MATHEMATICS LAB · GEOMETRY

Geometry Lab

Explore geometry in the Cartesian plane and in 3D space, from points to solids. Choose a tool, adjust the figure and discover how the results are obtained through dynamic constructions and explained formulas.

INTERACTIVE TOOLS

What would you like to explore?

Eighteen interactive tools, each with its own graph, formulas and ready-to-try examples.

EUCLIDEAN AND ANALYTIC GEOMETRY

Triangles and Notable Points

Build a triangle by dragging its vertices. Explore medians, altitudes, angle bisectors, perpendicular bisectors, and its incircle and circumcircle.

Centroid, orthocenter, incenter and circumcenterConstructions and MathJax calculationsFigures editable in real time

Open the Triangle Lab

ANALYTIC GEOMETRY

Lines in the Plane

Enter two line equations and view their graphs. Calculate intersections, distances, projections, angles, parallelism and perpendicularity.

Slope-intercept, implicit and vertical equationsAngles and intersection pointsDistances and orthogonal projections

Open the Lines Lab

PLANE GEOMETRY · POLYGONS

Polygons

Construct regular and concave polygons, and quadrilaterals, by dragging their vertices. Study sides, angles, diagonals, apothems, circles and areas using the shoelace formula.

Regular and irregular polygons with draggable verticesQuadrilaterals: diagonals, altitudes and symmetryFormulas and ready-to-try examples

Open the Polygons Lab

CONIC SECTIONS · ANALYTIC GEOMETRY

Circles

Enter a circle equation and explore its center and radius, intersections with lines or other circles, and tangents from a point.

Standard and general equationsIntersections and relative positionsTangents with step-by-step constructions

Open the Circles Lab

CONIC SECTIONS · ANALYTIC GEOMETRY

Parabolas

Visualize a parabola’s vertex, focus, directrix and axis. Explore its opening direction, line intersections, tangents and normal.

Vertex and polynomial formsDraggable vertex, focus and point TExplained intersections and tangents

Open the Parabolas Lab

CONIC SECTIONS · ANALYTIC GEOMETRY

Ellipses

Explore an ellipse by changing its center and semiaxes. Examine foci, directrices, eccentricity, line intersections and tangents.

Standard and general equationsDraggable foci, semiaxes and pointsFocal properties and explained calculations

Open the Ellipses Lab

CONIC SECTIONS · ANALYTIC GEOMETRY

Hyperbolas

Change a hyperbola’s center and semiaxes. Discover its two branches, asymptotes, foci, directrices, intersections and tangents.

Standard and general equationsAsymptotes and foci with dynamic constructionsIntersections, tangents and normals

Open the Hyperbolas Lab

CONICS · GENERAL EQUATION

General and Rotated Conics

Study Ax²+Bxy+Cy²+Dx+Ey+F=0. Classify the curve, rotate its axes, remove the mixed term and find its standard form, including degenerate cases.

Rotated ellipses, hyperbolas and parabolasDiagonalization and matricesFoci, vertices, asymptotes and examples

Open the General Conics Lab

TRANSFORMATIONS · PLANE GEOMETRY

Geometric Transformations

Compare a figure with its transformed image. Build translations, rotations, reflections, point symmetries, homotheties and compositions of transformations.

Isometries and homothetiesDraggable centers and axesStep-by-step animationMatrices and explained calculations

Open the Transformations Lab

ANALYTIC GEOMETRY · THREE-DIMENSIONAL SPACE

3D Analytic Geometry

Rotate a three-dimensional Cartesian coordinate system. Study points, vectors, lines and planes, with explained distances, intersections and angles.

Rotatable 3D view and XY, XZ, YZ projectionsIntersecting, parallel and skew linesDihedral angles, perpendiculars, projections and cross products

Open the 3D Geometry Lab

SOLID GEOMETRY · THREE-DIMENSIONAL SPACE

Geometric Solids

Explore cubes, right and oblique prisms, pyramids, cylinders, cones and spheres. Animate a right prism becoming oblique, verify Cavalieri’s principle and study horizontal and inclined cross-sections.

Rotatable 3D visualizationSurface areas and volumes explainedHorizontal and oblique sections24 ready-to-use examplesOblique prisms and Cavalieri animation

Open the Solids Lab

SOLID GEOMETRY · FLAT NETS

Animated Nets of Solids

Unfold a cube, prism, pyramid or truncated pyramid by rotating its faces. Explore the nets of cylinders, cones and truncated cones.

Animation, pause and manual progressPolyhedral nets, sectors and frustumsArea formulas and ready-to-try examples

Open the Solid Nets Lab

ANALYTIC GEOMETRY · 3D SURFACES

Quadric Surfaces

Explore ellipsoids, paraboloids and hyperboloids in 3D. Move cutting planes parallel to XY, XZ and YZ, identifying ellipses, parabolas, hyperbolas and degenerate sections.

Five families of quadric surfacesSections, semiaxes and areas explainedRotatable view and ready-to-use examples

Open the Quadrics Lab

ANALYTIC GEOMETRY · GENERAL 3D QUADRICS

General and Rotated Quadrics

Study quadratic equations in 3D with mixed xy, xz and yz terms. Diagonalize the symmetric matrix and examine the principal axes and plane sections of rotated surfaces.

Ten coefficients and a 3×3 matrixFull classification and degenerate casesAnimated axes, sections and examples

Open the General Quadrics Lab

ANALYTIC GEOMETRY · PARAMETRIC SURFACES

3D Parametric Surfaces

Explore tori, generalized cylinders, paraboloids, catenoids and other surfaces of revolution. Change parameters and inspect sections and tangent vectors.

Six surface families and eight ready-to-use examplesMovable point, tangents and normalSections, areas and volumes explained

Open the Parametric Surfaces Lab

DIFFERENTIAL GEOMETRY · SPACE CURVES

3D Parametric Curves

Explore helices, circles, skew cubics, Lissajous curves and torus knots. Animate point P and examine the Frenet frame, curvature and torsion.

Six curve families and eight clickable examplesTangent, normal, binormal and osculating planeArc length, curvature and torsion

Open the Parametric Curves Lab

DIFFERENTIAL GEOMETRY · SURFACES

Differential Geometry of Surfaces

Study tangent planes and curvatures of spheres, cylinders, tori, paraboloids and catenoids. Move P and watch the normal and principal directions change.

First and second fundamental formsGaussian, mean and principal curvatureNine ready-to-use examples with MathJax calculations

Open the Differential Geometry Lab

EUCLIDEAN GEOMETRY · LOCI AND CONSTRUCTIONS

Geometric Loci and Constructions

Construct perpendicular and angle bisectors, verify distance ratios, generate parabolas, ellipses and hyperbolas as loci, and explore pencils of lines.

Bisectors and Apollonius circlesFocal loci and concurrent or parallel line families13 examples, movable point and explained calculations

Open the Geometric Loci Lab

Constructions and calculations, not just answers

Each tool updates its graph as you change the parameters and explains the mathematical steps. Existing features remain available in their respective labs.