The Angle
An angle is not a shape. It is a rotation. The number tells you how far.
Why does this exist?
Every time a door swings open, a clock hand moves, or a wheel turns — an angle is being made. Angles are not shapes drawn on paper. They are rotations. They measure how far one direction has turned away from another.
That is the only thing an angle does. And it turns out that is enough to hold all of geometry together. Triangles, circles, coordinates, trigonometry — every branch of this course grows from this single idea: a direction, and how far it has turned.
An angle is a turn
Fix one ray pointing right. Now rotate a second ray away from it, starting from the same point.
Canvas 400 420
Scale 52
Grid 0
Origin at 200 240
Pivot O at 0, 0
Point A at 3, 0
Point B at 0, 3
Circle guide center O radius 3
Hide guide
Ray arm1 from O through A
Ray arm2 from O through B
Color arm1 gray
Color arm2 blue
Angle alpha at O from A to B
Color alpha blue
Label alpha "α"
Show angle alpha
Label O "O"
Animate B around guide speed slow
Drag B
The gray ray is fixed. The blue ray rotates. The arc between them — that arc is the angle. The number in the panel is how far the turn has gone, measured in degrees.
Grab B and take control of the turn.
Degrees — a unit of rotation
A full turn — all the way around back to the start — is divided into 360 equal steps. Each step is one degree, written as 1°.
Why 360? Ancient astronomers noticed the sun takes roughly 360 days to complete its path across the sky. They divided the circle into 360 parts to match. The number stuck, and it turns out to be useful: 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12. Almost any equal division of a circle gives a whole number.
A quarter turn is 90°. A half turn is 180° — the two arms point in opposite directions, forming a straight line. A full turn is 360° — you are back where you started.
Feel the landmarks
Drag B around O. Find the special positions.
Canvas 400 380
Scale 52
Grid 0
Origin at 200 200
Pivot O at 0, 0
Point A at 3, 0
Point B at 0, 3
Circle guide center O radius 3
Hide guide
Ray arm1 from O through A
Ray arm2 from O through B
Color arm1 gray
Color arm2 blue
Angle alpha at O from A to B
Color alpha blue
Label alpha "α"
Show angle alpha
Label O "O"
Text at -3.8, -3.0 gray "drag B"
Snap B to Circle guide
Drag B
Notice what happens at 90°: the two arms are perpendicular — they meet at a perfect corner. At 180°: the arms point in opposite directions and the arc flattens into a straight line. At 0° or 360°: the arms coincide and the angle vanishes.
These are not arbitrary. They are the landmarks every angle is measured against.
The five families
Every angle belongs to one of five families, named by where it falls between the landmarks.
Canvas 400 480
Scale 26
Grid 0
Origin at 200 240
# Row 1 — Acute
Point O1 at -1.5, 7.0
Point R1 at 0.0, 7.0
Point L1 at -0.4, 8.1
Segment sa1 from O1 to R1
Segment sb1 from O1 to L1
Angle a1 at O1 from R1 to L1
Color sa1 blue
Color sb1 blue
Color a1 blue
Text at 1.5, 7.3 blue "Acute"
Text at 1.5, 6.7 gray "less than 90°"
# Row 2 — Right
Point O2 at -1.5, 4.0
Point R2 at 0.0, 4.0
Point L2 at -1.5, 5.5
Segment sa2 from O2 to R2
Segment sb2 from O2 to L2
Color sa2 blue
Color sb2 blue
Mark right-angle at O2
Text at 1.5, 4.3 blue "Right"
Text at 1.5, 3.7 gray "exactly 90°"
# Row 3 — Obtuse
Point O3 at -1.5, 1.0
Point R3 at 0.0, 1.0
Point L3 at -2.6, 2.1
Segment sa3 from O3 to R3
Segment sb3 from O3 to L3
Angle a3 at O3 from R3 to L3
Color sa3 blue
Color sb3 blue
Color a3 blue
Text at 1.5, 1.3 blue "Obtuse"
Text at 1.5, 0.7 gray "90° to 180°"
# Row 4 — Straight
Point O4 at -1.5, -2.0
Point R4 at 0.0, -2.0
Point L4 at -3.0, -2.0
Segment sa4 from O4 to R4
Segment sb4 from O4 to L4
Angle a4 at O4 from R4 to L4
Color sa4 blue
Color sb4 blue
Color a4 blue
Text at 1.5, -1.7 blue "Straight"
Text at 1.5, -2.3 gray "exactly 180°"
# Row 5 — Reflex
Point O5 at -1.5, -5.0
Point R5 at 0.0, -5.0
Point L5 at -0.75, -6.3
Circle gc5 center O5 radius 1.5
Segment sa5 from O5 to R5
Segment sb5 from O5 to L5
Arc arc5 on gc5 from R5 to L5
Color sa5 blue
Color sb5 blue
Color arc5 blue
Text at 1.5, -4.7 blue "Reflex"
Text at 1.5, -5.3 gray "180° to 360°"
Hide gc5
Hide O1
Hide R1
Hide L1
Hide O2
Hide R2
Hide L2
Hide O3
Hide R3
Hide L3
Hide O4
Hide R4
Hide L4
Hide O5
Hide R5
Hide L5
Row 2 shows no arc — just the corner square. That square is the universal mark for exactly 90°. You will see it throughout this course wherever a right angle appears.
Row 5 — the reflex angle — is the one that surprises students. It is a real angle. It just opens the wide way around. You will meet it again when we study circles and arcs.
The names
An acute angle is less than 90°. The word comes from Latin for sharp — the opening is narrow.
A right angle is exactly 90°. The two arms are perpendicular. This angle is so important it has its own mark: the small square at the vertex.
An obtuse angle is between 90° and 180°. From Latin for blunt — the opening is wide.
A straight angle is exactly 180°. The two arms point in opposite directions. It looks like a line — because it is one.
A reflex angle is between 180° and 360°. It is the arc that goes the long way around.
Explore all five yourself
One fixed arm. One arm you control. Drag B all the way around — watch the degree value and the name change as you cross each landmark.
Canvas 400 400
Scale 52
Grid 0
Origin at 200 210
Pivot O at 0, 0
Point A at 3, 0
Point B at 0, 3
Circle guide center O radius 3
Hide guide
Ray arm1 from O through A
Ray arm2 from O through B
Color arm1 gray
Color arm2 blue
Angle alpha at O from A to B
Color alpha blue
Label alpha "α"
Show angle alpha
Show classification alpha
Label O "O"
Text at -3.8, -3.5 gray "drag B all the way around"
Snap B to Circle guide
Drag B
The name changes at the exact degree boundaries — 90°, 180°, 360°. Notice how Straight feels like a line, because it is one. Notice how Reflex feels like the angle is opening the wrong way — it is going the long way around.
The seed planted here
360° is a full turn — you arrive back at the start.
That is not just a fact about angles. It is the definition of a circle. A point that turns through 360° around a fixed centre traces a circle. Angles and circles are the same idea viewed from different positions.
In two lessons from now, we will see exactly why.