Type on any line or blank, or switch Draw on to write with a finger, stylus or mouse. Your work is saved in this browser on this device only. Nothing is sent anywhere. The answer key after page 2 stays covered until you open it, and it never prints with your work: it has its own print button.

Practice sheet · Building & Architecture Path · Lens 1: Math

Area, volume & slope

Practice for The Math of Building & Architecture: scale, area, volume, Pythagoras, slope, stairs and loads (pp.8–14), plus the Algebra Level Ups (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

Area = length × width (m²). Volume = length × width × depth (m³): change cm to m first, so 10 cm = 0.1 m. Buy whole boxes and cans: always round up.

Pitch = rise : run, often written per 12 (4/12). Pythagoras: a² + b² = c² finds a diagonal or a rafter. A 3-4-5 triangle checks a square corner (p.11).

Warm-up

One step

Scale, area, volume and pitch.
1

Read the plan

On a 1:50 plan, a bedroom measures 6 cm by 5 cm. How big is the real room in metres?

2

Room size

A room is 5 m by 3.6 m. Find its floor area and its perimeter.

3

Shed pad

A concrete pad for a shed is 2.5 m by 3 m and 10 cm thick. How many cubic metres of concrete?

4

Roof rise

A roof has a 4/12 pitch over a 6 m run. How high does it rise?

Core

Two or three steps

Draw a sketch and label every length.
5

Flooring order

The 5 m × 3.6 m room needs flooring. Add 10% for cuts and waste. Flooring comes in boxes that cover 2 m². How many boxes?

6

Paint the walls

The same room is 2.4 m high. Leave out a 1.9 m² door and a 1.5 m² window. You need two coats, and 1 L covers 10 m². Paint comes in 3.78 L cans. How many cans?

Area Practice · Building & Architecture Path · Unizon1/2
Building & Architecture Path · Unizon · PracticeArea, volume & slope
7

Square corner

A deck frame is 9 m by 12 m. How long should the diagonal be if the corners are square?

8

Rafter length

A roof with an 8/12 pitch has a 3.6 m run. Find the rise, the rafter length (to 2 decimal places) and the roof angle (to 0.1°).

9

Ramp run

A wheelchair ramp at 1:12 must climb 0.75 m to a door. How long is its run?

10

Stair plan

Floor to floor is 2,800 mm and the target riser is about 180 mm. How many risers, and how tall is each? Then use 2R + T ≈ 630 mm to find the tread.

Level Up

Equations

A1–A2 (pp.17–18) and roof loads (p.14).
11

Budget

A builder charges an $800 call-out fee plus $150 per m² of tiling (A1 style). The bill is $4,100. How many m² were tiled?

12

Two unknowns

An order has 50 items: studs at $6 and plywood sheets at $40. It costs $980. How many of each?

13

Snow load

Wet snow weighs about 300 kg per m³. How many kg per m² is 0.4 m of it? On an 8 m × 10 m flat roof, what is the total?

14

Garden pen

You have 60 m of fence for a pen against a wall (3 sides). With sides x, x and 60 − 2x, the area is x(60 − 2x). Which x gives 400 m²? What is the biggest area?

Make your own problem

Your turn to be the site planner

  1. Measure a real room, or invent one, and sketch it to a scale like 1:50.
  2. Write a question about its area, volume, slope or stairs.
  3. Solve it, round up for anything you buy, and check the answer is sensible.

My problem, and my worked answer:

Area Practice · Building & Architecture Path · Unizon2/2
Building & Architecture Path · Unizon · PracticeArea, volume & slope

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Real projects follow the building code; these numbers are for practice.

Warm-up
1
3 m × 2.5 m 1 cm on the plan = 50 cm real. 6 × 50 = 300 cm = 3 m; 5 × 50 = 250 cm = 2.5 m.
2
18 m² · 17.2 m 5 × 3.6 = 18 m². Perimeter: 2 × (5 + 3.6) = 17.2 m.
3
0.75 m³ 10 cm = 0.1 m. 2.5 × 3 × 0.1 = 0.75 m³. Order a little extra, as a builder would.
4
2 m Rise = 4/12 of the run: 6 × 4 ÷ 12 = 2 m.
Core
5
10 boxes 18 × 1.10 = 19.8 m². 19.8 ÷ 2 = 9.9, so round up to 10 boxes.
6
3 cans Walls: 17.2 × 2.4 = 41.28 m². Minus 1.9 and 1.5 = 37.88 m². Two coats: 75.76 m² ÷ 10 = 7.576 L. Two cans hold 7.56 L, just short, so buy 3.
7
15 m √(9² + 12²) = √225 = 15. It is the 3-4-5 triangle times 3.
8
rise 2.4 m · rafter ≈ 4.33 m · ≈ 33.7° Rise: 3.6 × 8 ÷ 12 = 2.4 m. Rafter: √(3.6² + 2.4²) = √18.72 ≈ 4.33 m (before any overhang). Angle: tan⁻¹(8 ÷ 12) ≈ 33.7°.
9
9 m 1:12 means 12 m of run for each 1 m of rise: 0.75 × 12 = 9 m.
10
16 risers of 175 mm · tread 280 mm 2,800 ÷ 180 ≈ 15.56, so round up to 16. 2,800 ÷ 16 = 175 mm. Tread: 630 − 2 × 175 = 280 mm. Check the local building code.
Level Up
11
22 m² 800 + 150a = 4,100 → 150a = 3,300 → a = 22.
12
30 studs · 20 sheets s + p = 50 and 6s + 40p = 980. Swap s = 50 − p: 300 + 34p = 980 → p = 20, s = 30.
13
120 kg/m² · 9,600 kg 0.4 × 300 = 120 kg on each m². The roof is 80 m²: 80 × 120 = 9,600 kg. Never climb a roof to clear snow; that’s a job for adults with safety gear.
14
x = 10 m or 20 m · biggest 450 m² x(60 − 2x) = 400 → x² − 30x + 200 = 0 → (x − 10)(x − 20) = 0. The hill peaks halfway, at x = 15: 15 × 30 = 450 m².
Make your own: sketch it, write the units, change everything to metres before you multiply, and round up when you buy.
Area Practice · Building & Architecture Path · UnizonAnswer key