Hands-on Activity Using the Pythagorean Theorem to Catch the Perfect Radio Waves!

Quick Look

Grade Level: 11 (9-11)

Time Required: 4 hours 30 minutes

(55 minutes on five different days)

Expendable Cost/Group: US $0.50

This activity requires some non-expendable (reusable) supplies, such as sensors, multimeters, screwdrivers, battery holders, batteries, wire connectors, alligator clips, suction cup hooks, and screw eyes, ~$65 per group; see the Materials List for details.

Group Size: 3

Activity Dependency: None

Subject Areas: Geometry

The image shows clip art of a satellite orbiting moon capturing radar images. Text on the image says: "Enceladus 'E-16' Flyby Radar Looks at Enceladus" and the image is dated Nov. 6, 2011.
NASA's Cassini spacecraft acquires the first detailed radar images of Saturn's moon Enceladus during a 2001 flyby.
Copyright © NASA http://www.nasa.gov/audience/foreducators/plantgrowth/reference/Eng_Design_5-12.html


Students learn the importance of the Pythagorean theorem as applied in radar imaging. They use a sensor unit with IRED (infrared emitting diode) to measure triangle distances and the theorem to calculate and verify distances. Student groups calibrate the sensor units to ensure accurate distance measurements. A "pretend" outdoor radar imaging model is provided to groups for sensor unit testing.

Engineering Connection

Math, science and engineering play an indisputable role in the creation of modern measuring devices. Without some of these devices, it would be extremely difficult or even impossible to measure distances due to bad weather, hostile territory or remote location. Examples devices include synthetic aperture radars (SAR) developed by electrical engineers. SAR systems use radio waves and echoes to measure slant range (distance from radar to target) and ground range (horizontal or ground distance from radar to target). SAR technology provides structural terrain information for mineral exploration, reconnaissance and targeting information for military operations, and oil spill boundaries on water to help with clean-up. A right triangular model can be used to illustrate the distance relationships formed from the radar, target and targets altitude. Similar to SARs, we can use the geometric concept of the Pythagorean theorem to calculate the distances in any right triangle model. Without an accurate measurement of distance, SARs would not be able to produce high-resolution images of remote locations.

Learning Objectives

After this activity, students should be able to:

  • Demonstrate a basic understanding of a synthetic aperture radar (SAR) system.
  • Formulate a tabular and graphical model that describes the relationship between variables.
  • Use the Pythagorean theorem to calculate side lengths of right triangles.

Educational Standards

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All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

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Materials List

Each group needs:

To share with the entire class:

Worksheets and Attachments

Visit [www.teachengineering.org/activities/view/utpa_perfectwaves_activity1] to print or download.

Pre-Req Knowledge

Students should be very comfortable working with the following concepts: the distance formula, types of triangles, and relationships between measurement of triangle sides and angles. Students should have prior knowledge of the triangle inequality theorem and recognize the distance and angle relationships formed from the radar, target and target's altitude. (The triangle inequality theorem may be explained while teaching the Pythagorean theorem equation since the equation clearly shows that one side of a triangle is always shorter than the sum of the other two sides.) All concepts, with the exception of the triangle inequality theorem are not required, but enhance th students' learning experiences. Use the pre-requisite knowledge to scale the activity for higher levels.


(Before students enter the classroom, with the projector off and classroom lights dimmed, open the Catching the Perfect SAR Wave Presentation [a PowerPoint file] to the title slide. From the title slide, click on "Play Multimedia" located on the lower part of the slide. This link plays sounds being generated from radio emissions from Saturn.)

As you walked in the classroom, were you intrigued by the sounds? What might these sounds be? What type of animals, instruments or natural phenomena might be making these sounds? (Give students to time to brainstorm ideas. Encourage them to use their imaginations. Turn on the classroom lights and have students share their ideas. After some guesses have been presented to the entire class, turn off the sounds.) Believe it or not, the sounds you heard are radio emissions coming from Saturn!

