MSP:MiddleSchoolPortal/Triangles: From Three Sides

From Middle School Portal

Triangles: From Three Sides - Introduction

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At least in part, geometry is an exploration of shapes, and the triangle, the simplest of polygons, provides a surprising variety of explorations for the middle school learner. As a geometric figure, it offers opportunities to study fundamental concepts of congruence and similarity as well as that critical discovery of ancient mathematicians, the Pythagorean theorem. As a source of problems for classroom practice, triangles appear at work in the context of measurement and construction. And triangles are featured in famous roles, such as the Sierpinski triangle and Pascal’s triangle.

Each set of resources in this publication opens to one of these distinct views of the triangle. We begin with background information for the teacher, followed by online resources on triangles as geometric figures, as workers in solving real-world application problems, and as well-known geometric objects. We hope your classes will enjoy all three sides of the triangle!

Background Information for Teachers

Although geometry is not usually associated with middle school math, teachers are increasingly asked to introduce the subject informally. Triangles from Three Sides includes topics that you may want to investigate further, such as proof and similarity, in introducing your students to geometry. These resources offer background on the topics.

The Pythagorean Theorem In an online course for K-8 teachers, you will look at a few proofs and several applications of one of the most famous theorems in mathematics. The course also offers well-illustrated, classroom-worthy applications of the theorem. This free course is part of Annenberg Media’s Learning Math: Geometry.

Similarity In this workshop session, you will explore similar triangles, geometric similarity, and basic ideas in trigonometry. Ideas are explained through hands-on interactive work as well as through diagrams and text. Practice problems, with accompanying solutions, help build the learning. The workshop is from Learning Math: Geometry, a free online course for K-8 teachers.

Pythagorean Theorem More than 70 proofs of the Pythagorean theorem! Not as dull as you might think! Many of the proofs present diagrams and algebraic outlines, but several are presented through interactive applets or as slide shows. One colorful diagram offers a "proof without words." There is interesting work here for students as well as their teachers. This page is from Interactive Mathematics Miscellany and Puzzles.

An excellent book on teaching geometry at the middle school level, Navigating Through Geometry in Grades 6-8, is available from the National Council of Teachers of Mathematics. The emphasis is on geometric thinking and hands-on exploration. An accompanying CD-ROM offers interactive activities as well as further readings for teachers and student activity sheets. Find ordering information at

Triangles as Geometric Figures

These resources offer online opportunities to explore several aspects of the triangle, from ideas of congruence to reflections to relationships among the triangle’s angles and sides. The final three resources deal with the the right triangle and the theorem that has proved so useful in measurement throughout centuries.

Triangle Geometry: Angles This interactive exploration of triangles begins at the beginning—with angles and their classification. Students can practice their understanding and then move on to construct triangles and consider the sum of the angles of any triangle. Finally, they explore the special relationship among the sides of a right triangle—the Pythagorean theorem, demonstrated here through a Java slide show.

Congruence of Triangles (Grades 6-8) With this virtual manipulative, students arrange sides and angles to construct congruent triangles. They drag line segments and angles to form triangles and flip the triangles as needed to show congruence. Options include constructing triangles given three sides (SSS), two sides and the included angle (SAS), and two angles and an included side (ASA). But the option that will motivate most discussion is constructing two triangles given two sides and a nonincluded angle (SSA). The question in this case is: Can you find two triangles that are not congruent?

Two Equilateral Triangles If your students have learned the theorems on triangle congruence, you may be looking for a challenging application. Here students are presented with an intricate figure showing two overlapping equilateral triangles. Because this resource is an applet, students can rotate the figure and easily see that two triangles in the figure are congruent. The challenge is to prove the triangles are congruent! The applet is from Manipula Math with Java.

Platonic Solids (Grades 6-8) Using this virtual manipulative, students can examine the five platonic solids, each a polyhedron with identical regular polygonal faces. Of the five such solids discovered by the Greeks, three consist entirely of congruent triangles: the tetrahedron, the octahedron, and the 20-sided icosahedron. The applet allows students to rotate the solids, mark each face, edge, and vertex, and study how simple triangles make up these unique 3-dimensional figures.

Transformations—Reflections Here students can manipulate one of six geometric figures on one side of a line of symmetry and observe the effect on its image on the other side. A triangle may be selected and then translated and rotated. The line of symmetry can be moved as well, even rotated, giving more hands-on experience with reflection as students observe the effect on the image of the triangle. This applet is one of many from the National Library of Virtual Manipulatives, a collection of activities that encourage middle school students to explore mathematical relationships. Each activity offers how-to’s as well as instructional ideas for the teacher.

Inner Center This applet enables students to see a fact about triangles, and then try to put the fact into words. Each triangle starts out with three small circles, one in each corner. As the student moves each circle to make it tangent to all three sides, the circle expands its radius depending on where it is moved. In this way, the student finds that no matter where the three circles start out, when they are tangent to all three sides of the triangle, they all end up in the same location. The center of that circle is called the inner center of the triangle. In this demonstration, the student sees that there is only one such center in any triangle.

Pythagoras’ Theorem In a quick but thorough demonstration, this applet explains each move in one proof of the theorem. The student can not only follow the steps but also back up to see an earlier step whenever necessary. Each step is a visual movement as well as an explanation in words. This page appears in Interactive Mathematics Miscellany and Puzzles.

Three Squares: When Do Two Squares Make a New Square? This activity opens with a diagram of two unequal squares and challenges students to find a way to construct a third square from the two. Students are encouraged to model the problem using squares of paper. The activity cleverly leads into a hands-on application of the Pythagorean theorem. The solution explains the puzzle through text and diagrams. Related questions introduce practice on the Pythagorean triples.

