Why Do All Quadrilaterals Have 360°? | Interior & Exterior Angles Made Simple

Have you ever looked at a square, a rectangle, a kite, or even a weird, lopsided four-sided shape and wondered: “Do all of these really have the same total angle sum?”

Yes — they do. And in this post, I’ll walk you through two beautiful ideas in geometry:

1. Why the interior angles of any quadrilateral always add up to 360°
2. Why the exterior angles of any quadrilateral also add up to 360°

I’ve also made two simple videos to go with this post, so you can watch the proofs and see example problems step by step.

---

1. Interior Angles of a Quadrilateral = 360°

The Big Idea

No matter how a quadrilateral looks — regular or irregular, convex or concave (as long as it’s simple), the sum of its four interior angles is always 360°.

Why Is This True? (Simple Proof)

Take any quadrilateral ABCD. Now draw one diagonal, say AC.

This splits the quadrilateral into two triangles:

• △ABC
• △ADC

We already know:

The sum of angles in a triangle = $180^\circ$

So:

• In $\triangle ABC$: $\angle BAC + \angle ABC + \angle BCA = 180^\circ$
• In $\triangle ADC$: $\angle DAC + \angle ADC + \angle DCA = 180^\circ$

Add both equations:

$$(\angle BAC + \angle DAC) + \angle ABC + \angle ADC + (\angle BCA + \angle DCA) = 180^\circ + 180^\circ$$

Notice:

• $\angle BAC + \angle DAC = \angle A$
• $\angle BCA + \angle DCA = \angle C$

So we get:

$$\angle A + \angle B + \angle C + \angle D = 360^\circ$$

---

2. Exterior Angles of a Quadrilateral = 360°

What Are Exterior Angles?

At each vertex of a quadrilateral, if you extend one side, the angle formed outside between the extended side and the adjacent side is called an exterior angle.

🎥 Watch the full explanation + a mixed problem here:

The Surprising Fact

Just like the interior angles, the sum of all four exterior angles of any quadrilateral is also 360° — no matter its shape.theproblemsite

Why Is This True? (Simple Proof)

Let the interior angles of quadrilateral $ABCD$ABCD be $\angle A, \angle B, \angle C, \angle D$.
Let the corresponding exterior angles be $\angle 1, \angle 2, \angle 3, \angle 4$.

At each vertex, interior + exterior form a linear pair, so they add to $180^\circ$180∘:

$$\begin{aligned} \angle 1 + \angle A &= 180^\circ \\ \angle 2 + \angle B &= 180^\circ \\ \angle 3 + \angle C &= 180^\circ \\ \angle 4 + \angle D &= 180^\circ \end{aligned}$$∠1+∠A,∠2+∠B,∠3+∠C,∠4+∠D​=180

Add all four equations:

$$(\mathrm{\angle }1+\mathrm{\angle }2+\mathrm{\angle }3+\mathrm{\angle }4)+(\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C+\mathrm{\angle }D)=4\times {180}^{\circ }={72}^{}$$

(∠1+∠2+∠3+∠4)+(∠A+∠B+∠C+∠D)=4×180∘=720∘

But we already know:

$$\angle A + \angle B + \angle C + \angle D = 360^\circ$$

So: (Sum of exterior angles)+360∘=720∘⇒Sum of exterior angles=360∘

Beautiful, right? Both interior and exterior angle sums are 360°, just in different ways.