A square pyramid is a three-dimensional shape that has a square base and four triangular faces that meet at a single point called the apex. The base of the pyramid is a square, and the four sides are triangles that meet at the apex.

Formulas for Square Pyramid

Surface Area of a Square Pyramid = Base Area + (1/2) * Perimeter of Base * Slant Height

Base Area of a Square Pyramid = (side length of base)^2

Volume of a Square Pyramid = (1/3) * (Base Area) * Height

How to Calculate Surface Area of a Square Pyramid

Find the area of the base by squaring the length of one side of the base.
Find the perimeter of the base by multiplying the side length by 4.
Find the slant height, which is the distance from the apex to the midpoint of any side of the base.
Use the formula: Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height

How to Calculate Volume of a Square Pyramid

Find the area of the base by squaring the length of one side of the base.
Find the height of the pyramid, which is the distance from the apex perpendicular to the base.
Use the formula: Volume = (1/3) * (Base Area) * Height

Example Problem

Find the surface area and volume of a square pyramid with a base side length of 8 units and height of 10 units.

Base Area = 8^2 = 64 square units

Slant Height can be calculated using the Pythagorean theorem: Slant Height^2 = (8/2)^2 + 10^2

Slant Height = sqrt((8/2)^2 + 10^2) = sqrt(68) = 8.246 units

Surface Area = 64 + (1/2) * 8 * 8.246 = 64 + 32.768 = 96.768 square units

Volume = (1/3) * 64 * 10 = 213.333 cubic units

So, the surface area of the square pyramid is 96.768 square units and the volume is 213.333 cubic units.

Practice Problems

1. Find the surface area and volume of a square pyramid with a base side length of 6 units and height of 7 units.

2. A square pyramid has a base side length of 12 units and height of 9 units. Find the surface area and volume.

3. A square pyramid has a base side length of 5 units and height of 10 units. Calculate the surface area and volume.

Practice solving these problems to strengthen your understanding of square pyramids!

◂Math Worksheets and Study Guides Eighth Grade. Linear equations

The resources above cover the following skills:

Algebra (NCTM)

Represent and analyze mathematical situations and structures using algebraic symbols.

Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

Grade 8 Curriculum Focal Points (NCTM)

Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Connections to the Grade 8 Focal Points (NCTM)

Geometry: Given a line in a coordinate plane, students understand that all 'slope triangles' - triangles created by a vertical 'rise' line segment (showing the change in y), a horizontal 'run' line segment (showing the change in x), and a segment of the line itself - are similar. They also understand the relationship of these similar triangles to the constant slope of a line.