Triangles at Work

Triangles do their share of work in construction as well as in measurement of distances, areas of irregular figures, and even the circumference of the Earth. These exploratory problems will give your students an insight into how triangles work in the everyday world.

Triangle Explorer Triangles are used in finding the area of irregular figures, which can be broken into rectangles and triangles and then handled in a piecemeal fashion. Also, the formulas for the areas of regular hexagons, octagons, and so forth are based on their triangular component. The formula for the area of a triangle is memorized by students in elementary school. What this applet brings to their learning is a way to consider the area of a triangle visually. Students can actually derive the formula themselves when the exercise is limited to the "easy" level, which shows only right-angled triangles. Higher levels of difficulty challenge students to find the area of scalene triangles, given helpful hints.

Measuring by Shadows In a letter to Dr. Math, a student asks, “How can I measure a tree using its and my shadows?” What follows is an explanation of how to draw the similar triangles and how to calculate the ratio of the shadow lengths. This is no better than the explanation you have given in class, but sometimes one more voice can make the difference for the struggling student.

Measuring the Height of a Building Using Shadows An out-of-the-ordinary question on the same subject: “What time of day is best to use a shadow to measure the height of a building by using triangles?” Dr. Math answers this one again, bringing in the critical consideration of error estimate. The question is offered here as a real-world problem for your class.

Go Fly a Kite! Looking for a challenging problem? "If you're flying a kite 75 yards away from 70-foot tall trees, how much line would it take to get to the top of the trees?" Your students can apply the Pythagorean theorem here, but the next level of the problem requires basic ideas of trigonometry: "What if the kite moved up so that it was 65 yards away? What would the angle be?" There are more levels to the problem, plus solutions given by students.

Measuring the Circumference of the Earth This is an awesome Internet project! The site presents all the information you need to re-create the measurement of the circumference of the Earth as done by the Greek librarian Eratosthenes more than 2,000 years ago. Every step in Eratosthenes’ method is explained, illustrated, and diagrammed; applets and handouts reinforce the learning. You can sign up your class to participate in the annual Noon Day Project. Students from around the country take shadow measurements at high noon on a designated day in March; these measurements are posted online and used to calculate the circumference. A teacher’s guide outlines the procedure, from posting measurements to finding a class estimate of the Earth’s circumference.

Triangles in the Limelight

Certain triangles have become famous! Most accessible to middle school students are the Sierpinski triangle, a fractal students can create themselves, and Pascal’s triangle, a source of patterns and real-world applications. These resources explore the iteration underlying fractals and the significant patterns in both triangles. The resources also offer problems that may engage your students in critical thinking.

Fractals: What Comes Next? Here is a problem based on the Sierpinski triangle: What is the relationship between the number of triangles and the sum of the triangle perimeters in each of the first three iterations of the Sierpinski triangle fractal? In other problems, students are challenged to find the amount of paint needed to cover the triangles created in the first few iterations of the fractal and then formulate a general rule. The activity also includes information about the fractal, such as the fact that it was named after the Polish mathematician Waclaw Sierpinski, who developed it around 1915.

Pascal's Triangle (Grades 9-12) Although this resource is marked for grades 9-12, the applet offers activities for middle school students as well. The applet shows the first 27 rows of Pascal’s triangle, more than sufficient for students to determine the rule behind the order of numbers in the triangle. They can also color in, one by one or all at once, the multiples of 2, or 3, or 4, and so forth. Older students can discover the relationship between the number of the row in the triangle and the sum of the numbers in that row. Included are instructions for using the manipulative, background on the triangle, and teaching suggestions.

Coloring Multiples in Pascal's Triangle An excellent exercise in finding multiples! The applet shows up to 15 rows of Pascal’s triangle, sufficient to contain numbers in the thousands. Students color numbers in the triangle by rolling a number and then clicking on all entries that are multiples of the number rolled, thereby practicing multiplication tables and division. Each entry is immediately noted as correct or incorrect, and the number of multiples remaining to find is given. Surprising are the number patterns created by the multiples!

The Smithville Families In this lesson from PBS Teacher Source, students consider the total number of possible girl/boy combinations in a five-child family. The lesson begins with a review of Pascal’s triangle and the creation of its first eight rows. Next, the Smithville families are generated by each group of students tossing a coin five times--heads for a girl, tails for a boy. Patterns of gender are examined and found, surprisingly, to have a relationship to the number sequences of Pascal’s triangle. The site provides a detailed procedure for the lesson, questions for class discussion, and worksheets. You can watch an online video of the lesson at

SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities. Visit the virtual learning experience on Triangles & Circles and The Pythagorean Theorem.


The FunWorks Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

NCTM Standards

When students arrive at middle school prepared to build on their informal knowledge of geometric figures, they can profit from the rich contexts provided by triangles for the development of visualization, practical measurement, and mathematical reasoning.

The NCTM Geometry Standard for grades 6-8 states that "middle grades students should explore a variety of geometric shapes and examine their characteristics" (NCTM, 2000, p. 233). As envisioned by the Principles and Standards for School Mathematics, students at this level would gain experience in working with concepts of congruence and similarity. In these investigations, students would connect mathematics to the world they live in as they study the geometry embedded in nature, the sciences, measurement, and even art.

Students would also apply geometry in problem solving, sharpening fundamental skills through "visualization, spatial reasoning, and geometric modeling to solve problems" (p. 237). They would explore flips, turns, and slides and their effects on geometric objects, because experiences with these transformations "should help students develop a strong understanding of line and rotational symmetry, scaling, and properties of polygons" (p. 235).

If you would like to see the goals of the Geometry Standard set out in detail, go to


National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Author and Copyright

Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

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Copyright January 2007 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.