Matemáticas
Equilateral Triangle: Three Equal Sides and Three 60-Degree Angles
Three equal sides, three 60° angles, three lines of symmetry. Plus the exact height — √3 over 2 times the side — and where those surds come from.
Equilateral triangle
An equilateral triangle has three sides of equal length and three equal interior angles, each of which measures exactly 60 degrees.
- También llamado
- Regular triangle
- Dónde lo encuentra el alumnado
- Grade 3 mathematics, when triangles are named by their sides, and again at GCSE Higher tier where the exact values of sin 60° and cos 60° are derived from one.
La respuesta corta
An equilateral triangle has three sides of equal length and three equal angles, each measuring 60 degrees. It has three lines of symmetry and rotational symmetry of order 3. Its height is not a whole number: for a side of length s the exact height is (√3/2)s, about 0.866 of the side.
Un ejemplo
h = (√3 ⁄ 2) × 6 = 3√3 ≈ 5.20 cm
An equilateral triangle of side 6 cm has a height of 3√3 cm, roughly 5.20 cm — noticeably less than the 6 cm side, which is the check most students skip. Its area is (√3/4) × 36 = 9√3 cm², about 15.59 cm². Both results are irrational, so an exact answer written as a surd is more accurate than any rounded decimal and is usually what the mark scheme wants.
Three of everything
Equal sides force equal angles, and since the three angles of any triangle add to 180 degrees, each one must be 60. This is the only triangle whose angles are fixed by its name — you are told all three the moment the word is used.
It is also the regular polygon with the fewest sides. Regular means equal sides and equal angles together, and for triangles the two conditions arrive as a package: any triangle with three equal sides automatically has three equal angles, and the reverse holds too.
- Three sides of equal length
- Three interior angles of 60 degrees each
- Three lines of symmetry, one from each vertex to the midpoint of the opposite side
- Rotational symmetry of order 3 — it lands on itself after a turn of 120 degrees
- Height (√3/2) × side; area (√3/4) × side²
The height is never as neat as the side
Drop a line from one vertex to the midpoint of the opposite side and the triangle splits into two right-angled halves with a base of s/2 and a hypotenuse of s. Pythagoras then gives the height as (√3/2)s, which is irrational for every whole-number side.
That is why a question giving a side of 8 cm expects 4√3 cm rather than a tidy decimal, and why the area of an equilateral triangle almost always ends up written with a surd in it.
Where the 60 degrees comes back later
The exact trigonometric values students are asked to memorise are not arbitrary. Take an equilateral triangle of side 2, cut it in half, and the right-angled triangle left behind has sides 1, √3 and 2. Every exact value for 30 and 60 degrees reads straight off it.
The same triangle is the reason hexagons behave so well. A regular hexagon divides into six equilateral triangles meeting at the centre, which is where its area formula and its 120-degree interior angles come from.
Preguntas frecuentes
Why is every angle in an equilateral triangle 60 degrees?
Because equal sides force equal angles, and the three angles of any triangle add to 180 degrees. Three identical angles summing to 180 leaves only one possibility: 60 each. No measuring is involved — the result follows from the definition, so it holds for a triangle of any size.
What is the height of an equilateral triangle with side 10 cm?
5√3 cm, which is about 8.66 cm. Halving the base gives 5 cm, and Pythagoras on a right-angled triangle with hypotenuse 10 and base 5 gives a height of √75, which simplifies to 5√3. Leaving it in surd form keeps the answer exact for any later working.
Is an equilateral triangle a regular polygon?
Yes, and it is the smallest one. A regular polygon has all sides equal and all angles equal, and an equilateral triangle satisfies both. The next two are the square and the regular pentagon. Triangles are the only family where equal sides guarantee equal angles automatically.
How many lines of symmetry does an equilateral triangle have?
Three, each running from a vertex to the midpoint of the side opposite. It also has rotational symmetry of order 3, meaning it looks unchanged after a turn of 120 degrees. An isosceles triangle has just one line of symmetry, and a scalene triangle has none at all.
Fuentes
- National curriculum in England: mathematics programmes of study — Department for Education
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