Calculating the Painted Area

Oct 13, 2025 by magicalceylontours.com 38 views

Introduction

When it comes to calculating the surface area of complex shapes, mathematicians often rely on the principles of geometry. In this article, we will explore the problem of finding the painted area of a pendant formed by a cylinder and a cone. We will use mathematical formulas and techniques to express the result in terms of the given dimensions.

The Shape of the Pendant

The pendant is formed by a cylinder and a cone, as shown in the diagram below. The cylinder has a radius of 10 mm and a height of 20 mm. The cone has a radius of 10 mm and a height of 20 mm. When assembled, the pendant has a total height of 40 mm.

The Cylinder

The cylinder has a radius of 10 mm and a height of 20 mm. To calculate the surface area of the cylinder, we need to find the area of the two circular bases and the lateral surface area.

The area of the two circular bases is given by:
- 2 * π * r^2
- = 2 * π * (10)^2
- = 200π mm^2
The lateral surface area of the cylinder is given by:
- 2 * π * r * h
- = 2 * π * (10) * (20)
- = 800π mm^2

The Cone

The cone has a radius of 10 mm and a height of 20 mm. To calculate the surface area of the cone, we need to find the area of the circular base and the lateral surface area.

The area of the circular base is given by:
- π * r^2
- = π * (10)^2
- = 100π mm^2
The lateral surface area of the cone is given by:
- π * r * l
- = π * (10) * (20)
- = 200π mm^2

Calculating the Painted Area

To calculate the painted area, we need to find the total surface area of the pendant. This includes the surface area of the cylinder and the cone.

The total surface area of the cylinder is given by:
- 2 * π * r^2 + 2 * π * r * h
- = 2 * π * (10)^2 + 2 * π * (10) * (20)
- = 200π + 800π
- = 1000π mm^2
The total surface area of the cone is given by:
- π * r^2 + π * r * l
- = π * (10)^2 + π * (10) * (20)
- = 100π + 200π
- = 300π mm^2
The total surface area of the pendant is given by:
- 1000π + 300π
- = 1300π mm^2

Expressing the Result in Terms of π

To express the result in terms of π, we can use the fact that π is approximately equal to 3.14.

The total surface area of the pendant is given by:
- 1300π
- ≈ 1300 * 3.14
- ≈ 4082 mm^2

Conclusion

In this article, we calculated the painted area of a pendant formed by a cylinder and a cone. We used mathematical formulas and techniques to find the total surface area of the pendant, which includes the surface area of the cylinder and the cone. The result is expressed in terms of π, which is approximately equal to 3.14.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for Engineers and Scientists" by Donald R. Hill
[3] "Calculus: Early Transcendentals" by James Stewart

Introduction

In our previous article, we explored the problem of finding the painted area of a pendant formed by a cylinder and a cone. We used mathematical formulas and techniques to express the result in terms of the given dimensions. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the formula for the surface area of a cylinder?

A: The formula for the surface area of a cylinder is given by:

2 * π * r^2 + 2 * π * r * h

Where r is the radius of the cylinder and h is the height of the cylinder.

Q: What is the formula for the surface area of a cone?

A: The formula for the surface area of a cone is given by:

π * r^2 + π * r * l

Where r is the radius of the cone and l is the slant height of the cone.

Q: How do I calculate the total surface area of the pendant?

A: To calculate the total surface area of the pendant, you need to find the surface area of the cylinder and the cone, and then add them together.

The surface area of the cylinder is given by:
- 2 * π * r^2 + 2 * π * r * h
- = 2 * π * (10)^2 + 2 * π * (10) * (20)
- = 200π + 800π
- = 1000π mm^2
The surface area of the cone is given by:
- π * r^2 + π * r * l
- = π * (10)^2 + π * (10) * (20)
- = 100π + 200π
- = 300π mm^2
The total surface area of the pendant is given by:
- 1000π + 300π
- = 1300π mm^2

Q: Can I use a calculator to calculate the surface area of the pendant?

A: Yes, you can use a calculator to calculate the surface area of the pendant. Simply enter the values of the radius and height of the cylinder, and the radius and slant height of the cone, and the calculator will give you the result.

Q: What is the significance of π in the calculation of the surface area of the pendant?

A: π is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. In the calculation of the surface area of the pendant, π is used to calculate the area of the circular bases and the lateral surface area of the cylinder and the cone.

Q: Can I use a different shape to form the pendant?

A: Yes, you can use a different shape to form the pendant. However, the calculation of the surface area of the pendant will be different depending on the shape used.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the painted area of a pendant formed by a cylinder and a cone. We provided formulas and techniques for calculating the surface area of the cylinder and the cone, and the total surface area of the pendant. We also discussed the significance of π in the calculation of the surface area of the pendant.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for Engineers and Scientists" by Donald R. Hill
[3] "Calculus: Early Transcendentals" by James Stewart