# Area of solids

Finding area of rectangular solids and cylinders by cutting them into flat pieces and adding the areas.

**A lesson plan for grades 7–12 Mathematics**

## Learning outcomes

Students will:

- learn the concepts of lateral area and total surface area as applied to rectangular solids and cylinders, and how mathematical formulas for surface areas of solids relate to these concepts.
- discuss application of these concepts and formulas in real life.

## Teacher planning

### Time required for lesson

1.5 hours

### Materials/resources

- Twelve pack box of soft drink cans
- Measuring device(ruler or tape measure)
- Graph paper or plastic graph sheet (can be purchased at Wal-Mart; about the size of a placemat)

### Technology resources

Calculator

## Pre-activities

Formulas for area of a rectangle and circle.

## Activities

- Students gently tear open the carton surrounding the cans, opening all the seams until the cardboard will lie flat.
- Students measure the carton one piece at a time, recording the dimensions of each piece.
- Students transfer the measurements to the graph paper and count the units on the graph paper, then find the total area of the solid by using individual area formulas for rectangle, square, circle, etc.
- Students use established formulas for surface area and lateral area of solids to find the area of the carton another way.
- Students compare the answers obtained by the two different methods.
- Students compare answers with each other.
- Repeat the same procedure for the soft-drink cylinders (as an alternative, use a cardboard cylindrical container, such as an oatmeal container).
- Let the students figure out the amount of cardboard and/or aluminum needed and the approximate cost of producing the containers (you can assume a per-square-foot cost of each material, or have curious students research this on the internet or by contacting soft-drink companies). Discuss how this affects the cost of the drinks.

## Assessment

Students must turn in their graphs and worksheets showing the formula problems. Also, one student from their group would make an oral presentation.

## Supplemental information

### Comments

Lessons similar to this are found in the Cord material.

- Common Core State Standards
- Mathematics (2010)
#### Grade 6

- Geometry
- 6.G.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

#### Grade 7

- 7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

- Geometry
#### High School: Geometry

- Modeling with Geometry
- GEO.M.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

- Modeling with Geometry

## North Carolina curriculum alignment

### Mathematics (2004)

#### Grade 7

**Goal 2**: Measurement - The learner will understand and use measurement involving two- and three-dimensional figures.**Objective 2.02**: Solve problems involving volume and surface area of cylinders, prisms, and composite shapes.

#### Grade 8

**Goal 3**: Geometry - The learner will understand and use properties and relationships in geometry.**Objective 3.01**: Represent problem situations with geometric models.

#### Grade 9–12 — Geometry

**Goal 2**: Geometry and Measurement - The learner will use geometric and algebraic properties of figures to solve problems and write proofs.**Objective 2.04**: Develop and apply properties of solids to solve problems.

#### Grade 9–12 — Integrated Mathematics 1

**Goal 2**: Geometry and Measurement - The learner will use properties of geometric figures to solve problems.**Objective 2.01**: Use the length, area, and volume of geometric figures to solve problems. Include arc length, area of sectors of circles; lateral area, surface area, and volume of three-dimensional figures; and perimeter, area, and volume of composite figures.

#### Grade 9–12 — Technical Mathematics 1

**Goal 2**: Geometry and Measurement - The learner will measure and apply geometric concepts to solve problems.**Objective 2.03**: Use the length, area, and volume of geometric figures to solve problems. Include arc length, area of sectors of circles; lateral area, surface area, and volume of three-dimensional figures; and perimeter, area, and volume of composite figures.