Classifying Triangles

Name a triangle by its sides (equilateral, isosceles, scalene) and by its angles (acute, right-angled, obtuse), and use the name to find a missing length.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 13 real past-paper questions below

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Names by Sides

The 1-mark question: write down the mathematical name of a triangle. First look at the sides. Small ticks mark sides that are equal.

Name	Sides	Angles
Equilateral	all 3 equal	all 60°
Isosceles	2 equal	2 equal
Scalene	none equal	none equal

Key idea

Equal sides always come with equal angles. The equal angles sit opposite the equal sides.

Key idea

An isosceles triangle has one line of symmetry. It goes from the top corner to the middle of the base, meets the base at a right angle and cuts it in half. An equilateral triangle has three.

Common mistake

Calling a triangle isosceles because it looks tall and thin. You need two equal sides, or two equal angles.

Names by Angles

Then look at the biggest angle.

Name	Biggest angle
Acute	less than $9 0^{\circ}$
Right-angled	exactly $9 0^{\circ}$ (a small square)
Obtuse	more than $9 0^{\circ}$

Worked example

1
$8 0^{\circ}, 6 0^{\circ}, 4 0^{\circ}$ — The question gives three angles. Look for equal angles first, because they tell you about the sides.
2
$no two angles equal$ — Equal angles sit opposite equal sides. No angles match, so no sides match: the triangle is scalene.
3
$biggest: 8 0^{\circ} < 9 0^{\circ}$ — Now the angle name. Only the biggest angle matters: if it is under 90°, all three are.
4
$acute$ — All three angles are less than 90°, so it is an acute triangle.
5
$scalene and acute$ — Both names are true. Give the one the question asks for: by its sides or by its angles.

Key idea

Names can combine. Angles of $9 0^{\circ}, 4 5^{\circ}, 4 5^{\circ}$ make a right-angled isosceles triangle.

Use the Name to Find a Length

The 2-mark question: the name tells you which sides are equal. Use that to find a missing side.

Triangle $A B C$ is isosceles with $A B = A C$ . $B C$ is 7 cm and the perimeter is 31 cm. Find $A B$ .

Worked example

1
$A B = A C$ — This tells you the two equal sides. The third side, BC = 7 cm, is the odd one out.
2
$31 - 7 = 24$ — Take the odd side off the perimeter. The 24 cm left is shared by the two equal sides.
3
$A B = 24 \div 2 = 12 cm$ — The two equal sides share the 24 cm evenly, so halve it.
4
$12 + 12 + 7 = 31 ✓$ — Check: the three sides add up to the perimeter, so 12 cm is right.

Sometimes you are given an equal side and asked for the third side. Each equal side is 9 cm and the perimeter is 25 cm.

Worked example

1
$2 \times 9 = 18$ — There are two sides of 9 cm, so together they make 18 cm.
2
$25 - 18 = 7 cm$ — The third side is whatever is left of the perimeter.

Key idea

Write the working, such as $(31 - 7) \div 2$ . That line earns a mark on its own, even if the final number slips.

Common mistake

Dividing 31 by 3. Only an equilateral triangle has three equal sides.

Name It From a Grid

Some questions plot the three corners on a grid and ask for the triangle's name. Count squares instead of trusting your eye.

Worked example

1
$P (1, 1), Q (3, 5), R (5, 1)$ — The question gives three corners. For each side, count how far across and how far up or down it goes.
2
$P Q : 2 across, 4 up$ — From P to Q: x goes 1 → 3 (2 across), y goes 1 → 5 (4 up).
3
$QR : 2 across, 4 down$ — From Q to R: x goes 3 → 5, y goes 5 → 1. The same moves, only going down, so QR is as long as PQ.
4
$P R : 4 across, 0 up$ — PR is flat and 4 squares long. PQ goes 4 up as well as 2 across, so PQ is longer than 4: a different length.
5
$P Q = QR \Rightarrow isosceles$ — Exactly two sides match, so the triangle is isosceles.

Key idea

A side 2 across and 4 up is as long as a side 4 across and 2 up. The order of the moves does not change the length.

Common mistake

Naming it by how it looks. A tilted isosceles triangle can look scalene, so always count the squares.

Which Method? Read the Question

Read the diagram's clues in this order: sides first, then angles.

You see	You do
Ticks on two sides	isosceles
Ticks on all three sides	equilateral
Corners plotted on a grid	count across and up for each side
A small square in a corner	right-angled
An angle over 90°	obtuse
Find a missing angle	Angles in a Triangle
Name a four-sided shape	Quadrilateral Properties

Key idea

If the diagram gives angles, not ticks, use them: two equal angles also means isosceles.

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