Triangles turn up everywhere — the warning sign at a road junction, a boat's triangular sail, a wedge of watermelon on a hot day. But they come in all sorts of shapes, and to talk about them clearly we first need names for them.

A triangle is the simplest flat shape there is: three straight sides and three corners. Yet no two triangles need look alike — some are tall and pointy, some squat and wide, some perfectly even. To talk about them clearly, mathematicians sort every triangle in two completely separate ways: by its sides and by its angles. Learn both, and you can name any triangle you meet.

Sorting by sides

Every triangle has three sides, and the first way to sort them is by which sides are equal:

• an equilateral triangle has all three sides equal — and, as a result, all three angles equal too, each exactly 60^\circ;
• an isosceles triangle has two sides equal (and the two angles opposite those sides — its base angles — are equal as well);
• a scalene triangle has no sides equal — every side, and every angle, a different size.

Sides known to be equal are marked with little tick marks: sides carrying the same number of ticks have the same length. So an equilateral triangle shows one tick on each of its three sides, an isosceles shows a tick on its two matching sides, and a scalene shows none.

Sorting by angles

The second way to sort a triangle is by its biggest angle:

• an acute triangle has all three angles less than 90^\circ — every corner is sharp;
• a right-angled triangle has exactly one 90^\circ angle, the square "corner of a book" angle (marked with a small square);
• an obtuse triangle has one angle greater than 90^\circ — one corner is open and blunt.

Named two ways at once

Here is the key idea: the side-name and the angle-name are independent, so a single triangle usually wears both labels. A triangle can be "isosceles and right-angled", or "scalene and obtuse". You pick one word from the side list and one word from the angle list.

• A triangle with sides 5, 5, 8 has two equal sides, so it is isosceles; its angles all stay below 90^\circ, so it is also acute — an isosceles acute triangle.
• A triangle with angles 90^\circ, 45^\circ, 45^\circ has a right angle, so it is right-angled; the two 45^\circ corners sit opposite two equal sides, so it is also isosceles — an isosceles right triangle (think of a square cut along its diagonal).
• A triangle with sides 3, 4, 5 has three different sides, so it is scalene; it also hides a perfect right angle, so it is right-angled too — a scalene right triangle.

The whole family

Step through the gallery to meet one of each. The top row is sorted by sides, the bottom row by angles — but remember the two labels combine, so a single triangle might be, say, "isosceles and right-angled".

Name the mystery triangle

Here is a fresh triangle, drawn with its markings: tick marks on any equal sides, an arc on marked angles, and a small square on a right angle. Read off its two names — one for the sides, one for the angles — then check against the caption. Press Refresh for a brand-new triangle to name.