How is it possible to hear something from such a very great distance? (Expected responses: Baby monitors, telephones, two-way radios, antennas, satellites, and car radio and stereo.) You are all correct and all these devices use waves as the medium for sound! Is it possible to see an object far away with something blocking your view? (Student might take a little bit longer to answer, but expect answers such as: airport body scanners, MRI scanners, medical x-rays, and thermal imaging. Great, these are also all correct! Today we will be focusing on radar systems, which are used to see objects or topography from great distances. Radar systems also use waves as the medium to capture images. Radar imaging applications include examples such as military surveillance, terrain mapping and weather reporting. Radar imaging technology, developed by electrical engineers, provides structural terrain information to geologists for mineral exploration, reconnaissance and targeting information for military operations, and oil spill boundaries on water to environmentalists.

Today we will learn about the importance of the Pythagorean theorem in radar imaging systems! (At this point, start the presentation and let the journey of radar imaging begin.)



The Catching the Perfect SAR Wave Presentation provides the necessary background knowledge for the activity. The supplemental multimedia links on slides 4, 6 and 9 are strongly recommended, and it is suggested that teachers familiarizing themselves with these links before presenting them to students. The multimedia link on slide 2 requires java virtual machine plug-in. The slide 6 multimedia link takes you to the NASA Mission: Science website that contains an introductory video on the electromagnetic spectrum and individual waves. Slide 9 contains a direct link to an interactive radar image of Washington, DC. Please note that the multimedia might open in the background and you may need to switch over to your Internet browser. Radar diagrams and correct explanation of the geometry is crucial in order for students to make connections between concepts. Once you have completed the presentation with the class, distribute the Understanding the Problem Worksheet and have students complete the answers before beginning the hands-on activity.

A circular diagram shows these steps: 1. identify the problem, 2. research the problem, 3. brainstorm possible solutions, 4. select a promising solution, 5. create a prototype, 6. test and evaluate the prototype, 7. improve and redesign as needed. After step 7, the cycle continues to step 1.
Figure 1. The steps of the engineering design process
Copyright © 2019 Teach Engineering
The activity is divided into three different days, but can be extended to a fourth day if additional time is required. During all the activity days, students work in groups of three. Choose the groups wisely. Students are reading instructions, assembling hardware with tools, moving objects and gathering data. The activity days are organized as follows: Day 1: "radar" system construction, Day 2: sensor calibration, Day 3: sensor testing and evaluation. Reserve an optional Day 4 if additional time is needed.

Students follow a partial engineering design process, similar to the NASA engineering design process shown in Figure 1. During the introduction and accompanying presentation, students develop an understanding of the problem. In the last slide of the PowerPoint presentation, the problem is stated, but students may reference their notes to restate the problem in their own words. During the first two days of the activity, students work on step 2 of the design process as they learn about the components, tools and activity constraints. After students identify the criteria and constraints, they build a functional "radar" system prototype, notably step 7 of the design process. Finally, students refine their designs by examining and evaluating their prototypes on the third day, thereby completing step 8.

Pythagorean theorem equation: a2 + b2 = c2 where c represents the length of the hypotenuse (slant range) and a (altitude) and b (ground range) represent lengths of the other two sides of a right triangle. The hypotenuse c = (a2 + b2)^(1/2).

Before the Activity

Photo shows a clear gallon-sized plastic zip-lock bag with battery holder, batteries, sensor, alligator clips, two screws, two hex nuts, and two wire connectors. The image is labeled '"Radar" System Unit Kit.
Prepare for each group a radar system unit activity kit.
Copyright © RET-ENET Program, University of Texas-Pan American

