Geometry Simulations
Introduction
Below are the 8 GeoGebra 3D simulations that were custom built by GeoGebra for this project. You can interact with them on 2D devices like laptops, phones, or tablets by clicking on each image. You can also download our AR or VR device builds. Below each simulation is a list of some conjectures that could be explored with that simulation for middle or high school geometry.
Geometry simulations
parallel lines
A simulation opens with a transversal cutting across two parallel lines. Students can manipulate all the lines (in three dimensions), but the two parallel lines stay parallel and co-planar, and the transversal line remains on the same plane as the parallel lines. Students can toggle on/off having angle measurements shown. In Lineland, there does not need to be any functionality where players can draw new lines or points themselves.
conjectures
When two parallel lines are intersected by a transversal, alternate interior angles are congruent.
If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are congruent.
If two parallel lines are cut by a transversal, the pairs of corresponding angles are congruent.
Pairs of angles across from each other formed by 2 lines intersecting are congruent.
Triangle
A simulation opens up with a triangle. The triangle can be manipulated in three dimensions (manipulate vertices individually or rotate/translate/dilate the whole triangle), and measurements for angles/sides can be toggled on and off. Vertices must remain coplanar. Functionality includes the ability to draw line segments and points (e.g., draw the vertical height of the triangle that is perpendicular to the base; draw a midsegment), and toggle on the display of appropriate measurements. Functionality includes the ability to construct side and angle bisectors of the triangle. Functionality to rotate triangle by a certain number of degrees about a center, and reflect the triangle over a line.
conjectures
An angle bisector of a triangle also bisects the opposite side.
Given that you know the measures of all three angles of a triangle, there is NOT one unique triangle that can be formed with these angle measurements.
The segment that joins the midpoints of two sides of any triangle is parallel to the third side.
Any side of a triangle must be shorter than the sum of the other two sides.
If two sides of a triangle are congruent, the angles opposite those sides are congruent.
parallelogram
A simulation opens up with a parallelogram. Opposite sides stay parallel, but the parallelogram can otherwise be manipulated in three dimensions (dilation, rotation, translation, change angles, extend/shrink one set of sides) as long as the vertices stay coplanar. Measurements for angles/sides can be toggled on and off. Functionality includes the ability to draw line segments and points (e.g., draw the diagonals of the parallelogram, or draw a vertical height of the parallelogram that is perpendicular to the base), and toggle on the display of appropriate measurements.
Buttons to transform parallelogram into square, rhombus, rectangle.
conjectures
If the diagonals in a parallelogram are congruent then the parallelogram is a rectangle.
The diagonals of a rhombus bisect the angles at all four vertices.
Opposite angles in a parallelogram are congruent.
circle
A simulation opens up with a circle. The circle can be manipulated in three dimensions (made bigger/smaller, rotated), and measurement of circumference/radius can be toggled on and off. All points on the circle must remain co-planar. Functionality includes the ability to draw line segments and points (e.g., draw a diameter, a central angle, an inscribed angle, or a chord; show an arc length), and toggle on the display of appropriate measurements. Functionality includes the ability to draw perpendicular bisectors of lines.
conjectures
The perpendicular bisector of any chord always goes through the center of the circle.
The measure of inscribed angles that intersect the same arc is equal.
Parallel lines intersecting a circle intercept congruent arcs.
The measures of opposite angles of a cyclic quadrilateral add up to 180.
Sphere
A simulation opens up with a sphere. The sphere can be translated and dilated using gestures. Functionality to draw points, radii, diameters, and great circles on the sphere. Ability to draw lines/planes that are tangent to the sphere.
conjectures
It is always possible to find a great circle through any two points in a given sphere.
There is only one unique great circle through any two points in a given sphere (FALSE)
For any sphere section, its lateral surface will equal that of the cylinder with the same height as the section and the same radius of the sphere.
A line intersects a sphere at most 2 points.
The ratio of the surface area of a sphere to its volume is less than the ratio of the surface area of a cube to its volume. In other words, a sphere holds more inside of it for less packaging outside.
Cylinder
A simulation opens up with a right cylinder. The cylinder can be manipulated in three dimensions using gestures (dilation, rotation, translation), but it stays a right cylinder. It can also be resized with respect to one dimension (e.g., made taller or fatter). Measurements of height, radius, surface area, and volume can be toggled on/off. Functionality to inscribe a sphere in a cylinder. Functionality to unfold the cylinder into its net.
conjectures
Given a sphere inscribed in a cylinder, the surface area of the sphere is equal to the lateral surface area of the cylinder.
Given a sphere inscribed in a cylinder, the sphere has ⅔ the volume of its circumscribing cylinder.
Given a cylinder with radius r and height h, the cylinder can be unrolled to include a rectangle with length h and width 2𝝅r
prism
A simulation opens up with a cube. The cube can be manipulated in three dimensions using gestures, but it stays a cube. Measurements of side lengths, surface area, and volume can be toggled on/off. Functionality to unfold the cube into its net.
conjectures
If the length, width, and height of a cube are each doubled, then the volume increases by a factor of 6. (False)
If the length, width, and height of a cube are each doubled, then the volume increases by a factor of 8 (i.e., is multiplied by 8).
Parallel cross-sections of a prism are congruent (False if not parallel to base).
The lateral surface area of a cube is equal to the perimeter of a side multiplied by the height.
Along with being unrolled into a “cross” shape, a cube could also be unrolled into a “T” shape and into a straight row of squares.
pyramid/Cone
A simulation opens up with a right pyramid or cone. The pyramid/cone can be manipulated in three dimensions using gestures (dilation, rotation, translation), but it stays a pyramid/cone. It can also be resized with respect to one dimension (e.g., made taller or fatter). Measurements of height, sides of the base (or radius If a cone), surface area, and volume can be toggled on/off. Functionality to inscribe a pyramid/cone in a cube or cylinder, and to move the apex such that the pyramid/cone is not a right pyramid/cone. Functionality to fill the volume of the pyramid/cone to some fraction (e.g., half of the total height).
conjectures
Take a regular pyramid/cone and push the sides against the interior walls of a cube so you make an oblique pyramid/cone, then the volume is unchanged.
Parallel cross-sections of a pyramid are similar.
The volume of a pyramid/cone is ⅓ the volume of a prism/cylinder with the same base and height.
In a pyramid, the number of vertices is always equal to the number of faces.
Given a cone with radius r and height h, the cone can be unrolled into a circle with radius r and a triangle with height h.