In this lesson, students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems. Students also identify and describe the properties of two-dimensional figures. Students are challenged to set a table for 22 using one unit lengths of paper (each unit length represents one student sitting at the table). They are asked to pretend that they all want to sit together for a dinner, and the table has to be rectangular. They must figure out what the table will look like if it must be rectangular and if everybody must sit at the table. They must determine what's the biggest table going to look like where everybody sits, and what's the smallest table going to look like where everybody sits. Students work to conceptually figure out relationships between perimeter and area in a creative and interactive way. They brainstorm ideas in groups, discuss as a class to determine the best options, and then physically set the two tables using strips of paper that represent "one unit" in length per person.

Students construct "a tetrahedron and describe the linear, area and volume using non-traditional units of measure. Four tetrahedra are combined to form a similar tetrahedron whose linear dimensions are twice the original tetrahedron. The area and volume relationships between the first and second tetrahedra are explored, and generalizations for the relationships are developed." (from NCTM Illuminations)

The purpose of this task is for students to decompose a figure into rectangles and then find the total area by adding the area of all of its smaller, non-overlapping rectangles.

In this activity, students learn how engineers design faucets. Students will learn about water pressure by building a simple system to model faucets and test the relationship between pressure, area and force. This is a great outdoor activity on a warm day.

Students explore volume by trying to pack a moving truck with boxes of specified sizes.
The lesson is structured in the following way:
* Before the lesson, students attempt the Packing It In task individually. You review their responses and formulate questions that will help them improve their work.
* At the start of the lesson, students read your comments and consider ways to improve their work.
* In pairs or threes, students work together to develop a better solution, producing a poster to show their conclusions and their reasoning.
* Then, in the same small groups, students look at some sample student work showing different approaches to the problem. They evaluate the strategies used and seek to improve the arguments given.
* In a whole-class discussion, students compare different solution methods.
* Finally, students reflect individually on their learning.

Challenged with a hypothetical engineering work situation in which they need to figure out the volume and surface area of a nuclear power plant’s cooling tower (a hyperbolic shape), students learn to calculate the volume of complex solids that can be classified as solids of revolution or solids with known cross sections. These objects of complex shape defy standard procedures to compute volumes. Even calculus techniques depend on the ability to perform multiple measurements of the objects or find functional descriptions of their edges. During both guided and independent practice, students use (free GeoGebra) geometry software, a photograph of the object, a known dimension of it, a spreadsheet application and integral calculus techniques to calculate the volume of complex shape solids within a margin of error of less than 5%—an approach that can be used to compute the volumes of big or small objects. This activity is suitable for the end of the second semester of AP Calculus classes, serving as a major grade for the last six-week period, with students’ project results presentation grades used as the second semester final test.

Students apply their knowledge of scale and geometry to design wearables that would help people in their daily lives, perhaps for medical reasons or convenience. Like engineers, student teams follow the steps of the design process, to research the wearable technology field (watching online videos and conducting online research), brainstorm a need that supports some aspect of human life, imagine their own unique designs, and then sketch prototypes (using Paint®). They compare the drawn prototype size to its intended real-life, manufactured size, determining estimated length and width dimensions, determining the scale factor, and the resulting difference in areas. After considering real-world safety concerns relevant to wearables (news article) and getting preliminary user feedback (peer critique), they adjust their drawn designs for improvement. To conclude, they recap their work in short class presentations.

This video takes students through a real world problem solving process from start to finish. Students investigate the process of changing the shape of freeze-dried dog food patty from a cylinder to a square prism without changing the volume, density or weight of the patty’s materials. Will this change increase efficiency? Will this change decrease production cost?

This lesson from Illuminations helps students to begin to develop an understanding of area by using a Java applet on the computer. Learners use 12" × 12" paper squares to measure the area of a door and determine whether area increases or decreases as the length or width of their door changes. The lesson includes discussion questions, extensions, and other teacher support. It is supported by the applet IGD: Areas in Geometry (cataloged separately).