Three corners, any shape you like — and they always add up to exactly 180°. Don't take my word for it. Grab a corner and try to break it.
Start herePick any triangle — a tall skinny one, a squashed flat one, a perfect little pyramid — and add up its three inside corners. You'll always land on the same number: 180°.
That's a half-turn: the angle you'd sweep through if you spun on the spot from facing straight ahead to facing straight behind. Every triangle in the universe is built from exactly that much "corner," shared out three ways. One corner can hog most of it while the other two stay tiny, or all three can split it evenly — but the grand total never changes.
Three sides and three corners: that's the entire definition of a triangle, the simplest flat shape you can draw with straight edges. And out of that plainness falls a guarantee so dependable that engineers, mapmakers and game designers lean on it without a second thought. By the end of this page you'll have felt the rule with your own hands, watched a proof of it slide into place, and learned to use it to uncover any angle a triangle is trying to keep secret.
Two quick words, and then we play. A vertex is simply a corner — the point where two sides of the triangle meet. A triangle has three of them, so people often label them with letters like A, B and C. (The plural, if you ever need it, is "vertices.")
An interior angle is the amount of "turn" tucked inside the triangle at one of those corners — how widely the two sides spread apart there. "Interior" just means inside; there's an outside angle too, and we'll meet it later. A wide-open corner is a big angle; a sharp, pinched corner is a small one. We measure that opening in degrees (°): a full spin all the way around is 360°, a quarter-turn — a perfectly square corner — is 90°, and a flat, straight line is 180°. Hold onto that last one, because it's the whole story of this page.
The little curved mark you'll see drawn between two sides is just a way of pointing at the angle and saying "this one, right here." So when we ask what a triangle's angles add up to, we mean: take the opening at A, the opening at B and the opening at C, and tally them. That single question has one stubborn answer — and you're about to test it.
Three corners (vertices) A, B and C. The orange arc inside each one marks its interior angle. Our whole question is: what do those three angles add up to?
Spoiler from the next section: it never matters how you shove the corners around. The answer is locked.
Reading that the angles "always add to 180°" is one thing; feeling it refuse to break is another. So this is the heart of the page. Grab any corner with your finger or mouse and haul it around. The three angle readouts will swing wildly — push a corner out and that angle balloons while the other two shrink to make room; tuck a corner in and watch the opposite thing happen. But keep one eye on the golden box: the total never leaves 180°.
Think of it as a fixed budget. There are 180 "degrees of corner" to share between A, B and C, and dragging only changes who gets how much — never the size of the pot. Prefer the keyboard? Choose a corner with the A / B / C buttons, then move it with the two sliders or the arrow keys (and press 1, 2 or 3 to switch corners). The angle readouts below are the real, accessible report card.
The total is computed live from the corner positions — and rounded so the three numbers always add to a true 180°.
The drag demo shows you that it's always 180°. This one shows you why — and it's a proof you can do at home with nothing but paper and scissors. Cut out any triangle you like. Now tear off all three corners, the way you'd rip the crusts off a sandwich. Slide those three torn corners together so their sharp points all meet at one spot, sitting edge to edge with no gaps and no overlaps. Every single time, the three of them line up flat — they make a perfectly straight line. And a straight line is exactly 180°. Press the button and watch the corners march down and assemble.
Together the three corners fill a straight angle: a half-turn, 180°.
Why does this work for every triangle and not just lucky ones? The grown-up reason uses parallel lines — two lines that run in the same direction forever and never meet, like railway tracks. Draw a line through the top corner that runs parallel to the base. The two slanted sides of the triangle now cut across both the base and this new line. When a line crosses a pair of parallels, it copies angles from one to the other: the two bottom corners reappear up at the top as matching "Z-shaped" angles (mathematicians call them alternate angles). And up there, those two copies plus the original top corner sit snugly side by side along the straight top line. Three angles, one straight line, no scissors required — always 180°. The paper version and the parallel-lines version are the same truth told two ways.
This is where the rule earns its keep, and where it shows up most in tests and real problems. If you know two of a triangle's angles, the third isn't a mystery you have to measure with a protractor — it's already decided. Because all three are locked into a total of 180°, the missing one has to be whatever is left over: 180° minus the two you already have. That's the entire method. Add the two you know, subtract from 180, done. Here are three worked examples — read how each one is set up — and then it's your turn to drive.
One corner is a square 90° corner, and another is 35°.
The hidden angle is 55°.
You measure 70° and 70° at two corners.
The third must be 40°.
A corner is 120°, another is 25°.
Even with a fat 120° corner, it still balances.
