Compute the perimeter, area and volume of common shapes and solids, with the right units.
Why this matters
How much cable runs around a room? How much paint covers a wall? How many litres does a tank hold? Those three questions are perimeter, area and volume, and they show up in civil and mechanical engineering, lab work, electrical installation, game levels and even data-centre planning. The formulas are short; the skill is choosing the right one and keeping the units straight.
Three kinds of measurement
| Measures | Units | Think of it as |
|---|---|---|
| Perimeter | length (m, ft) | the distance around the edge |
| Area | length² (m², ft²) | how many unit squares cover it |
| Volume | length³ (m³, ft³) | how many unit cubes fill it |
The unit power tells you what you have computed. If an "area" comes out in metres, something went wrong.
Formulas for flat shapes
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle (length l, width w) | 2l + 2w | l × w |
| Square (side s) | 4s | s² |
| Triangle (base b, height h) | sum of the three sides | ½ × b × h |
| Circle (radius r) | 2πr (the circumference) | πr² |
The triangle's height is measured perpendicular to the base, not along a slanted side. The circle's radius is half the diameter; mixing the two is a very common slip.
Formulas for solids
| Solid | Volume |
|---|---|
| Rectangular box (l, w, h) | l × w × h |
| Cylinder (radius r, height h) | πr²h |
| Cone | ⅓ πr²h |
| Sphere | (4/3) πr³ |
A pattern: a box and a cylinder are both "area of the base × height". If you remember that idea, you can work out the volume of any prism.
Worked example 1: flooring
A room is 4.2 m by 3.5 m. Flooring is sold in boxes covering 2.2 m².
- Area: 4.2 × 3.5 = 14.7 m²
- Boxes: 14.7 ÷ 2.2 ≈ 6.68, so buy 7 boxes (round up, and installers usually add about 10% for cuts and waste).
- Skirting board around the edge: perimeter = 2(4.2) + 2(3.5) = 15.4 m.
Worked example 2: a cylindrical tank
A tank has radius 0.5 m and height 1.2 m. How many litres does it hold?
- Volume: π × 0.5² × 1.2 = π × 0.25 × 1.2 ≈ 0.942 m³
- 1 m³ = 1000 L, so about 942 litres.
If the radius had been given in centimetres (50 cm), working in cm gives π × 50² × 120 ≈ 942,478 cm³, and 1 L = 1000 cm³ gives the same 942 L. Pick one unit system and convert at the end.
Worked example 3: composite shapes
An L-shaped patio can be split into two rectangles: 6 m × 2 m and 3 m × 2 m. Total area = 12 + 6 = 18 m². Most irregular shapes can be broken into rectangles, triangles and circles, then added (or subtracted for holes).
Scaling: why doubling is not just double
Double every length of a shape and:
- perimeter doubles (×2)
- area becomes 4 times larger (×2²)
- volume becomes 8 times larger (×2³)
This matters in practice. A pipe with twice the diameter has four times the cross-sectional area, and a scale model's weight grows with the cube of its size.
In code
import math
def circle_area(r):
return math.pi * r ** 2
def box_volume(l, w, h):
return l * w * h
print(round(circle_area(3), 2)) # 28.27
print(box_volume(2, 3, 4)) # 24
Use math.pi rather than 3.14 when precision matters; 3.14 is already wrong in the third digit.
Common mistakes
- Using diameter where the formula needs radius.
- Using a slanted side as a triangle's height.
- Forgetting the ½ in the triangle area or the ⅓ in a cone.
- Mixing units (one side in cm, another in m).
- Giving area in m or volume in m². Check the unit power.
Quick recap
- Perimeter is length, area is length², volume is length³.
- Rectangle area l × w, triangle ½bh, circle πr²; prism volume = base area × height.
- Break odd shapes into simple ones.
- Scaling lengths by k scales area by k² and volume by k³.
Try it
In the Do tab, add the cylinder volume formula to a small geometry toolkit, then use it to check the tank from example 2.