▶ 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
Drag any vertex, keeping the pentagon convex; the readout under the canvas shows the sum of the exterior angles and the sum of the interior angles. Learn more about why it holds
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?Extend every side of the pentagon the same way round; the angle between a side and the next side's extension is an exterior angle. At each vertex the interior and the exterior angle make a straight angle, 180°; five of them make 900°. The interior angles add up to 540° by the polygon angle sum, so the exterior angles add up to 900° − 540° = 360°. For n sides: n · 180° − (n − 2) · 180° = 360°. Drag any vertex, keeping the shape convex.Polygon exterior angles
Given Every side of the convex pentagon ABCDE is extended beyond its far end, all the same way round. The angle between each extension and the next side is an exterior angle: α′ at A, β′ at B, γ′ at C, δ′ at D, ε′ at E. To prove: α′ + β′ + γ′ + δ′ + ε′ = 360°, and the same for every convex polygon.
Step through the proof, then drag any vertex of the pentagon, keeping it convex, and watch the five exterior angles change while their sum in the readout stays at 360°. 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
Proof?Polygon exterior anglesDisplayLabelsMeasuresDrag any vertex (keep the pentagon convex)
Given Every side of the convex pentagon ABCDE is extended beyond its far end, all the same way round. The angle between each extension and the next side is an exterior angle: α′ at A, β′ at B, γ′ at C, δ′ at D, ε′ at E. To prove: α′ + β′ + γ′ + δ′ + ε′ = 360°, and the same for every convex polygon.
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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 — every vertex A to E moves freely while the pentagon stays convex; the extensions follow
4Readout — the line under the figure: the sum of the five exterior angles and the sum of the five interior angles, 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 5 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 5 of 5
Proof?Polygon exterior anglesDisplayLabelsMeasuresDrag any vertex (keep the pentagon convex)
Given Every side of the convex pentagon ABCDE is extended beyond its far end, all the same way round. The angle between each extension and the next side is an exterior angle: α′ at A, β′ at B, γ′ at C, δ′ at D, ε′ at E. To prove: α′ + β′ + γ′ + δ′ + ε′ = 360°, and the same for every convex polygon.
2α + α′ = 180° at every vertex
Linear pair At A the interior angle α and the exterior angle α′ lie on either side of the side AB, along the straight line through E, A and A1. Together they make a straight angle: α + α′ = 180°. The same at every vertex: β + β′ = γ + γ′ = δ + δ′ = ε + ε′ = 180°.
3Five straight angles: 900°
Adding Add the five equations: (α + α′) + (β + β′) + (γ + γ′) + (δ + δ′) + (ε + ε′) = 5 · 180° = 900°. One straight angle for each vertex, n · 180° for an n-gon.
4α + β + γ + δ + ε = 540°
Polygon angle sum The interior angles of a convex pentagon add up to (5 − 2) · 180° = 540°, by the polygon angle sum (diagonals from one vertex cut it into three triangles). For an n-gon the interior angles add up to (n − 2) · 180°.
5α′ + β′ + γ′ + δ′ + ε′ = 360°
Subtracting Take the interior sum away from the total of the straight angles: α′ + β′ + γ′ + δ′ + ε′ = 900° − 540° = 360°. For an n-gon, n · 180° − (n − 2) · 180° = 360°, whatever n is: the exterior angles of every convex polygon add up to one full turn. ∎
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1The exterior angles — α′ to ε′, between each side and the extension of the side before it, all the same way round
2The extensions — each side drawn a little past its far end
3The interior angles — α to ε, filling the straight angle with the exterior angle at each vertex
5Replay — at the last step Play becomes Replay and Next is off
The Statement
Walk round a convex polygon and extend each side beyond the vertex you arrive at. The angle between that extension and the next side is the exterior angle at the vertex: it is the turn you make there. The theorem says that the exterior angles, one at each vertex and all taken the same way round, add up to a full turn:
α′+β′+γ′+δ′+ε′=360°
for the pentagon in the player, and 360° for every convex polygon, however many sides it has. The triangle version has its own page; this page proves the general statement from the polygon angle sum.
