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Lesson: The Math of Sailboat Design

Sailboat(Lesson courtesy the Cooper-Hewitt National Design Museum  Educator Resource Center and Tonya Adison). Level: High School. Time  Required: Two 60-minute class periods.

Designing the Sail of a Boat


Why do sailboats have triangular sails? This lesson introduces the basic postulate of right triangle trigonometry — the Pythagorean Theorem — and demonstrates how this Theorem can be tested and proved. Students  learn about sailboat design and how the shape of sails affects their movement. They apply what they learn about a hands-on, engineering math lesson  that requires them to design a boat’s sail. With the opportunity to explore and prove the Pythagorean Theorem, students gain a better grasp of how to take accurate measurements and how to read a standard ruler. They also exhibit creativity in designing/decorating their sailboats. Through this process, they come to understand this famous theorem.


Students will:

  • investigate the design of sailboat sails
  • take accurate measurements using a ruler
  • identify right triangles based on given measurements of sides
  • find the missing side of right triangle
  • solve word problems involving right triangles
  • draw diagrams to show a visual representation of a written problem
  • make connections to sailboat design
  • make connections between the theorem and real world situations

National Standards

Mathematics (2007):

  • recognize reasoning and proof as fundamental aspects of mathematics
  • make and investigate mathematical conjectures
  • compute fluently and make reasonable estimates
  • represent and analyze mathematical situations and structures using algebraic symbols
  • analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
  • specify locations and describe spatial relationships using coordinate geometry and other representational systems
  • understand measurable attributes of objects and the units, systems, and processes of measurement
  • apply appropriate techniques, tools, and formulas to determine measurements
  • build new mathematical knowledge through problem solving
  • solve problems that arise in mathematics and in other contexts
  • apply and adapt a variety of appropriate strategies to solve problems
  • communicate their mathematical thinking coherently and clearly to peers, teachers, and others
  • recognize and use connections among mathematical ideas
  • understand how mathematical ideas interconnect and build on one another to produce a coherent whole
  • recognize and apply mathematics in contexts outside of mathematics





  • rulers
  • Post-its
  • calculators
  • markers
  • crayons
  • colored Pencils
  • push pins/thumb tacks or magnets


Right triangle–a polygon with three vertices and three straight line segment sides (a triangle) that has one 90-degree angle.

Hypotenuse–the side opposite the 90 degree angle and the longest side of a right triangle.

Leg–a side of the right triangle that is not the hypotenuse.

Pythagorean Theorem–a2 + b2 = c2 or the sum of the squares of two sides of a right triangle are equal to the square of the hypotenuse.



Begin a class discussion asking the students if they have ever seen a sailboat. Assuming there are some “yes” responses in the group, ask what students noticed about the design of the sailboat. Someone should observe that their sails are triangular. Ask why the sails have this shape. After allowing students to offer their ideas, explain that the design allows a boat to take advantage of winds at 90 degree angles by way of “tacking.” The sail design enables the boat to move in previously inconceivable ways. Pass out a copy of “How does a Sailboat Move Upwind?”  to each student and hold a class discussion about how the sail design works. Write key points on the board.

Day 1

  • Define a right triangle and identify the legs and the hypotenuse of the triangle.
  • Provide the formula for the Pythagorean Theorem, a2 + b2 = c2, and identify a and b as the legs and c as the hypotenuse.
  • Give the students about 10-15  minutes to work on the Student Practice problems.
  • Invite a few students to put their answers to the problems on the board and explain their work to the class.
  • Ask students if they agree with the work on the board and if they solved the problems in a similar fashion.
  • Answer any remaining questions about the problems.
  • Tell the students that they will be using the information they learned today to design sailboat sails in the next lesson.

Day 2

Slide1Introduce the activity as a follow-up to the previous day’s work, explaining that the class will conduct an experiment to determine if the Pythagorean Theorem really works. “We have learned about sailboat design and learned how to calculate the missing side of a right triangle, but how can we be sure this is true?”

  • Instruct each student to measure 2 sides of each of the triangular sails on the handout and record their answers on their post-its.

  • Students calculate the measure of the third side of the triangle using the Pythagorean Theorem.

  • Students then measure the third side of the triangle and compare their answer to the one they got in the previous step.

  • Students record and summarize what they notice.

  • Students decorate their sailboats as they choose and write a brief summary of why their sailboat sail design works.


Depending on your classroom, allow students to display their work for a “gallery walk” where they are able to see what other students created.


Check the measurements and subsequent calculations, paying special attention to the substitution in the formula for a, b, and c. You may also wish to special attention to the summary portion of the activity. Is the sailboat decorated creatively? In the summary of how the sail on a sailboat works, did the student grasp why the shape and design of the sail help the boat move across distances?


Extension Activities:

The distance formula: This formula  is basically the Pythagorean Theorem reorganized. You will need a sheet of graph paper, a transparency sheet of graph paper, and a map (preferably of your town/area with some landmarks). Students place the transparency over the map and plot the locations of two places (they need to be on corners). Students are instructed to draw a right triangle, drawing a vertical and horizontal line to complete this task. They then use the Pythagorean Theorem to find the distance between these points or the hypotenuse of the triangle. Introduce the distance formula and show how its pieces are derived from the Pythagorean Theorem. This could also be used with any map activity.

Advice for Teachers:

Be sure to spend a few minutes with the students checking to see that they understand how to read a ruler accurately. Students may not grasp the purpose of the activity at first, but it should make more sense to them once they work through it and compare their answers with those of other students.

Author: Tonya Adison

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