Geometry
Geometry: 49 free interactive math resources. Shapes, size, and spatial relationships — explore by dragging figures.
Guide
About this topic
The geometry pillar gathers interactives for triangles, circles, coordinate geometry, and transforms. Drag figures first, then write formulas.
Topic map
- Triangles: angle sum, similarity, Pythagoras
- Circles: area intuition
- Coordinate geometry and symmetry
- 3D volume models when available
Subtopics
Key formulas
a² + b² = c²πr²Triangle angles sum to 180°
FAQ
Why learn geometry interactively?
Dragging vertices shows how sides and angles co-vary.
What does the circle-area viz teach?
Rearranging sectors into a rectangle derives πr².
How do I write the formula afterward?
Open the equation editor or symbol sheet.
Resources in this topic
49 free interactive resources in 7 groups
Triangles
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Triangle angle sum
Drag any vertex — the interior angles always sum to 180°.
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Exterior angle of a triangle
See that an exterior angle equals the sum of the two remote interior angles.
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Equilateral triangle
Equal sides, 60° angles; altitude, median, and bisector coincide.
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Isosceles triangle
Equal sides ↔ equal base angles; apex bisector, median, and altitude coincide.
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Right-triangle properties
See that a leg opposite 30° is half the hypotenuse.
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Congruent triangle criteria
Explore SSS, SAS, and ASA corresponding parts.
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Pythagorean theorem
Drag a right triangle and watch the square areas satisfy a² + b² = c².
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Converse of Pythagoras
Use side lengths to decide whether a triangle is acute, right, or obtuse.
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Angle bisector property
Points on the bisector are equidistant from the sides.
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Perpendicular bisector property
Points on the perpendicular bisector are equidistant from the endpoints.
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Similarity & congruence models
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Similar triangles
Drag vertices and watch corresponding side ratios.
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AA similarity
Two triangles with two pairs of equal corresponding angles are similar.
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Similarity models (A / figure-8)
Proportional segments in the A-frame and figure-8 configurations.
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Circle of Apollonius
Locus of points with PA/PB = k (constant) is a circle.
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Apollonius circle applications
On a circle, construct M so k·PB becomes PM; minimize PA + k·PB.
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Homothety locus (“melon–bean”)
Driver on a circle; follower at fixed ratio about A also traces a circle.
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Homothety variant (rotate + scale)
Follower from rotation plus scaling — locus remains a circle.
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Weighted path minimum (Snell-type)
Minimize AP + k·PB for k < 1 (different “costs” on two segments).
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Circles
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Inscribed angle theorem
Drag points on a circle: an inscribed angle is half the central angle on the same arc.
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Tangent properties
A tangent is perpendicular to the radius at the point of contact.
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Perpendicular chord theorem
A diameter perpendicular to a chord bisects the chord (and its arcs).
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Circle area
Slice a circle into sectors and rearrange toward a rectangle: S = πr².
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Hidden circle I: fixed chord, fixed angle
Points seeing a fixed segment at a fixed angle lie on an arc.
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Hidden circle II: angle in a semicircle
Right angles subtended by a segment lie on the circle with that diameter.
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Circular-track pursuit
On a loop, the faster runner gains one full lap to catch up.
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Polygons & parallel lines
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Parallelogram properties
Drag vertices: opposite sides parallel and equal, opposite angles equal, diagonals bisect.
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Rectangle, rhombus, square
Compare side, angle, and diagonal features of special parallelograms.
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Interior angle sum of a polygon
Split from one vertex into triangles to see (n − 2)×180°.
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Parallel lines: tests and properties
Drag a transversal and compare corresponding, alternate interior, and consecutive interior angles.
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Transforms & symmetry
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Reflection (axial symmetry)
Drag the mirror line and figure to see corresponding points.
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Translate, rotate, scale
Compare translation, rotation, and homothety images.
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Two moving points on a line
Two points move on a number line; track positions and distance vs time.
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Shortest path to a river (reflection)
Use reflection to minimize AP + PB.
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Straight-line pursuit
Faster runner catches a slower one: positions vs catch-up time.
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Space & coordinate geometry
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Orthographic views
Relate a solid to front, left, and top projections.
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Cube nets
Watch a cube unfold along edges into a plane net.
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Rectangular prism volume
Volume = length × width × height — pack unit cubes.
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Coordinate plane intro
Quadrants, coordinates, and where a point sits.
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Definition of an ellipse
Locus of points with PF₁ + PF₂ = 2a (constant).
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Definition of a parabola
Equidistant from a focus and a directrix.
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Plane vector operations
Vector addition (parallelogram rule) and scalar multiplication.
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Clock-hand angles
Minute hand 6°/min, hour hand 0.5°/min — find the angle between them.
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Rectangle area
Area = length × width — tile with unit squares.
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Exam geometry: folding a rectangle
Fold rectangle along BE and study image C′ and equal lengths.
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Surfaces z = f(x, y)
3D graphs of z = f(x, y): rotate to see peaks, valleys, and level sets.
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Cylinder and cone volume
Compare V=πr²h with V=(1/3)πr²h — same base and height, the cone is one-third of the cylinder.
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Laws of sines and cosines
On the circumcircle: the chord midpoint splits a, and a right triangle shows sin A = (a/2)/R.
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Sphere surface area and volume
Sphere: area 4πr², volume (4/3)πr³ — double r → area ×4, volume ×8.
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