Shapes are everywhere — in windows, wheels, pizza slices, and the stars in the sky. Let's meet some of them, count their sides, paint them, and play a little game at the end!
A round shape with no corners. Like a pizza!
Three sides. Like a slice of toast.
All four sides the same. Like a chessboard tile.
Two long sides, two short. Like a door.
A square standing on one corner — a kite!
Six sides. Like a honeycomb in a beehive!
A shape that shines! Five points reaching out.
A stretched circle. Like an egg!
Here's a secret every shape knows: the number of sides a shape has is always equal to the number of corners! Triangle: 3 sides, 3 corners. Square: 4 and 4. Hexagon: 6 and 6!
A polygon is just a fancy word for any flat shape with straight sides. Triangles, squares, and hexagons are all polygons. A circle is not a polygon because it has no straight sides — it's a curve!
Pick a color. Tap the shape. Make it yours!
Every building, every bridge, every circuit board starts with a sketch and an angle. Here's everything you need to know — and you can try it live.
When two straight lines meet at a point, the space between them is an angle. The point where they meet is the vertex. Angles are measured in degrees (°). A full turn is 360°.
Small and sharp. Like a pizza slice.
A perfect corner. Like the edge of a book.
Wider than a right angle. Lazy and stretched.
A flat line. Half of a full turn.
Bigger than straight. Long way around.
A complete circle. Back to start.
Drag the slider. Watch the angle — and its name — change live.
A protractor measures angles, a ruler draws straight lines, and a sharp pencil keeps things crisp. Engineers sketch by hand before touching a computer.
Use your ruler. Draw a straight horizontal line. Mark a dot — your vertex.
Line up the center point with your vertex. 0° on the baseline.
Read along the curved edge. Make a tiny mark where it says 60°.
Remove the protractor. Use your ruler to connect vertex through the mark.
Triangles come in flavors. But one rule is sacred: the three angles always add up to 180°. Always.
All angles < 90°
One angle = 90°
One angle > 90°
All sides equal. 60° each.
Two sides equal.
No sides equal.
| Rule | What it says |
|---|---|
| Angles on a straight line | Add up to 180° |
| Angles around a point | Add up to 360° |
| Angles in a triangle | Add up to 180° |
| Angles in a quadrilateral | Add up to 360° |
| Vertically opposite angles | Always equal |
| Alternate angles (parallel lines) | Equal — "Z shape" |
| Co-interior angles (parallel) | Add up to 180° — "C shape" |
| Corresponding angles (parallel) | Equal — "F shape" |
Identify any O-Level graph from its shape, read its equation, spot its features — turning points, roots, asymptotes — and sketch by hand. Live sliders show how parameters change the curve.
The simplest graph of all — a straight line. m is the gradient (steepness), c is the y-intercept (where it crosses the y-axis).
| Gradient | Coefficient of x |
| Y-intercept | The constant c |
| X-intercept | Set y = 0 → x = −c/m |
| Parallel lines | Same gradient (m) |
| Perpendicular lines | m₁ × m₂ = −1 |
Square x and you get a U-shape. The sign of a decides direction: a > 0 opens up (smile), a < 0 opens down (frown).
| Turning point | Min (a>0) or max (a<0). x = −b/(2a) |
| Line of symmetry | Vertical: x = −b/(2a) |
| Y-intercept | The constant c |
| Roots | Solve ax² + bx + c = 0 |
| Discriminant b²−4ac | >0: 2 roots · =0: 1 root · <0: no real roots |
A cubic has an x³ term. The graph can wiggle — turn twice, creating a local max and min. a > 0 rises left-to-right. a < 0 falls left-to-right.
| End behaviour | a>0: up-right, down-left · a<0: reversed |
| Turning points | At most 2 |
| Roots | Up to 3 — crosses x-axis |
| Y-intercept | The constant c |
Divide by x. As x → 0, y → infinity. As x grows big, y → 0. Two curves in opposite quadrants.
| Vertical asymptote | y-axis (x = 0). Never touches. |
| Horizontal asymptote | x-axis (y = 0). Never touches. |
| Quadrants | a>0: Q1 & Q3 · a<0: Q2 & Q4 |
| Symmetry | Rotational, order 2 about origin |
| Roots | None |
Exponentials describe things growing or shrinking at a rate proportional to their size — bacteria, interest, radioactive decay. b > 1: growth. 0 < b < 1: decay.
| Y-intercept | The value a |
| Horizontal asymptote | y = 0 (x-axis) |
| Growth (b>1) | Rises steeply right |
| Decay (b<1) | Falls to right |
| Always positive | If a>0, always above x-axis |
Trig graphs are the waves of mathematics — repeating patterns that describe sound, light, tides, pendulums, and every rotating thing in existence. Once you see the shape, you'll spot them everywhere.
Change the four parameters. Watch the wave transform.
| A (amplitude) | How tall the wave is. Bigger A = taller peaks and deeper troughs. |
| B (frequency) | How squished the wave is. Bigger B = more waves in the same space. |
| C (phase) | Slides the wave left or right. |
| D (vertical shift) | Moves the whole wave up or down. |
| Graph | Range | Period | Notes |
|---|---|---|---|
| y = sin x | −1 ≤ y ≤ 1 | 360° | Smooth wave, crosses origin |
| y = cos x | −1 ≤ y ≤ 1 | 360° | Same wave, starts at y = 1 |
| y = tan x | All real numbers | 180° | Has asymptotes at 90°, 270°, etc. |
Time on x-axis, distance on y-axis. Gradient = speed. Steeper = faster. Flat horizontal = stopped.
On speed-time graphs, gradient = acceleration. Area under = distance travelled. Flat line = constant speed. Upward = accelerating. Downward = decelerating.
Look at the graph. Match it to its equation. One guess per question — choose carefully!
Your progress saves automatically (in this session). Every item is from the official Cambridge 4024 syllabus.