  • In advance, prepare an activity kit for each group; this expedites the distribution process and helps students to focus on the activity and learning objectives. Make a kit by placing a sensor, battery holder, batteries, screw, hex nut, two wire connectors, two alligator clips, precision screwdriver and two-inch shield wires (obtained from excess wires from sensor) in a zip lock bag (or any type of clear plastic bag).
  • Wall space or boxes are needed for the calibration of the "radar" system on day 2.
  • Your class and group sizes determine the number of evaluation sites to prepare. Table 1 shows six set-up lengths for altitude and ground range. The altitude and ground range values represent the height and base of a right triangle. Students use the Pythagorean theorem to solve for an altitude or ground range (legs of a right triangle). Set up the evaluation sites after the completion of "Radar System Construction," preferably not during class time.
  • To set up the evaluation sites, simply mark an altitude/height on a wall and corresponding ground range/base on the floor. At each marked location, place a clear plastic suction cup containing a screw eye on the top end. Connect both suction cups using a piece of fishing line or string. Tape a model airplane on top of the wall suction cups. Refer to Figures 2 and 3.
    A three-column table provides values for altitude (cm), ground range (cm) and the corresponding slant range (cm). The values correspond to the 3 side lengths of a right triangle. In the format (altitude, ground range, slant range), the values provided are: (30, 20, 36.05), (87, 55, 102.9), (70, 48, 84.8), (60, 36, 69.9), (42, 32, 52.8), and (35, 28, 44.8).
    Table 1: Set-up lengths for the altitude and ground range, with corresponding slant ranges.
    Copyright © RET-ENET Program, University of Texas-Pan American

A diagram shows placement of clear suction cups, screw eyes, airplane clipart and string, forming a right triangle in which the ground range is the triangle base, the altitude is the triangle height and the string is the triangle hypotenuse (the slant range). The angle between the ground range and the altitude is 90°. A clipart airplane is positioned at the intersection of the altitude and the slant range.
Figure 2. Evaluation site set-up.
Copyright © RET-ENET Program, University of Texas-Pan American
A diagram shows the altitude, ground range and slant range right triangle. All three lengths form a right triangle. The altitude is the triangle height, the ground range is the base, and the slant range is the hypotenuse. The angle between the altitude and the ground range is a 90° right angle. A clipart airplane is positioned at the intersection of the altitude and the slant range, and a clipart radar system receiver is positioned at the intersection of the ground range and slant range.
Figure 3. Radar system geometry.
Copyright © RET-ENET Program, University of Texas-Pan American

With the Students

Day 1:

(Note: Prior to day 1, students should have seed the PowerPoint presentation and and completed the worksheet.)

  1. Divide the class into groups of three students each.
  2. Distribute the "radar" system unit kits with all the materials to construct the "radar" system prototype.
  3. Have groups construct prototypes. Choose to walk the students step-by-step through the prototype construction or have them work on their own, collaboratively as a team. Refer to the Radar System Construction as a manual to construct the prototypes.
    Photo shows a constructed "radar" system using a Sharp GP2Y0A02YK0F sensor and two alligator clips connected to the sensor with wires.
    "Radar" system prototype.
    Copyright © RET-ENET Program, University of Texas-Pan American

Day 2:

(Note: By day 2, students should have completed construction of the "radar" system prototype.)

  1. Distribute multimeters and the Mathematical Model to each group. Have students use this worksheet to record results during the calibration.
  2. Assign each team member a role as follows: system keeper, system analyst and system maintenance leader. The system keeper holds on to the "radar" system while using the multimeter to obtain voltage readings. The system analyst uses the given template to record the voltage readings and distance of the system to the target (a box). The system maintenance leader is responsible for ensuring the target (wall or box) is aligned with the "radar" system. Explain to each role and model the set-up for system calibration.
    Photo shows a multimeter, measuring tape, and box to calibrate a Sharp GP2Y0A02YK0F Sensor. The sensor is facing the wall is connected to the multimeter by two wires, one red and one black with alligator clips.
    "Radar" system calibration set-up.
    Copyright © RET-ENET Program, University of Texas-Pan American
  3. Use the Radar System Calibration as a manual for calibrating the prototypes.
  4. Expect student groups to complete the Mathematical Model during the calibration.

Day 3:

  1. Give each group a copy of the Radar System Evaluation. Identify the evaluation sites created with the corresponding number on the evaluation paper. Have students use the "radar" system and system calibration recordings to determine each missing distance. Have students take multiple readings within the allotted time. Readings may vary slightly according to personal calibration. Refer to the Sample Radar Voltage Readings.
  2. Direct students to use the Pythagorean theorem to verify that the missing distance captured by the "radar" system is correct.
    Photo shows a multimeter, airplane cutout, and Sharp GP2Y0A02YK0F sensor aimed at a wall. While the distance to the wall is unknown, the vertical distance along the wall to a clipart airplane is shown as 35 cm and the diagonal (slant range) distance from the clipart airplane to the sensor is shown as 44.8 cm.
    Indoor "radar" system evaluation with unknown ground distance, height of 35 cm, and hypotenuse of 44.8 cm.
    Copyright © RET-ENET Program, University of Texas-Pan American


altitude: The height of an object measured vertically from the ground.

