What does the theorem of Pappus say?

A theorem from Euclid’s Elements (c. 300 bc) states that if a line is drawn through a triangle such that it is parallel to one side (see the figure), then the line will divide the other two sides proportionately; that is, the ratio of segments on each side will be equal.

What does the second theorem of Pappus indicate?

The second theorem of Pappus states that the volume of a solid of revolution obtained by rotating a lamina about a non-intersecting axis lying in the same plane is equal to the product of the area of the lamina and the distance traveled by the centroid of.

Why is Pappus theorem used?

Theorem of Pappus lets us find volume using the centroid and an integral. where V is the volume of the three-dimensional object, A is the area of the two-dimensional figure being revolved, and d is the distance traveled by the centroid of the two-dimensional figure.

What is first theorem of Pappus Guldinus?

The Pappus–Guldin Theorems Suppose that a plane curve is rotated about an axis external to the curve. Then 1. the resulting surface area of revolution is equal to the product of the length of the curve and the displacement of its centroid; 2.

What is Guldinus rule?

It states that the volume of each solid of revolution is equal to the area of its base multiplied by the circumference of the circle in which the center of gravity of that figure is revolved. This is the Theorem of Pappus (or the Pappus-Guldin Theorem).

Who invented Pappus Theorem?

Paul Guldin
In mathematics, Pappus’s centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus’s theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin.

Which is an example of pappus’theorem?

The six vertices of the hexagon can be arranged on the two lines in any particular order. With each arrangement of the vertices, we have an instance of Pappus’ theorem. Below is an example: Now, why don’t you try to draw for yourself your own picture of Pappus’ theorem?!

When did pappus’s centroid Theorem appear in print?

The Pappus’s statement of this theorem appears in the print for first time in 1659, but it was known before, by Kepler in 1615 and Guldin in 1640.

How is P appus theorem similar to Pascal’s theorem?

P appus’ theorem looks very similar to Pascal’s hexagon theorem. In Pascal’s theorem, we have a hexagon inscribed in a circle and the intersection points of the three pairs of opposite sides of the hexagon lie on a straight line.

How did Blaise Pascal generalize the theorem of Pappus?

About 1300 years after Pappus wrote the Collection, an interesting generalization of Pappus’ Theorem was discovered by Blaise Pascal based on the ideas of Girard Desargues. This generalization is discussed in the note on Pascal’s Mystic Hexagram. Return to MathPages Main Menu