Geometry Labs


by Henri Picciotto

See below for revisions, extensions, connections, and corrections.

Geometry Labs is a book of hands-on activities that use manipulatives to teach important ideas in geometry. These 78 activities have enough depth to provide excellent opportunities for discussion and reflection in both middle school and high school classrooms.

  • Middle school teachers will find many labs that help prepare students for high school geometry by getting them to think visually and become familiar with fundamental concepts, figures, and vocabulary.
  • Teachers of high school geometry —whether traditional, inductive, or technology-based— will find many labs that approach key topics in their curriculum from a different point of view. In some cases, the labs in this book can replace corresponding lessons in your textbook. Other labs can be used to preview or review material that you teach in more traditional ways.
  • Trigonometry teachers will find a new approach, using the CircleTrig geoboard, to introduce basic right-triangle and unit-circle trigonometry concepts. This approach also makes trigonometry accessible to younger students.

You may download The whole book (3.2 MB) or individual sections
for your non-commercial use.

For a preview of some labs from the book, see:


The book uses tangrams, pattern blocks, cubes, mirrors, plus two manipulatives I designed, the CircleTrig Geoboard, and the Geometry Labs Template.


CircleTrig Geoboard

On one side, an 11 by 11 geoboard. Useful for many things, especially activities about slope, area, distance, the Pythagorean theorem, and simplifying radicals. On the other side, a circle geoboard, with radius 10 cm, and a 360° protractor. Useful for activities leading to the inscribed angle theorem, and to introduce basic trig, both on the unit circle (the unit being the decimeter) and in right triangles. Most of these activities of course can be done on paper, but it's more fun to start on the geoboard.

Geoboard Activities
Virtual 11 by 11 Geoboard | Virtual Circle Geoboard

(Where to purchase the CircleTrig Geoboard.)


Geometry Labs Template

Eight different triangles, ten different quadrilaterals, seven different regular polygons, and all the pattern blocks — shapes you need for basic geometry activities. The template is great for exploring the Archimedean tilings. It also includes centimeter and inch rulers, plus a numberless protractor. You can keep it in a 3-ring binder. See it as a PDF.

The template is also useful in introducing translations, reflections, and rotations. See this worksheet.

(Where to get the Geometry Labs Template.)


Individual Sections

Connections, Corrections, Extensions, and Revisions

You are welcome to contribute to this page. Send me e-mail.

Table of contents, Introduction
Section 1: Angles
Google slide for group record-keeping of Lab 1.1 (Angles Around a Point)
Possible intro to Lab 1.7 (Angles and Triangles in a Circle), by Michael Pershan.
Consider doing Lab 1.10 (Soccer Angles) before Lab 1.7, as suggested by Alex Wilson.
Extension of Lab 1.10 (Soccer Angles), using interactive geometry software.
Section 2: Tangrams
Possible preview: Tearing Tangrams
Virtual tangrams
Section 3: Polygons
Correction to the Lab 3.9 solution, by Austin Shapiro and his students at Proof School, SF
(Lab 3.9 itself has now been corrected. The above only applies if you used its original version.)
Section 4: Polyominoes
Virtual Grid Paper
Extension of Lab 4.7 (Minimum Covers), on my blog.
A book of Polyomino Lessons (Some overlap with this section.)
More on polyforms: Geometric Puzzles in the Classroom
Geometric Puzzles Home Page
Section 5: Symmetry
Slight edit of Lab 5.6, an attempt to make the table headings easier to understand.
Online applet to support Lab 5.6: Symmetry with Pattern Blocks
Some Lab 5.6 photos for discussion.
Extension of Lab 5.6: Cover the Big Dodecagon with pattern blocks, + connections.
Symmetry Home Page
Online unit: Symmetry Labs
Section 6: Triangles and Quadrilaterals
Lab 6.1: Use patty paper! Have students share the work and report what they discover.
John Golden's GeoGebra version of Lab 6.4: Making Quadrilaterals from Triangles
Section 7: Tiling
Tiling Home Page
Virtual tiles: Triangles and Quadrilaterals | Regular Polygons
Section 8: Perimeter and Area
Connection to Lab 8.4 or 8.6: Heilbronn's Triangle
Extension to Lab 8.5: Inscribing Geoboard Squares in Polyominoes
Extension to Lab 8.6: Proof of Pick's Formula
Section 9: Distance and Square Root
Revision and extension of Labs 9.1 and 9.6: Taxicab Geometry
Preview or review of Lab 9.2: Pythagorean Theorem home page
Desmos version of Labs 9.3 and 9.4, by Liz Caffrey
Extension to Lab 9.4: Ariadne's String
Section 10: Similarity and Scaling
Extension to Lab 10.2, #4: Include "tilted" rectangles in your search.
Connection to Lab 10.2, #5: No Three on a Line
Different version of Lab 10.4: Rep-Tiles, including an extension.
Extension to Section 10: Shrinky Dinks, by Rachel Chou (requires cm grid paper)
Section 11: Angles and Ratios
Trig intro: I now do 11.2 with the 10-cm circle before 11.1, and 11.5 before 11.4 (more info).
Connection and slides for Lab 11.8: π for circles, π for polygons?
Notes, Special Papers, and Answers
Virtual Grid Paper | Virtual Geoboard | Virtual Circle Geoboard
10cm Circle (PDF), to do Chapter 11 on paper.
10cm Circle in Cabri, and Sketchpad. (The latter contributed by Roger Gemberling.)


Related pages on this site:
Pattern Blocks
Virtual Tangrams
Virtual Grid Paper
Virtual Geoboard
Geometric Puzzles
Middle School
Geometric Transformations