Area, perimeter and volume without the formula sheet
A triangle is half a rectangle. A parallelogram is a rectangle that has been leaned over, which changes nothing about its area. A trapezium is a rectangle whose width is the average of the two parallel sides.
Nearly all of it comes back to length times width, and the exceptions are the ones with a circle in them. Seeing that is worth more than memorising the list.
Most shapes are a rectangle with something done to it
Base times height gives the area of a rectangle. A triangle on the same base and height is exactly half of it, which is where the half comes from — not from anywhere more mysterious.
A parallelogram has the same area as the rectangle with the same base and perpendicular height, because leaning a shape over moves area sideways without creating or destroying any. The catch is that the height has to be the perpendicular height, not the sloping side.
A trapezium is the interesting one. Its area is the average of the two parallel sides times the height, which is precisely the rectangle you would get by levelling the sloping top out.
The circle is the genuine exception, and it is worth simply knowing: area is pi times the radius squared, circumference is pi times the diameter.
Work it out: Area Calculator →
Doubling a dimension does not double the answer
Area grows with the square of the dimensions. Double the radius of a circle and the area quadruples, which is why a sixteen inch pizza is more than twice the food of a twelve inch one rather than a third more.
Volume grows with the cube. Double every dimension of a box and the volume goes up eightfold while the surface area only goes up fourfold. That single relationship explains a surprising amount of the physical world — why large animals struggle to lose heat and small ones struggle to keep it, why a scale model of a building is not a scale model of its structure, and why a big pan of food takes so much longer to cool.
It is also the most common estimating error in the building trades. A room that is twenty per cent larger in each direction needs forty-four per cent more flooring and seventy-three per cent more concrete.
Circle r=5 area 78.54, circumference 31.42
Double r area x4, circumference x2
Box, all dimensions doubled: volume x8, surface x4
Work it out: Circle Calculator →
The units come out squared and cubed
This sounds too obvious to state and it is where most real mistakes happen. Measure in centimetres and the area is in square centimetres and the volume in cubic centimetres. Mixing feet and inches in the same calculation is the single commonest cause of an answer that is wrong by a factor of twelve.
A cubic centimetre is a millilitre, so a thousand of them is a litre — which is the conversion that turns a volume into something you can buy. A cylinder three units across and ten tall is 282.74 cubic units, or 0.283 litres in centimetres.
Get into the habit of writing the unit next to every number. It is the cheapest error check there is, and it catches the factor-of-twelve mistakes before they become an order.
Work it out: Volume Calculator →
The one to know by heart
A triangle with sides of 3 and 4 has a hypotenuse of exactly 5. It is worth knowing because it works in reverse: if three measured sides satisfy a squared plus b squared equals c squared, the corner between the two short sides is exactly square.
That is how a foundation, a deck or a garden bed gets squared up without a set square. Measure three units along one edge, four along the other, and adjust until the diagonal is exactly five. Any multiple works, and bigger is more accurate — 6-8-10 or 9-12-15 on a large job.
The usual slip is putting the hypotenuse in as one of the short sides. It is always the longest, and it is always opposite the right angle. Given the hypotenuse and one side you subtract rather than add.
Work it out: Pythagorean Theorem Calculator →
Questions
Why is the area of a triangle half the base times the height?
Because a triangle is exactly half of the rectangle with the same base and perpendicular height. Two copies of the triangle fit that rectangle exactly.
Does a parallelogram have the same area as a rectangle?
Yes, if the base and the perpendicular height match. Leaning a shape over moves area sideways without changing how much there is — but you must use the perpendicular height, not the sloping side.
If I double the radius, what happens to the area?
It quadruples, because the radius is squared. That is why a sixteen inch pizza is much more than twice a twelve inch one.
What happens to volume when I double every dimension?
It goes up eightfold, while surface area only goes up fourfold. That ratio is why large objects hold heat and small ones lose it.
How do I check a corner is square without a set square?
Measure 3 units along one edge and 4 along the other; if the diagonal between those marks is exactly 5, the corner is square. Use a larger multiple like 6-8-10 for better accuracy on a big job.
How many millilitres in a cubic centimetre?
One. So a thousand cubic centimetres is a litre, which is what turns a calculated volume into something you can actually buy.