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Equilateral Triangle: All Angles 60°

How to use
  1. ▶ Play runs the proof, Next → and ← Back step one scene at a time, Reset restores the opening scene and the original figure. Learn more about the controls
  2. Drag AA or BB to turn and resize the triangle; the readout under the canvas shows the three angles and the common side. Learn more about why it holds
  3. Labels toggles the point names, Measures writes each marked angle's size beside its arc. The Step explanations log keeps every step so far. Learn more about reading a scene
Proof?AB = AC makes the triangle isosceles with apex A, so the angles at B and C are equal; BA = BC makes it isosceles with apex B, so the angles at A and C are equal. All three angles are equal and add up to 180°, so each is 60°. Conversely, if the three angles are equal, the sides facing them are equal. Drag A or B.Equilateral triangle
Display
Drag A or B to turn and resize the triangle
αβγABC
α = 60°, β = 60°, γ = 60°, AB = BC = CA = 4.00
Step 1 / 6

Step explanations

1A triangle with three equal sides
Given
ABC is equilateral: AB = BC = CA (one tick each). Its angles are α at A, β at B and γ at C. To prove: α = β = γ = 60°, and conversely a triangle with three equal angles has three equal sides.
Drag A or B: C keeps the sides equal. Learn more about the statement

Getting Started with the Visualizer
The Statement
Apex at A
Apex at B
Three Equal Angles
Sixty Degrees
Why It Holds
Dragging the Figure
Consequences



Getting Started with the Visualizer

DemoA run through the proof player
Step 0 of 7
Step through the proof, then drag AA or BB to turn and resize the triangle, and watch the three angles in the readout stay at 60°60°. The picture below is the player with its parts numbered; the other tabs list what can be dragged, what each control does and what the step log shows.
Opening scene: the controls
1234567
1Proof — the theorem loaded in the player; the ? gives its idea in one line
2Display — Labels shows the point names, Measures writes each marked angle's size beside its arc
3Draggable points — A and B move freely and turn or resize the triangle; the hollow point C follows so all sides stay equal
4Readout — the line under the figure: α, β, γ and the common side, live while you drag
5Back, Play, Next, Reset — step through the proof or run it; Reset returns to step 1 and restores the original figure
6Speed and step counter — how fast Play advances, and which of the 6 steps is on screen
7Step explanations — every step reached so far, with its justification tag and a link to the section that explains it; click an entry to jump back to it
Proof complete: step 6 of 6
12345
1The three angles — α, β, γ, each 60° at the end
2The equal sides — one tick on each side
3The apex at C — γ, equal to both base angles
4Justification tags — Given, Isosceles base angles (twice), Combining, Angle sum, Equal angles equal sides
5Replay — at the last step Play becomes Replay and Next is off

The Statement

A triangle is equilateral when its three sides are equal, and equiangular when its three angles are equal. The theorem says the two are the same thing, and gives the angle:

AB=BC=CA  ⟺  α=β=γ=60°AB = BC = CA \iff \alpha = \beta = \gamma = 60°


An equilateral triangle is an isosceles triangle in three ways at once, with each vertex as the apex, and the whole proof is the isosceles base angles theorem applied twice, followed by the angle sum.

In the player AA and BB are free and CC is placed so that the triangle stays equilateral; the readout shows the three angles and the common side.
αβγABC
Opening scene, frozen

The equilateral triangle ABC, one tick on each side.


Apex at A

Take AA as the apex: the sides ABAB and ACAC are equal, so the triangle is isosceles and its base angles, at BB and CC, are equal:

β=γ\beta = \gamma
αβγABC
Apex at A, frozen

AB = AC: the base angles β and γ are equal.


Apex at B

Now take BB as the apex: BA=BCBA = BC, so the base angles at AA and CC are equal:

α=γ\alpha = \gamma
αβγABC
Apex at B, frozen

BA = BC: the base angles α and γ are equal.


Three Equal Angles

Put the two together: β=γ\beta = \gamma and α=γ\alpha = \gamma, so

α=β=γ\alpha = \beta = \gamma


Every equilateral triangle is equiangular. (Taking CC as the apex would give α=β\alpha = \beta as well, but it is not needed.)
αβγABC
Three equal angles, frozen

α = β = γ.


Sixty Degrees

The angles of a triangle add up to 180°180°:

α+β+γ=3α=180°,α=β=γ=60°\alpha + \beta + \gamma = 3\alpha = 180°, \qquad \alpha = \beta = \gamma = 60°


The size of an equilateral triangle does not matter: big or small, every angle is 60°60°.
αβγABC
Sixty degrees, frozen

3α = 180°: every angle is 60°.


Why It Holds

The converse runs the other way. If a triangle has α=β=γ\alpha = \beta = \gamma, then the sides facing equal angles are equal: α=β\alpha = \beta gives BC=CABC = CA, and β=γ\beta = \gamma gives CA=ABCA = AB. (If two sides facing equal angles were unequal, the larger side would face the larger angle, a contradiction.) So

equilateral  ⟺  equiangular\text{equilateral} \iff \text{equiangular}


The end-of-proof mark ∎ in the log closes the argument. The tags name the facts used: the given figure, isosceles base angles twice, combining, the angle sum, and equal angles facing equal sides.

For four or more sides the two ideas come apart: a rhombus has equal sides without equal angles, and a rectangle has equal angles without equal sides. The triangle is the only polygon where either one forces the other.
αβγABC
The conclusion, frozen

Equilateral exactly when equiangular.


Dragging the Figure

Drag AA or BB: the triangle turns and grows or shrinks, CC follows so that the three sides stay equal, and the three angles in the readout stay at 60°60°. Only the common side length changes.

The player keeps the side long enough for the angle marks and the whole triangle inside the frame; a drag that would push a vertex out is refused.
αβγABC
After dragging B, frozen

B moved up and in: the triangle turned and shrank, every angle still 60°.


Consequences

• The regular hexagon. Six equilateral triangles meet round a point, 6⋅60°=360°6 \cdot 60° = 360°, and make a regular hexagon whose side equals its radius.
• Regular polygons. The equilateral triangle is the regular polygon with n=3n = 3; the regular polygon angle 180°−360°/n180° - 360°/n gives 60°60°.
• Tiling. Since 60°60° divides 360°360°, equilateral triangles tile the plane, six at every vertex.
• Constructing 60°. Euclid's first proposition builds an equilateral triangle with two circles of the same radius; that gives a 60°60° angle with compass and ruler, and bisecting it gives 30°30°.
• The 30-60-90 triangle. The altitude of an equilateral triangle cuts it into two right triangles with angles 30°30°, 60°60°, 90°90°, whose short leg is half the hypotenuse; it gives sin⁡30°=12\sin 30° = \tfrac12 and cos⁡30°=32\cos 30° = \tfrac{\sqrt3}{2}.