BLOG

The interior angles of a polygon grow with every side you add. The exterior angles do not: they total 360° for a triangle, a hexagon and a hundred-sided shape alike. Walking round the outline shows why in one sentence. Find the missing exterior angle in an interactive playground.

In a regular polygon every angle is the same, so one number describes the whole shape. You can find it by sharing out the angle sum, or far faster by going through the exterior angle. Compare the two routes on six regular polygons in an interactive playground.

A parallelogram is defined by one thing only: both pairs of opposite sides are parallel. Equal opposite sides, equal opposite angles, supplementary neighbours and bisecting diagonals all follow from it rather than being separate rules. Work the angles out in an interactive playground.

A rhombus is a parallelogram with all four sides equal, so it inherits everything a parallelogram has. What it adds is about the diagonals: they meet at right angles and each one bisects the two corners it runs between. Use both facts in an interactive playground.

A rectangle is a parallelogram with a right angle, so it inherits equal opposite sides and bisecting diagonals without asking. What it adds is that the two diagonals are the same length, which turns the shape into four isosceles triangles. Use that in an interactive playground.

A trapezium has one pair of parallel sides instead of two; a kite has two pairs of equal sides that sit next to each other rather than opposite. Each keeps part of what a parallelogram offers and loses the rest, and each has its own angle rule. Use both in an interactive playground.

Parallelogram, rhombus, rectangle and square are not four separate shapes but one family, each adding a condition to the one before. The clearest way to tell them apart is to ignore the outline and look at what the diagonals do. Name them from the diagonals alone in an interactive playground.

The circumference is the distance all the way round a circle, and it is always π times the diameter — a little over three diameters, whatever the size. The only decision is whether you were given the diameter or the radius. Practise both in an interactive playground.

π is not a rule someone invented; it is what you get when you measure round a circle and across it and divide. Every circle gives the same answer, which is why one number works for all of them. Use it in reverse to find a circle's width in an interactive playground.

An arc is part of a circle's rim, and its length is that same fraction of the whole circumference. Find the fraction from the angle at the centre, then apply it — two steps, no new formula to learn. Practise on six sectors in an interactive playground.

A parallelogram's area is its base multiplied by its perpendicular height, and the reason is a single cut: slice a triangle off one end, slide it to the other, and you have a rectangle. The one thing to get right is which number is the height. Practise in an interactive playground.

A trapezium's parallel sides are different lengths, so it has no single width — but the average of the two behaves exactly like one. Add them, halve, and multiply by the perpendicular height. See why that works and practise it in an interactive playground.

Rhombuses and kites are awkward to measure by base and height, but both have diagonals that cross at right angles — and that gives a simpler route. Multiply the two diagonals and halve. See why one rule covers both shapes, in an interactive playground.

Most real shapes have no area formula at all. They do not need one: cut them into rectangles and triangles, find each piece, and add. The skill is choosing the cut and working out the lengths the drawing does not give you directly. Practise both in an interactive playground.

A sector is a slice of a circle, and its area is that same slice of the circle's area. Find the fraction from the angle, then take that fraction of πr². The trap is different from the arc-length one: here the radius is used as it stands. Practise in an interactive playground.

Similar shapes have equal angles and sides in the same ratio: one is an exact enlargement of the other. Two rectangles can look very alike and fail the test outright. Sort the genuine enlargements from the near-misses in an interactive playground.

The scale factor is the number every length is multiplied by when a shape is enlarged. Find it by dividing a new length by the matching old one — and expect an answer below 1 when the shape has shrunk. Practise both directions in an interactive playground.

Similar triangles share one scale factor across every pair of matching sides, so one complete pair is enough to find any other length. Work out the factor, then multiply. Practise the two steps — and the matching up that comes before them — in an interactive playground.

Enlarging a shape by a scale factor multiplies every length by that factor — but the area goes up by the factor squared. Double a shape and it holds four times as much. See why, and why it explains a great deal about pizza and paint, in an interactive playground.

Letters are not there to make maths harder. They are there because some rules have to cover every number at once, and you cannot write them all down. Build the rule for a growing pattern in an interactive playground.