Surface Area

Surface area is the total amount of space that the surface of a three-dimensional object occupies. It is measured in square units, such as square inches or square meters. Finding the surface area of different three-dimensional objects requires different formulas, but the basic concept is to find the total area of all the surfaces that make up the object.

Formulas for Calculating Surface Area

Here are some common formulas for calculating the surface area of various three-dimensional shapes:

Rectangular Prism: 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height of the prism.
Cube: 6s², where s is the length of one side of the cube.
Cylinder: 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder.
Sphere: 4πr², where r is the radius of the sphere.
Cone: πr² + πr√(r² + h²), where r is the radius of the base and h is the height of the cone.
Triangular Prism: (2 * base area) + (perimeter of base * height), where the base area is the area of the triangle base and the perimeter is the perimeter of the base.

Example Problems

Let's work through a couple of example problems to illustrate how to calculate the surface area of different shapes.

Example 1: Rectangular Prism

Find the surface area of a rectangular prism with the following dimensions: length = 5 cm, width = 3 cm, and height = 4 cm.

Surface Area = 2lw + 2lh + 2wh

Surface Area = 2(5)(3) + 2(5)(4) + 2(3)(4)

Surface Area = 30 + 40 + 24

Surface Area = 94 square cm

Example 2: Sphere

Find the surface area of a sphere with a radius of 6 cm.

Surface Area = 4πr²

Surface Area = 4 * π * 6²

Surface Area = 4 * π * 36

Surface Area ≈ 452.39 square cm

Study Tips

When studying surface area, it's important to understand the different formulas for each type of three-dimensional object. Practice applying the formulas to various examples and familiarize yourself with the units of measurement (e.g., square centimeters, square inches) that are used for surface area. Additionally, visualize the objects and their surfaces to better understand how the surface area is calculated.

Remember to pay attention to the specific details provided in each problem, such as dimensions or measurements, as these will be essential for accurately calculating the surface area of a given object.

By mastering the formulas and practicing with a variety of examples, you'll become proficient in finding the surface area of different three-dimensional shapes.

◂Math Worksheets and Study Guides Eighth Grade. Displaying data

Organizing & Displaying Data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

Select and use appropriate statistical methods to analyze data.

Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

Develop and evaluate inferences and predictions that are based on data.

Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.