Notice the trick works even when a corner looks alarming. A 120° corner is obtuse — fatter than a square corner — yet the method doesn't blink: it just leaves less for the other two to share. And if you ever spot that two angles are equal (a hint that two sides match), you can sometimes find all three from a single given angle. Keep that in your back pocket for the next section.
Now you try. Two angles are given; work out the third, type it in, and check yourself. Press Show me if you get stuck, and New triangle for a fresh one as many times as you like — there are plenty.
(angles labelled, not drawn to scale)
The 180° rule never changes, but a few triangles are so neatly built that their angles fall into tidy patterns worth memorising — patterns that turn "find the missing angle" into something you can often do in your head. Tap through the four most famous ones and watch the diagram and the numbers update together.
All three sides equal, so all three angles are equal too. Split 180° three ways and every corner is 60°.
Once the 180° rule is yours, it hands you a second rule for free. Take one side of the triangle and keep drawing it straight on, past a corner. The angle that opens up between that extension and the next side is the exterior angle — the angle on the outside of that corner. Here's the lovely part: an exterior angle always equals the sum of the two interior angles at the other two corners — the two it's "facing away" from. So you can find it instantly by just adding those two, no subtraction needed.
The orange ext angle equals a + c (the two far corners), not just one of them.
Why? The exterior angle and the corner right next to it (b) sit on a straight line, so they add to 180°. But a + b + c is also 180°. Both equal 180, so the exterior angle must equal a + c. It's the 180° rule wearing a disguise.
This is the trap almost everyone falls into at least once. It feels like a giant triangle should hold more angle than a tiny one — bigger shape, bigger everything, right? But angle and size are two different ideas. An angle measures how much two sides turn away from each other; it doesn't care how long those sides are. Growing a triangle stretches its sides — the lengths get bigger — but it never opens or closes the corners. A blown-up triangle is the very same shape, just zoomed in, like the same photo printed poster-size. Its three angles are identical to the small version's, and they still total a flat 180°. Drag the slider: the side length climbs and climbs, while the angle total sits there, unbothered, at 180°.
The angles here are locked at 50°, 60° and 70°. Change the size all you want — the total stays 180°.
Two triangles that have the same three angles but different sizes are called similar — they're scaled copies of each other, like a photo and its thumbnail. That's the deeper lesson hiding here: angles capture a triangle's shape, while side lengths capture its size, and the 180° rule is purely about shape. So a triangle drawn on a postage stamp and one chalked across a basketball court can be the exact same triangle as far as their angles are concerned.
Everything on this page assumes your triangle is drawn on a flat surface — paper, a screen, a tabletop, a whiteboard. On any flat surface, the total is exactly 180°, full stop. That's the world you live in for school geometry, and it's the world the 180° rule was made for.
There's a beautiful twist, though. Draw a triangle on a curved surface — say, a giant one on the surface of the Earth, with corners at the North Pole and two points on the equator — and its angles can add up to more than 180°. The curve of the ground bulges the corners open. It's not that the rule is broken; it's that "flat" was a hidden ingredient all along. For every triangle you'll cut, sketch or solve in this lesson, the surface is flat, so the answer stays a rock-solid 180°. Knowing the condition is what turns a memorised fact into real understanding.
The 180° rule isn't a classroom trick to be forgotten after the test — it's load-bearing, sometimes literally holding things over your head.
Builders frame bridges and roofs out of triangles because a triangle is the one shape that can't be pushed out of square — its angles are pinned by its side lengths. When an engineer designs the zig-zag of beams, the angles in each triangle have to add to 180°, so knowing two lets them cut the third beam at exactly the right slant.
Triangles tile a flat surface with no gaps. Slide six identical equilateral triangles around a single point and their corners (60° each) fill 6 × 60 = 360°, a complete turn — which is exactly why that pattern lies perfectly flat. The 180° rule is hiding inside every tiled floor and honeycomb-style design.
It shows up in navigation and surveying too. Split any straight-sided field, plot of land or map region into triangles, measure a couple of angles and one length, and the 180° rule lets you fill in the rest — working out distances and directions you could never reach with a tape measure. Surveyors have mapped whole countries this way, triangle by triangle. The same idea quietly powers the maths behind 3D video games, where every curved hero and sprawling landscape is secretly built from thousands of tiny triangles. One small, unbreakable rule — and an enormous amount of trust placed in it.
Any triangle's three interior angles add to exactly 180° — a straight line, a half-turn.
Know two angles? The third is just 180° minus those two. No measuring needed.
Stretch or shrink a triangle and the total never changes — only the side lengths do.