Surface Area of A Pyramid



In this lesson, we will learn

  • how to find the surface area of any pyramid.
  • how to find the surface area of a regular pyramid.
  • how to find the surface area of a square pyramid.
  • how to find the surface area of a pentagonal and a hexagonal pyramid.
  • how to find the surface area of a pyramid when the slant height is not given.
Related Topics: More Geometry Lessons

Surface area of a Pyramid

A pyramid is a solid with a polygonal base and several triangular lateral faces. The lateral faces meet at a common vertex. The number of lateral faces depends on the number of sides of the base. The height of the pyramid is the perpendicular distance from the base to the vertex.

A regular pyranid has a base that is a regular polygon and a vertex that is above the center of the polygon. A pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle base.

pyramid

We can find the surface area of any pyramid by adding up the areas of its lateral faces and its base.

Surface area of any pyramid = area of base + area of each of the lateral faces


If the pyramid is a regular pyramid, we can use the formula for the surface area of a regular pyramid.

Surface area of regular pyramid = area of base + halfps

where p is the perimeter of the base and s is the slant height.


If the pyramid is a square pyramid, we can use the formula for the surface area of a square pyramid.

Surface area of square pyramid = b2 + 2bs

where b is the length of the base and s is the slant height.

Worksheet Calculate the volume of square pyramids. Worksheet Calculate the volume of prisms & pyramids.



Example:

Calculate the surface area of the following pyramid.

square pyramid 

Solution:

Sketch a net of the above pyramid to visualize the surfaces.

net pyramid 

Since the given pyramid is a square pyramid, we can use any of the above formulas.

Using the formula for the surface area of any pyramid

Area of base = 6 × 6 = 36 cm2

Area of the four triangles = × 6 × 12 × 4 = 144 cm2

Total surface area = 36 + 144 = 180 cm2

 

Using the formula for a regular pyramid

Surface area of regular pyramid = area of base + halfps

= 6 × 6 + half × 6 × 4 × 12 = 180 cm2

 

Using the formula for a square pyramid

Surface area of square pyramid = b2 + 2bs

= 6 × 6 + 2 × 6 × 12 = 180 cm2




Surface area of pyramid by adding up the area of each surface

The following videos show how to find the surface area of a pyramid by adding up the area of each surface.






Surface area of square pyramid by using a formula

These videos show how to find the surface area of a square pyramid using the formula
Surface area = 2bs + b2 where b is the length of the base and s is the slant height.

 





Surface area of regular pyramid by using a formula

These videos show how to calculate the surface area of a regular pyramid using the formula
surface area = area of base + 1/2 × perimeter of base × slant height

 







Surface area of pentagonal and hexagonal pyramids

This video provides a specific example of how to find the surface area of a pyramid, given base edge and height. The base is a pentagon. It shows how to find the apothem and slant height.

 

How to find the surface area and volume of a pentagonal pyramid with the known apothem.



How to find the surface area and volume of a hexagonal pyramid with known side length.






Surface area of regular pyramid when the slant height is not given

This video shows how to find the lateral surface area of a pyramid when the slant height is not given. (Part 1)

 

This video shows how to find the total surface area of a pyramid when the slant height is not given. (Part 2)

 

This video shows how to calculate the surface area of a rectangular pyramid when the slant height is not given.







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