Step-by-Step Solutions for Pappus Theorem Problems
Pappus theorem (also known as Pappus - Guldinus theorem, pappus theorem) is named after the Greek Mathematician Pappus of Alexandria (4th century A.D.) It is a theorem that deals with the area of surface of revolution and volume of surfaces and solids of revolution.
The Pappus Guldinus theorem helps to find the surface area and volume of surfaces and solids of revolution.
The statement of the Pappus theorem was introduced in the print for the first time in 1659, but it was known before by Kepler in 1659, and Guldin in 1640.
The First Pappus Guldinus Theorem
Let C be any curve in the plane. The area of the surface (A) obtained when C is revolved around an external axis is equivalent to the product of the arc length L of C and distance (d) travelled by the Centroid of C.
\[A = Ld\]
For example, the surface area of torus with minor and major radius r and R respectively is: V = 2\[\pi r\]+ 2\[\pi R\]= 4\[\pi^{2}r R\]
The Second Pappus Guldinus Theorem
Let R be any region in the plane. The volume of the solid obtained when R is revolved around a central axis and is equivalent to the product of the area A of R and the distance covered (d) by the centroid of R.
\[V = Ad\]
For example, the volume of torus with minor and major radius r and R respectively is:
\[V = (\pi r^{2})(2\pi R) = 2\pi^{2}Rr^{2}\]
This special case was introduced by Johannes Kepler using infinitesimals.
Examples
Let us consider the cylinder obtained by revolving a rectangle with vertical side h and horizontal side r around one of its vertical sides (say its left side). The surface area of the cylinder, excluding the top and bottom, is calculated using the Pappus theorem as the surface is obtained by revolving its right side around its left side. The length of the arc of its right side is h and the distance travelled by its centroid is simply 2r. Therefore, the area of the circular cylinder will be 2\[\pi rh\].
The volume of the cylinder is the distance travelled by its centroid multiplied by the area rh of the rectangle. The centroid of the rectangle is considered as its centre, which is a distance \[\frac{r}{2}\] from the axis of revolution. Hence, it covers a distance of 2\[\pi\] \[\frac{r}{2}\] as it revolves. Therefore, the volume of the cylinder is \[(\pi r)\] \[(rh)\]=\[\pi r^{2}h\].
Pappus And Guldinus Theorem Examples With Solutions
Example 1:
The surface area and volume of torus can easily be calculated using the Pappus theorem. Considering the radius of circle r, and distance from the centre of the circle to the axis of its revolution is R, then the surface area of torus can be calculated using the first Pappus theorem as:
\[A = Ld\] = \[2 \pi r\].\[2\pi R\]= \[4\pi^{2}rR\]
(Image will be uploaded soon)
The volume inside the torus can be calculated using the second Pappus theorem as:
\[V = Ad\] = \[\pi r^{2}\].\[2\pi R\]= \[2\pi^{2}r^{2}R\]
Example 2:
Find the volume of the right circular cone with radius r and height (h) = 10
The Pappus theorem for volume is defined as\[V = Ad\]. Before calculating the volume, it is important to find the area of the triangle to which we are revolving. The right circular cone shape can be considered as a triangle rotated around an axis. The formula for the area of triangle is given as:
\[\frac{1}{2}\]\[base \times height\]
The base of the triangle will be considered as the radius (r) = 6 and the height (h) of the triangle = 10.
A = \[\frac{1}{2} \times 6 \times 10\]
A = 30
The next step is to solve for distance (d). Distance will include the relationship of the centroid of the triangle and its rotation.
