Mensuration — Regular Right Pyramid with Triangular or Square Base
What is Mensuration — Regular Right Pyramid with Triangular or Square Base?
A polygon that is equiangular (all angles are equal) and equilateral (all sides are equal).
Key formula / rule: Volume of a Pyramid
Key points
- Understand the properties of a regular right pyramid.
- Calculate the volume of such pyramids.
- Calculate the lateral and total surface area.
- Apply formulae to solve problems involving these pyramids.
Common exam trap
Confusing slant height with perpendicular height.
Definitions
- Term
Regular Polygon
- Meaning
A polygon that is equiangular (all angles are equal) and equilateral (all sides are equal).
- Term
Right Pyramid
- Meaning
A pyramid where the apex is directly above the centroid of its base.
- Term
Apex
- Meaning
The point (vertex) opposite the base of a pyramid.
- Term
Slant Height (l)
- Meaning
The height of each triangular lateral face, measured from the midpoint of a base edge to the apex.
- Term
Perpendicular Height (h)
- Meaning
The shortest distance from the apex to the plane of the base.
- Term
Apothem (r)
- Meaning
The distance from the center (centroid) of a regular polygon to the midpoint of one of its sides.
Learning objectives
Understand the properties of a regular right pyramid.
Calculate the volume of such pyramids.
Calculate the lateral and total surface area.
Apply formulae to solve problems involving these pyramids.
Formulae
- Name
Volume of a Pyramid
- Note
h is the perpendicular height from the apex to the base.
- Expression
V = (1/3) * Base Area * Height
- Name
Lateral Surface Area of a Pyramid
- Note
l is the slant height (height of a lateral triangular face).
- Expression
LSA = (1/2) * Perimeter of Base * Slant Height
- Name
Total Surface Area of a Pyramid
- Note
- Expression
TSA = Base Area + Lateral Surface Area
- Name
Area of Square Base
- Note
- Expression
Abase = side2
- Name
Perimeter of Square Base
- Note
- Expression
Pbase = 4 * side
- Name
Area of Equilateral Triangle Base
- Note
- Expression
Abase = (√(3)/4) * side2
- Name
Perimeter of Equilateral Triangle Base
- Note
- Expression
Pbase = 3 * side
- Name
Relationship between heights and slant height
- Note
r is the apothem (distance from centroid to midpoint of a base side). For a square, r = side/2. For an equilateral triangle, r = side/(2*√(3)).
- Expression
l2 = h2 + r2
Prerequisites
Basic geometry of triangles and squares.
Area and perimeter of regular polygons.
Pythagorean theorem.
Understanding of 3D shapes.
Common mistakes
Confusing slant height with perpendicular height.
Incorrectly calculating the base area for equilateral triangles.
Using the wrong formula for lateral surface area.
Assuming the pyramid is regular or right when it is not specified.
Keywords
Pyramid
Regular Pyramid
Right Pyramid
Volume
Surface Area
Slant Height
Perpendicular Height
Square Base
Triangular Base
Mensuration
Practice preview
A regular right pyramid has a square base with a side length of 10 cm. If its height is 12 cm, what is its volume?…
easy
A regular right pyramid has a square base with a side length of 6 cm and a height of 10 cm. What is its volume?…
easy
The volume of a regular right pyramid with a square base is 400 cubic cm. If the side length of its base is 10 cm, what is the height of the pyramid?…
medium