Area, volume, and perimeter

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 perimeterThe distance around the edge of a shape, measured in length units such as m., areaHow many unit squares cover a surface, measured in squared units such as m². and volumeHow many unit cubes fill a solid, measured in cubed units such as m³., 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

The unit power tells you what you have computed. If an "area" comes out in metres, something went wrong.

Formulas for flat shapes

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

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 prismA solid with the same cross-section all the way along, whose volume is base area × height..

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.