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High School Geometry: Tutoring Solution14 chapters | 159 lessons

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Lesson Transcript

Instructor:
*Karin Gonzalez*

Karin has taught middle and high school Health and has a master's degree in social work.

This lesson provides an informative overview of the properties of square pyramids. Along with providing the definition for a square pyramid, the lesson includes the formulas for finding base area, volume, and surface area.

You need not travel to Egypt to know that it hosts some of the world's most marvelous wonders: the Pyramids of Giza. These pyramids were built between 2589 and 2504 BC, so it is a true wonder that they still stand strong and tall today in the expansive desert just outside of Cairo. The Pyramids of Giza are one of oldest examples of square pyramids.

Like all pyramids, **square pyramids** share the property of being a polyhedron with a polygonal base and triangular sides reaching up towards a point, called an apex. A **polyhedron** is a three-dimensional figure with flat polygons as sides, and a **polygon** is a flat, enclosed, plane shape with at least three sides and angles.

From this information it can be concluded that not all pyramids are square pyramids. All pyramids are named after their base. So, a **square pyramid** is a pyramid with a square base, four triangular sides, five vertices, and eight edges.The base of this pyramid is a square, and the top point is the apex.

It was mentioned that a square pyramid is a polyhedron. But to be more specific, it could be called a **pentahedron** because it is a polyhedron that has five sides. Remember that *penta* means five in Latin.

If all triangular sides of the square pyramid are equilateral, meaning each triangle has equal sides and angles, then all the edges of the square pyramid will be equal in length. This is actually called a **Johnson solid**. Other geometric forms can also be Johnson solids if they are convex polyhedrons with regular polygon faces but not uniform.

There are formulas for calculating the base area, volume, and surface area of a square pyramid. First, study the diagram and look closely at how each dimension is labeled:

The dimensions of a square pyramid are as follows:

*a* = base side length*e* = lateral side length*h* = height*s* = slant height

Finding the base area of a square pyramid is simple, since it is finding the area of a square. If *a* is an edge of the base, then the formula to find base area of a square pyramid would be:

The volume of an object is the space that it occupies, or the space enclosed within its edges and borders. In order to find the volume of a square pyramid, the base area must be known or at least the measurement of the base edge, so that it can be squared. The formula to find the volume is:

The surface area of an object is the area of the outer layer of an object. So, for a square pyramid, it would be the area of all four triangular sides plus the area of the square base. The formula to find the surface area is:

All **pyramids** are polyhedrons with a polygonal base and triangular sides reaching up towards an apex. But what are the properties specific to square pyramids? All square pyramids have: a square base, four triangular sides, five vertices, eight edges. A square pyramid can be called a pentahedron because it has five faces: four triangle faces and one square face. There is a special type of square pyramid called a Johnson solid: when all triangular sides of the square pyramid are equilateral and all the edges of the square pyramid will be equal in length. Using formulas and the dimensions from a square pyramid, the base area, volume, surface area can be calculated.

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High School Geometry: Tutoring Solution14 chapters | 159 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:16
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Angle of Elevation: Definition, Formula & Examples 4:50
- Centroid: Definition, Theorem & Formula 6:15
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