What Is The Sum Of The Exterior Angles Of A Triangle?
The sum of the exterior angles of a triangle equals $360°$, when you take exactly one exterior angle at each vertex. An exterior angle is the angle formed between one side of the triangle and the extension of an adjacent side. Every triangle has three vertices, so you take three exterior angles, one per corner, and they always total $360°$.
$$e_1 + e_2 + e_3 = 360°$$
This is true for any triangle. It does not depend on the triangle being equilateral, right-angled, or anything special. A long thin scalene triangle and a neat equilateral triangle both give exactly $360°$.
One clarification that saves a lot of confusion: at each vertex you can actually draw two exterior angles (by extending either of the two sides), and they are equal to each other because they are vertically opposite. The $360°$ rule counts one per vertex — three in total. If you counted both at every vertex you would get $720°$, which is a different (and less useful) statement.
Why It Is 360° — The Proof
The result drops straight out of two facts you already know.
Fact 1 — Each interior and exterior angle form a linear pair
At each vertex, the interior angle and the exterior angle sit on a straight line, so they add to $180°$:
$$\angle A + e_1 = 180°$$ $$\angle B + e_2 = 180°$$ $$\angle C + e_3 = 180°$$
Fact 2 — Add all three equations
Adding the left sides and the right sides:
$$(\angle A + \angle B + \angle C) + (e_1 + e_2 + e_3) = 540°$$
Fact 3 — The interior angles of a triangle sum to $180°$
Substitute $\angle A + \angle B + \angle C = 180°$:
$$180° + (e_1 + e_2 + e_3) = 540°$$ $$e_1 + e_2 + e_3 = 540° - 180° = 360°$$
That is the whole proof. The exterior angles sum to $360°$ because three straight lines contribute $540°$ and the interior angles claim exactly $180°$ of it, leaving $360°$.
Examples Of The Sum Of Exterior Angles Of A Triangle
These build from a plain check, through the interior-angle link, to a wrong-turn example. Each stays inside the exterior-angle idea.
Example 1
An equilateral triangle has three equal exterior angles. What is each one, and do they sum to $360°$?
Each interior angle of an equilateral triangle is $60°$.
Each exterior angle is its linear-pair partner:
$$180° - 60° = 120°$$
Summing the three exterior angles:
$$120° + 120° + 120° = 360°$$
They add to $360°$, exactly as the rule promises.
Example 2
Two exterior angles of a triangle are $130°$ and $110°$. Guess the third quickly first.
The tempting shortcut is to reach for $180°$: subtract the two known angles from $180°$, giving $180° - 130° - 110° = -60°$. A negative angle is impossible, so that instinct is wrong.
The mistake was using the interior-angle total ($180°$) instead of the exterior-angle total ($360°$). Exterior angles sum to $360°$, not $180°$.
The correct calculation:
$$e_3 = 360° - 130° - 110° = 120°$$
The third exterior angle is $120°$. Whenever you subtract and get a negative or a suspiciously small angle, check whether you reached for the wrong total.
Example 3
A triangle has interior angles $50°$, $60°$, and $70°$. Find all three exterior angles and confirm the sum.
Each exterior angle is $180°$ minus its interior angle:
$$e_1 = 180° - 50° = 130°$$ $$e_2 = 180° - 60° = 120°$$ $$e_3 = 180° - 70° = 110°$$
Adding them:
$$130° + 120° + 110° = 360°$$
The sum is $360°$, confirming the rule for this scalene triangle.
Example 4
One exterior angle of a triangle is $95°$. What is its interior angle?
The interior and exterior angle at a vertex form a linear pair, so they add to $180°$:
$$\text{interior} = 180° - 95° = 85°$$
The interior angle is $85°$.
This uses the linear-pair fact on its own, without needing the $360°$ total.
Example 5
A right triangle has interior angles $90°$, $30°$, and $60°$. Do its exterior angles still add to $360°$?
Yes. Compute each exterior angle:
$$180° - 90° = 90°$$ $$180° - 30° = 150°$$ $$180° - 60° = 120°$$
Adding:
$$90° + 150° + 120° = 360°$$
The right angle does not change anything. The $360°$ rule holds for right triangles just as it does for every other kind.
Example 6
How does the exterior angle at one vertex relate to the two interior angles it is not next to?
By the exterior angle theorem, an exterior angle equals the sum of the two remote interior angles (the two interior angles not adjacent to it).
