THEOREMS OF PAPPO-GULDINO. Pappus of Alexandria was an ancient Greek mathematician active mainly in the field of geometry; various theorems have been attributed to him, including the theorems of Pappus-Guldino or also called the theorems of the centroid of Pappus.
The Pappo-Guidino theorems are two related theorems, useful for calculating the surface area and volume of a solid of revolution.
Solids of revolution are a particular class of solids that are obtained by a 360° rotation of:
- A flat figure around one of its sides or a straight line (even external to the figure itself, for example the torus);
- A straight or curved line around another straight line.
The main solids of rotation that we have to deal with in school are: the cylinder, the cone and the sphere; but there are solids of rotation with a much more particular geometry.
But let’s go back to the theorems of Pappo-Guldino. As already said, they allow us to calculate the volume and the lateral surface area by knowing the center of gravity of the plane figure (for the volume) and of the curve (for the surface area) that generates the solid.
First Pappus-Guldinus theorem:
“The area of a surface of revolution obtained from a plane curve rotated by an angle θ between 0° and 360° around an axis is equal to the product of the angle, the distance of the center of gravity of the curve from the axis of rotation and the length of the curve itself.”
In formulas we have:
A=θ?d?l
Where d is the distance from the center of gravity and l is the length of the curve.
Let’s take a known solid and try this formula. For example, we want to find the surface area of a cylinder.
From geometry we know that the lateral surface area will be equal to: S=2πr?h with r being the radius of the base circumference and h being the height of the cylinder.
Now let’s use the Pappus-Guldinus theorem. The cylinder is obtained by rotating a rectangle around one of its sides:
The rotation angle θ will be equal to 360° (or 2π);
The center of gravity is found on the curve that describes the solid and will have a distance d equal to the radius;
The length of the curve will be the height of the rectangle.
A=θ?d?l=2π?r?h
Second Pappus-Guldinus theorem:
“The volume of a solid of revolution obtained by rotating a plane figure through an angle θ between 0° and 360° around an axis is equal to the product of the angle of rotation, the distance of the center of gravity of the figure from the axis of rotation, and the area of the figure itself.”
In formulas:
V=θ?d?A
With d the distance from the center of gravity and A the area of the plane figure.
We always want to find the volume of a cylinder:
- The rotation angle θ is always 2π;
- We know that the center of gravity of the plane figure, which in this case is a rectangle, is ar/2 if we indicate with r the side of the rectangle (which coincides with the radius of the base circumference);
The area of the rectangle will be the base times the height, if we said the base is equal to r and the height of the figure h, we will have rh.
V=θ?d?A=2π?r/2?rh=πr 2 h
Which we know to be correct.