In other words, the distance (d) = \[2\pi \bar{x}\], where \[\bar{x}\] is the coordinate of the centroid and 2refers to the fact that the object is being rotated around an axis. Therefore, the equation for \[\bar{x}\] is given as:
\[\bar{x}\]= \[\frac{1}{A}\] \[\int_{a}^{b}\] \[x f(x) dx\]
Here, we are still missing the f(x), which is the third side of the triangle. There are different ways to calculate the third side of the triangle. But, we will use the pythagorean theorem which states that
H² = h² + b²
Here, H is the hypotenuse, b is the base, and h is the height
Now, we will solve for H and substitute the values as shown below:
H = \[\sqrt{h^{2} + b^{2}}\]
H = \[\sqrt{(10)^{2} + (6)^{2}}\]
H = \[\sqrt{136}\]
Note : H =\[f(x)\]
Hence, \[f (x)\] = 0
Let us now solve for \[\bar{x}\]
\[\bar{x}\] = \[\frac{1}{30}\] \[\int_{0}^{6}\] \[x \sqrt{136} dx\]
\[\bar{x}\] = \[\frac{\sqrt{136}}{30}\] \[\int_{0}^{6}\] \[x\] \[dx\]
\[\bar{x}\] =\[\frac{\sqrt{136}}{30}\] \[(\frac{x^{2}}{x})\] \[\int_{0}^{6}\]
\[\bar{x}\] =\[\frac{\sqrt{136}}{30}\] \[[\frac{(6)^{2}}{2} - \frac{(0)^{2}}{2}]\]
\[\bar{x}\] =\[\frac{6\sqrt{34}}{5}\]
Now, let us solve for distance d = 2\[\lambda\] \[\bar{x}\]
\[d = 2 \pi \frac{6\sqrt{34}}{5}\]
\[d = \frac{12\pi\sqrt{34}}{5}\]
At last, we will solve for volume using V = Ad
V = (20) \[(\frac{12\pi\sqrt{34}}{5})\]
V = 72\[\pi \sqrt{34}\]
Therefore, the volume of right circular cone with base radius 6 and height 10 is V = 72\[\pi \sqrt{34}\]
FAQs on Understanding Pappus Theorem: Concepts and Applications
1. What is the Pappus theorem?
Pappus's theorem refers to two important results in geometry, known as Pappus's Centroid Theorem and Pappus's Hexagon Theorem. Most commonly in mathematics, the theorem refers to the Pappus's centroid theorems, which relate the surface area and volume generated when a plane figure rotates about an external axis. The second Pappus theorem, for example, states:
- The volume generated by rotating a plane region about an external axis (which does not intersect the region) is equal to the product of the area of the region and the distance traveled by its centroid.
2. How to find volume using pappus theorem?
To find the volume using Pappus's theorem, you use the following steps:
- Identify the centroid of the given plane area (let’s call the area $A$).
- Determine the distance traveled by the centroid as the region revolves (this is the path’s circumference, $d = 2\pi r$, where $r$ is the distance from centroid to axis of rotation).
- Apply Pappus's second centroid theorem: $$\text{Volume} = (\text{Area of region}) \times (\text{Distance traveled by centroid})$$ or $$V = A \times (2\pi r)$$
3. What is the first theorem of Pappus?
The first theorem of Pappus (also called Pappus's first centroid theorem) states:
- The surface area generated when a plane curve of length $L$ is revolved about an axis (external and coplanar) equals the product of the curve’s length and the distance traveled by its centroid.
4. What are the axioms needed to prove Pappus' theorem?
The axioms used to prove Pappus's theorem include fundamental geometric and calculus concepts, such as:
- The parallel axis theorem and concept of centroids.
- The definition of arc length, area, and volume by integration.
- The movement of a point or region along a circular path in the plane.
5. What are practical applications of Pappus's theorem in real life?
Pappus's theorem finds applications in engineering, physics, and design:
- Volume and surface area calculations for tanks, pipes, and mechanical parts.
- Designing objects of revolution, such as bottles and domes.
- Structural analysis in civil and mechanical engineering.
6. How does Pappus's theorem simplify the calculation of solids of revolution?
Pappus's theorem allows you to bypass complex integration when finding the surface area or volume of solids generated by rotation. Instead of integrating, you can multiply the known area (or length) by the distance the centroid travels: $$\text{Volume} = \text{Area} \times (2\pi r)$$ This shortcut significantly speeds up solutions, and is a technique detailed in Vedantu’s step-by-step math tutorials.
7. What is the difference between Pappus's first and second centroid theorems?
The key differences are:
- Pappus's First Theorem: Calculates the surface area generated by revolving a plane curve around an external axis.
- Pappus's Second Theorem: Calculates the volume generated by revolving a plane area around an external axis.
8. Can Pappus's theorem be used for all types of curves and regions?
Pappus's theorem applies to cases where the region or curve is plane and the axis of rotation lies outside and in the same plane. It cannot be used if the axis crosses or passes through the region. The figure must be simple and the centroid must maintain a constant distance from the axis. In Vedantu’s classes, students learn which types of shapes and rotations qualify for direct application of Pappus’s theorem.
9. How do you find the centroid for use in Pappus's theorem?
To use Pappus’s theorem, you must first find the centroid ($\bar{x}$, $\bar{y}$) of the curve or region:
- For uniform shapes (like rectangles or circles), the centroid formulas are straightforward and included in Vedantu’s formula sheets.
- For irregular shapes, you calculate the centroid using integration: $$\bar{x} = \frac{\int x dA}{\int dA},\quad \bar{y} = \frac{\int y dA}{\int dA}$$