For a triangle with interior angles $50°$, $60°$, $70°$, the exterior angle at the $70°$ vertex is:
$$e_3 = 50° + 60° = 110°$$
This matches the $110°$ found in Example 3. The exterior angle theorem and the $360°$ sum are two views of the same angle relationships — one looks at a single vertex, the other adds up all three.
Why This Matters — "Turn all the way around and you land facing home"
The $360°$ result has a beautiful physical meaning that the algebra hides. Walk once around the outside of the triangle, and the exterior angle at each corner is exactly how far you turn at that corner. By the time you return to your start, facing the same direction you began, you have turned through one full circle — $360°$.
Robotics and navigation. A robot tracing a closed triangular path turns through its exterior angles at each corner; the turns must total $360°$ for it to end up pointing the way it started. This "turning angle" idea is how path-planning code checks that a loop actually closes.
It generalises to every polygon. The exterior angles of any convex polygon — quadrilateral, pentagon, hexagon — always sum to $360°$, for the same walk-around reason. The triangle is just the simplest case of a rule that never changes with the number of sides. See exterior angles of a polygon.
Where the maths is going. This links to the interior-angle sum $(n-2) \times 180°$ for a polygon with $n$ sides, covered in sum of angles in a polygon. The two facts together let you find any missing angle in any polygon. The idea that a full turn is $360°$ even underpins how a gyrocompass tracks heading through turns.
The algebra says $540° - 180° = 360°$; the geometry says one full turn brings you home. Same truth, two languages.
Mistakes To Watch For
Mistake 1: Using 180° as the exterior-angle total
Where it slips in: finding a missing exterior angle when the other two are given.
Don't do this: subtract the known exterior angles from $180°$, as if they behaved like interior angles.
The correct way: exterior angles sum to $360°$, so subtract from $360°$. Interior angles sum to $180°$; exterior angles sum to $360°$. The first instinct is to reuse the $180°$ interior total everywhere; the fix is to keep the two totals firmly apart. A negative or tiny answer is the tell-tale sign you grabbed the wrong one.
Mistake 2: Counting two exterior angles at the same vertex
Where it slips in: diagrams that show both sides extended at a corner, giving two exterior angles there.
Don't do this: add up all the exterior angles you can see and report $720°$.
The correct way: the $360°$ rule counts one exterior angle per vertex — three in total. The two exterior angles at a single vertex are equal (vertically opposite), so you pick one, not both. Three angles, one per corner.
Mistake 3: Confusing the exterior angle with the interior angle in the theorem
Where it slips in: applying the exterior angle theorem to find a remote interior angle.
Don't do this: set the exterior angle equal to the adjacent interior angle, or add the wrong pair of interior angles.
The correct way: the exterior angle equals the sum of the two remote (non-adjacent) interior angles — not the one beside it. In the real world this exact mix-up shows up in surveying and construction, where a misread turning angle at one station throws off the whole traverse; surveyors close a loop by checking that the exterior turning angles sum to $360°$, and a single swapped interior-for-exterior value fails the check.
Key Takeaways
The sum of the exterior angles of a triangle is always $360°$, one angle per vertex.
It holds for every triangle — scalene, isosceles, equilateral, acute, right, or obtuse.
Proof: three linear pairs give $540°$; the interior angles take $180°$; the exterior angles get the remaining $360°$.
Each exterior angle equals its adjacent interior angle's supplement ($180° -$ interior) and the sum of the two remote interior angles.
The same $360°$ rule holds for every convex polygon — the triangle is just the simplest case.
To take the sum of exterior angles further with a teacher, explore Bhanzu's geometry tutor sessions, a middle school math tutor for triangle angles, or general math classes online.
Try These Problems
Practice these problems to solidify your understanding. Work through them and check the answers below.
Two exterior angles of a triangle are $145°$ and $100°$. Find the third.
A triangle has interior angles $40°$, $75°$, and $65°$. Find all three exterior angles and confirm they sum to $360°$.
An exterior angle of a triangle is $118°$. What is its adjacent interior angle?
Answer to Question 1: $360° - 145° - 100° = 115°$. Answer to Question 2: $140°$, $105°$, $115°$; they sum to $360°$. Answer to Question 3: $180° - 118° = 62°$.
Want a live Bhanzu trainer to walk through more angle problems? Book a free demo class.
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