antenna: The part of a radar system that propagates the produced rf (radio frequency) pulses.

backscatter: The reflected electromagnetic energy from an object. It is also called echo.

electromagnetic spectrum: The range of all possible frequencies of electromagnetic radiation.

frequency: The number of repetitions per unit of time; cycles per second.

ground range: The ground distance from a radar to a target.

radar: A radio detection and ranging system detects targets by reflected electromagnetic energy.

receiver: The part of a radar system that captures, amplifies and demodulates rf pulses.

slant range: The diagonal distance from a radar to a target.

transmitter: The part of a radar system that produces rf (radio frequency) pulses of energy.

wave: A continuous oscillation of energy in space time.


Pre-Activity Assessment

Understanding the Problem: Have students complete the Understanding the Problem Worksheet, which is designed to reinforce their understanding about waves, the electromagnetic spectrum, radar systems and the Pythagorean theorem as it relates to engineering. At the end of the assessment, students are asked to restate their understanding of challenge/problem they are asked to solve in the activity.

Activity Embedded Assessment

A Mathematical Model: Have students complete the Mathematical Model, which provides a blank table and graph paper for student to collect and plot data. Student use their constructed sensor unit and record distance and the corresponding voltage reading. Independent and dependent variables are labeled on the table and graph.

Post-Activity Assessment

Radar System Evaluation: Have students complete the Radar System Evaluation, which is designed to evaluate students' knowledge of the Pythagorean theorem. They are given clipart of a radar, target and target's altitude, which all together form a triangular model. They use the Pythagorean equation with two known side lengths on the triangular model to calculate the missing side length.

Activity Extensions

Have students calculate the percent error of the distances obtains from reading voltages to the actual distances calculated using the Pythagorean theorem. The equation to determine the percent error can be found on the Radar System Evaluation.

Activity Scaling

  • For lower grades, re-teach the Pythagorean theorem by working out examples for each side of a right triangle. Make sure students have the Pythagorean theorem recorded in their notes. You may also want to model each step of the radar system construction.
  • For upper grades, give less guidance. After showing the presentation to provide students with background information, have them assemble the radar system with guidance from the manual. As part of their assessment, have students create evaluation sights using their sensors. You may also want to assign them more evaluation problems.


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"Viewing Radio Waves." October 31, 2011. Teachers' Domain. Accessed June 28, 2012. http://www.teachersdomain.org/resource/npe11.sci.phys.energy.viewradiowaves/

Photo Dept. NASA Headquarters, 300 E. St. SW, Washington, DC 20546 http://saturn.jpl.nasa.gov/photos/imagedetails/index.cfm?imageId=4399

"Radio Waves & Electromagnetic Fields." Updated April 19, 2007. Teachers' Domain. Accessed June 29, 2012. http://www.teachersdomain.org/resource/hew06.sci.phys.energy.radiowaves

"The Electromagnetic Spectrum" Updated June 20, 2012. NASA Mission: Science. Accessed June 29, 2012. https://mynasadata.larc.nasa.gov/ElectroMag.html

"Video Tour of Electromagnetic Specturm." Updated June 20, 2012. NASA Mission: Science. Accessed June 29, 2012. http://missionscience.nasa.gov/ems/emsVideo_01intro.html/

"Imaging with Radar." Updated January 29, 2004. Teachers' Domain. Accessed June 29, 2012. http://www.teachersdomain.org/resource/phy03.sci.phys.energy.radar/


© 2013 by Regents of the University of Colorado; original © 2012 The University of Texas-Pan American


Luis Avila; Mounir Ben Ghalia

Supporting Program

RET-ENET Program, Electrical Engineering Department, The University of Texas-Pan American


This activity was created through The University of Texas-Pan American's Electrical Engineering Research Experiences for Teachers in Emerging and Novel Engineering Technologies (RET-ENET) Program with support from National Science Foundation grant no. CNS 1132609. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: February 12, 2020

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