In the player the five vertices are free; each side is drawn extended a little beyond its far end, and the exterior angles are marked there. The readout shows the sum of the exterior angles and, for comparison, the sum of the interior ones.
Opening scene, frozen
The pentagon with every side extended the same way round; the five exterior angles marked.
Linear Pairs
At the vertex A the interior angle α (between the sides AE and AB) and the exterior angle α′ (between AB and the extension of EA) lie side by side along the straight line through E, A and the extension. Two angles that fill a straight line are a linear pair, and they add up to 180°, as on the vertical angles page:
α+α′=180°
Nothing is special about A: at every vertex the interior and the exterior angle make a straight angle, so β+β′=γ+γ′=δ+δ′=ε+ε′=180°.
Linear pairs, frozen
Interior and exterior angle at each vertex make a straight angle; α and α′ lit.
Five Straight Angles
Add the five equations, one per vertex:
(α+α′)+(β+β′)+(γ+γ′)+(δ+δ′)+(ε+ε′)=5⋅180°=900°
On the left are all five interior angles and all five exterior angles together. For a polygon with n sides the same addition gives n⋅180°.
Five straight angles, frozen
All ten angles lit: five straight angles, 900°.
The Interior Sum
The interior angles of a convex pentagon add up to (5−2)⋅180°=540°: the polygon angle sum page cuts the pentagon into three triangles from one vertex. For an n-gon the interior angles add up to (n−2)⋅180°:
α+β+γ+δ+ε=540°
The interior sum, frozen
The five interior angles: 540° by the polygon angle sum.
Why It Holds
Take the interior sum away from the total of the straight angles:
α′+β′+γ′+δ′+ε′=900°−540°=360°
For an n-gon, n⋅180°−(n−2)⋅180°=2⋅180°=360°: the number of sides cancels out, and the exterior angles of every convex polygon add up to one full turn. The end-of-proof mark ∎ in the log closes the argument. The tags name the facts used: the given figure, linear pairs, an addition, the polygon angle sum and a subtraction.
There is also a picture behind the algebra: walking once round the polygon you turn by each exterior angle in turn, and when you are back at the start facing the way you began you have turned through exactly 360°.
The conclusion, frozen
The five exterior angles: 900° − 540° = 360°.
Dragging the Figure
Drag any vertex: the two sides at it swing, the exterior angle there and at its two neighbours change, and the sum in the readout stays at 360° while the interior sum stays at 540°. Pull a vertex outwards and its exterior angle grows; push it towards the centre and the exterior angle shrinks towards 0° as the two sides straighten out.
The player keeps the pentagon convex (every corner turning the same way), with sides long enough for the angle marks and all the extensions inside the frame. A drag that would make the pentagon dent inwards, or carry a point out of the frame, is refused; for a non-convex polygon the statement needs signed turns, and the player keeps to the convex case.
After dragging A, frozen
A pulled to the right: the exterior angles at E, A and B change, the sum is still 360°.
Consequences
• Regular polygons. In a regular n-gon every exterior angle is 360°/n, so every interior angle is 180°−360°/n: 108° for the pentagon, 120° for the hexagon, 135° for the octagon. • Only three regular tilings. The plane can be tiled by equal regular polygons only when the interior angle 180°−360°/n divides 360°, which happens for n=3,4,6 and no other n. • Turtle geometry. A turtle that repeats "forward, turn by t" closes a regular polygon after 360°/t steps; the exterior angle sum is why. • The triangle case. For n=3 the theorem is the sum of exterior angles of a triangle, proved there directly from the exterior angle theorem. • Total curvature. For a smooth closed convex curve the same statement is that the tangent turns through 360° once round; the polygon theorem is its discrete form.