# Surface Area and Volume

### 6.4 and 6.7

## Surface Area, Volume & Cross Sectional Area

6.4 Surface area and volume of a prism and a pyramid (in particular, cuboid, cylinder, and cone). Surface area and volume of a sphere. Formulae will be given for the lateral surface area of a cylinder and a cone, the surface area of a sphere, and the volume of a pyramid, a cone, and a sphere.

6.7 Identify the shapes of two-dimensional cross sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

**Surface Area and Volume**

**Context:** Teaching Surface Area and Volume lends itself to easy, tangible connections. This can help make learning fun and easily accessible to students. However, the following concepts and skills can be developed when students are learning this topic:

- Use of formulas
- Calculator proficiency
- Rounding to given significant figures
- Accurate constructions
- Unit conversion
- 100
^{3}cm^{3 }= 1 m^{3}

- 100
- Recognizing what 3D shapes can approximate real-life objects
- Relationship between the growth of squared and cubed units

When teaching this unit, students should be challenged to track their thinking. A good strategy for students is to sketch their ideas in diagrams, then add formulas to those diagrams, then plug the numbers in (on paper), then use the calculator to evaluate. They should write down their unrounded number and then round to the appropriate place (typically three significant figures at this level).

Students need to understand that for prisms and cylinders, volume is the product of the base’s area (which is a cross section) and the perpendicular “height.” For other 3D shapes, formulas will be provided.

To help them visualize the volume of prisms use a large stack of printer paper. When holding up one sheet, it is practically a two-dimensional plane. It is unlikely that the volume of one sheet could be measured in a High School classroom. However, when there are 500 sheets, they create enough dimension in a 3^{rd} direction that volume could be calculated. Side Note: After a stack of 500 sheets has been calculated, students could figure out the volume of one sheet. This could be a good exercise!

Students also need the ability to see how a net can make a 3D shape. They should be able to calculate the volume of a prism from its net. They should also be able to draw a net from seeing a 3D shape.

**Big Idea**

Surface area is the outward facing surface. Like the skin on a person. Finding surface area is similar to finding the area of a compound shape.

Volume is how much three-dimensional space is occupied by an object. The units are three-dimensional.

**Key Knowledge**

Prerequisite: Students need to be proficient with finding the area of common 2D shapes.

New: How to draw a net. The volume of prisms, the formulas for surface area and volume of spheres, cones and rectangular pyramids.

**Pro-Tip**

(for students)

When finding the surface area of a 3D object, be sure to account for ALL sides. This is the easiest thing to over-look.

Click on the PDF icon to download a Lesson Guide that will pace the PowerPoint. Click on the PowerPoint icon to download the PowerPoint Lesson.

The lessons are intentionally over-built with more material than the time will allow. This way you have more options on how to best serve the needs of your students.

This is a week-long lesson. The direct instruction is only one day. For that day you will be provided with a Lesson Guide which will help you to pace the lesson and understand the objective of each slide in the PowerPoint. There is a homework assignment for that day. The rest of the week is a project where students will build two different prisms that must contain the same volume. The process of administering this project, along with documents for the project, are linked in the lesson plan, which can be downloaded by clicking the PDF icon above.

Below is a copy of one of the slides from the lesson.

There is one homework assignment for this unit. The rest of the learning will be done through the project outlined in the lesson plans.

To download the homework, click here.