Exterior Angles of a Triangle Calculator

What Is an Exterior Angle of a Triangle?

An exterior angle of a triangle is formed when one side of the triangle is extended outward from a vertex. The angle between the extended side and the adjacent side of the triangle is called the exterior angle.

For every vertex of a triangle, the interior angle and exterior angle lie on a straight line, which means they are supplementary and always sum to $180^\circ$ ($\pi$ radians):

$$\alpha + \alpha' = 180^\circ, \quad \beta + \beta' = 180^\circ, \quad \gamma + \gamma' = 180^\circ$$

The Exterior Angle Theorem

The fundamental Exterior Angle Theorem states that the measure of an exterior angle of a triangle equals the sum of the measures of the two opposite (remote) interior angles:

$\alpha' = \beta + \gamma$
$\beta' = \alpha + \gamma$
$\gamma' = \alpha + \beta$

Because the interior angles of any triangle in Euclidean geometry sum to $180^\circ$ ($\alpha + \beta + \gamma = 180^\circ$), substituting $\beta + \gamma = 180^\circ - \alpha$ immediately confirms that $\alpha' = 180^\circ - \alpha$.

Sum of the Exterior Angles of a Triangle

When taking one exterior angle at each vertex, the sum of all three exterior angles of any triangle is always constant and equal to $360^\circ$ ($2\pi$ radians):

$$\alpha' + \beta' + \gamma' = (180^\circ - \alpha) + (180^\circ - \beta) + (180^\circ - \gamma) = 540^\circ - (\alpha + \beta + \gamma) = 540^\circ - 180^\circ = 360^\circ$$

This universal polygon property holds true for all triangles, whether they are acute, right, obtuse, equilateral, isosceles, or scalene.

Step-by-Step Calculation Examples

Example 1 (Two Interior Angles Known): Suppose a triangle has interior angles $\alpha = 50^\circ$ and $\beta = 70^\circ$.

Find third interior angle: $\gamma = 180^\circ - (50^\circ + 70^\circ) = 60^\circ$.
Exterior angle at vertex A: $\alpha' = \beta + \gamma = 70^\circ + 60^\circ = 130^\circ$.
Exterior angle at vertex B: $\beta' = \alpha + \gamma = 50^\circ + 60^\circ = 110^\circ$.
Exterior angle at vertex C: $\gamma' = \alpha + \beta = 50^\circ + 70^\circ = 120^\circ$.
Check sum: $130^\circ + 110^\circ + 120^\circ = 360^\circ$.

Related Geometry & Trigonometry Tools

Explore other helpful triangle and angle calculators:

Triangle Calculator: Calculate all sides, angles, area, and perimeter of any general triangle.
Right Triangle Calculator: Solve right triangles using Pythagorean theorem and trigonometric ratios.
Isosceles Triangle Calculator: Determine equal sides, apex angle, and base angles.
Classifying Triangles Calculator: Classify triangles by sides and angles automatically.

Frequently Asked Questions

What is the Exterior Angle Theorem?

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite non-adjacent interior angles ($\text{exterior angle} = \text{interior}_1 + \text{interior}_2$).

What is the sum of all exterior angles of a triangle?

The sum of the exterior angles of any triangle (taking one exterior angle per vertex) is always $360^\circ$ or $2\pi$ radians.

How are interior and exterior angles related at the same vertex?

At any vertex, the interior angle and its corresponding exterior angle form a linear pair along a straight line, making them supplementary with a sum of $180^\circ$.

Can an exterior angle of a triangle be acute?

Yes. If an interior angle is obtuse (greater than $90^\circ$), its supplementary exterior angle will be acute (less than $90^\circ$). However, a triangle can have at most one obtuse interior angle, so it can have at most one acute exterior angle.

What are the exterior angles of an equilateral triangle?

In an equilateral triangle, each interior angle measures $60^\circ$. Therefore, each of the three exterior angles measures $180^\circ - 60^\circ = 120^\circ$, summing to $360^\circ$.

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