.
Each piece is cut so two ends meet at the corner. The cut from square is (180° − 135°) ÷ 2 = 22.5°. A shortcut: 180° ÷ 8 gives the same number.
Answer: Corners are 135°; set the saw to 22.5° on both ends of every piece.
A shed roof has a 24 ft span and a 6-in-12 pitch, with a 12 in horizontal overhang. Ignoring the ridge board, how long is each common rafter along the slope, from the ridge line to the end of the overhang?
Show the worked answer
The run is half the span: 24 ÷ 2 = 12 ft.
For every 12 in of run the roof rises 6 in, so the slope factor is √(12² + 6²) ÷ 12 = 1.118.
Rafter to the wall line: 12 × 1.118 = 13.416 ft. The 1 ft overhang adds 1 × 1.118 = 1.118 ft.
Total: 14.534 ft × 12 = 174.41 in.
Answer: About 174.4 in (14.53 ft) per rafter before ridge and plumb-cut adjustments, so a 14 ft board is too short. This is geometry only, not a sizing check.
Minimum dressed dry sizes used in problems 6 and 9 (2x6 and 2x10), and section 2.2, which measures board feet on nominal sizes.
Every answer is calculated with the same functions as the linked calculators and checked by automated tests. Prices are made up for the exercises and are not market prices.