The four types of triangle are equilateral (three equal sides, each angle 60°), isosceles (two equal sides and two equal base angles), scalene (no equal sides or angles), and right-angled (one angle of exactly 90°). These types can overlap: a right-angled triangle is also either isosceles or scalene.
What makes a triangle equilateral?
An equilateral triangle has three equal sides and three equal angles. Because all three angles must add to 180° and all are equal, each angle is 180° ÷ 3 = 60°.
Key properties:
- All three sides are the same length.
- All three angles are 60°.
- Three lines of symmetry (each from a vertex to the midpoint of the opposite side).
- Rotational symmetry of order 3.
An equilateral triangle can never be right-angled — none of its angles reach 90°.
What makes a triangle isosceles?
An isosceles triangle has exactly two equal sides and two equal base angles — the angles at the ends of the unequal (base) side.
If the two equal angles are each x, then: 2x + apex angle = 180°.
Example: An isosceles triangle has a top angle of 40°. Find the base angles.
2x = 180° − 40° = 140°, so x = 70°. Both base angles are 70°.
Properties of an isosceles triangle:
- Two equal sides (called the legs).
- Two equal base angles.
- One line of symmetry (from the apex to the midpoint of the base).
- Rotational symmetry of order 1 (it looks different if rotated, unless equilateral).
What makes a triangle scalene?
A scalene triangle has no equal sides and no equal angles. All three sides are different lengths and all three angles are different values — they still sum to 180°, but none are equal to each other.
Scalene triangles have:
- No lines of symmetry.
- Rotational symmetry of order 1 only.
Most triangles you encounter in real life are scalene.
What makes a triangle right-angled?
A right-angled triangle contains one angle of exactly 90° (a right angle), marked with a small square in diagrams. The side opposite the right angle is always the longest side and is called the hypotenuse.
Because the three angles must sum to 180°, the other two angles are always acute (less than 90°) and they must add to exactly 90°.
A right-angled triangle can be:
- Right-angled scalene: the two acute angles are different (e.g. 30°, 60°, 90°).
- Right-angled isosceles: the two acute angles are both 45° (e.g. 45°, 45°, 90°). The two shorter sides are equal.
How do all four types compare?
| Type | Equal sides | Equal angles | Lines of symmetry |
|---|---|---|---|
| Equilateral | 3 | 3 (all 60°) | 3 |
| Isosceles | 2 | 2 (base angles) | 1 |
| Scalene | 0 | 0 | 0 |
| Right-angled | 0 or 2 | see above | 0 or 1 |
Note that "right-angled" describes an angle property, while equilateral, isosceles, and scalene describe side properties. A single triangle can carry two labels: for example, a right-angled isosceles triangle is both right-angled and isosceles.
How do you identify the type from a diagram?
Look for the following clues:
- Tick marks on sides — single ticks show one pair of equal sides; double ticks show another. Three sets of single ticks indicate equilateral.
- Arc marks on angles — arcs on two angles show they are equal.
- The small square — always indicates a right angle of exactly 90°.
- No marks at all — the triangle is probably scalene (or you need to measure).
What are the most common mistakes?
- Thinking equilateral is a type of isosceles. Technically it is (it has two equal sides — in fact all three), but in exam questions "isosceles" almost always means exactly two equal sides. Follow the question's wording.
- Forgetting that right-angled and isosceles can coexist. A 45°–45°–90° triangle is both.
- Assuming a triangle with no marks is scalene. The diagram may simply lack markings. Measure or check angle values before labelling.
- Adding angles incorrectly. Always check that your three angles sum to exactly 180° — if they don't, you have made an arithmetic error.
Frequently asked questions
Can a triangle be equilateral and right-angled at the same time?
No. An equilateral triangle has all angles equal to 60°. A right-angled triangle requires one angle of 90°. Since 60° ≠ 90°, these two types cannot coincide. If all three angles were 90° they would total 270°, which is impossible — triangle angles must sum to 180°.
What is the hypotenuse and which type of triangle has one?
The hypotenuse is the longest side of a right-angled triangle, always located directly opposite the right angle. Only right-angled triangles have a hypotenuse. It is the side used in Pythagoras' theorem: a² + b² = c², where c is the hypotenuse.
How do you find a missing angle in an isosceles triangle?
Use the fact that the two base angles are equal. If you know the apex angle, subtract it from 180° and divide by 2: base angle = (180° − apex angle) ÷ 2. If you know one base angle, the other base angle is identical, and the apex angle is 180° − 2 × base angle.
What is the difference between acute, obtuse, and right-angled triangles?
These are classifications based on the largest angle. An acute triangle has all three angles less than 90° (equilateral and most isosceles triangles are acute). An obtuse triangle has one angle greater than 90°. A right-angled triangle has one angle of exactly 90°. Any triangle — equilateral, isosceles, or scalene — can be acute, obtuse, or right-angled (except equilateral, which is always acute).
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