Calculating the Base Radius of the Cone

Oct 13, 2025 by wbrum.org 91 views

Introduction

In this article, we will explore the properties of a cone formed by a sector of a circle. We will calculate the base radius of the cone, the height of the cone, the total surface area, the volume, and the vertical angle of the cone.

Calculating the Base Radius of the Cone

A sector of a circle with a radius of 7 cm and an angle of 120° at the center of the circle is used to form a cone. To calculate the base radius of the cone, we need to use the formula:

r = (θ / 360) × R

where r is the base radius of the cone, θ is the angle of the sector in degrees, and R is the radius of the circle.

Plugging in the values, we get:

r = (120 / 360) × 7 r = 0.3333 × 7 r = 2.33 cm

Calculating the Height of the Cone

To calculate the height of the cone, we need to use the formula:

h = R - r

where h is the height of the cone, R is the radius of the circle, and r is the base radius of the cone.

Plugging in the values, we get:

h = 7 - 2.33 h = 4.67 cm

Calculating the Total Surface Area of the Cone

The total surface area of the cone is the sum of the area of the base and the area of the curved surface. The area of the base is given by:

A = πr^2

where A is the area of the base, π is a mathematical constant approximately equal to 3.14, and r is the base radius of the cone.

The area of the curved surface is given by:

A = πrl

where A is the area of the curved surface, π is a mathematical constant approximately equal to 3.14, r is the base radius of the cone, and l is the slant height of the cone.

To calculate the slant height of the cone, we need to use the formula:

l = √(R^2 + h^2)

where l is the slant height of the cone, R is the radius of the circle, and h is the height of the cone.

Plugging in the values, we get:

l = √(7^2 + 4.67^2) l = √(49 + 21.81) l = √70.81 l = 8.43 cm

Now, we can calculate the total surface area of the cone:

A = πr^2 + πrl A = 3.14 × (2.33)^2 + 3.14 × 2.33 × 8.43 A = 3.14 × 5.42 + 3.14 × 19.59 A = 17.05 + 61.55 A = 78.6 cm^2

Calculating the Volume of the Cone

The volume of the cone is given by:

V = (1/3)πr^2h

where V is the volume of the cone, π is a mathematical constant approximately equal to 3.14, r is the base radius of the cone, and h is the height of the cone.

Plugging in the values, we get:

V = (1/3) × 3.14 × (2.33)^2 × 4.67 V = (1/3) × 3.14 × 5.42 × 4.67 V = 1.047 × 25.38 V = 26.55 cm^3

Calculating the Vertical Angle of the Cone

The vertical angle of the cone is the angle between the base of the cone and the slant height of the cone. To calculate the vertical angle of the cone, we need to use the formula:

θ = arcsin(h / l)

where θ is the vertical angle of the cone, h is the height of the cone, and l is the slant height of the cone.

Plugging in the values, we get:

θ = arcsin(4.67 / 8.43) θ = arcsin(0.555) θ = 32.5°

Conclusion

In this article, we calculated the base radius of the cone, the height of the cone, the total surface area, the volume, and the vertical angle of the cone formed by a sector of a circle with a radius of 7 cm and an angle of 120° at the center of the circle. The results are:

Base radius of the cone: 2.33 cm
Height of the cone: 4.67 cm
Total surface area of the cone: 78.6 cm^2
Volume of the cone: 26.55 cm^3
Vertical angle of the cone: 32.5°

Q: What is a sector of a circle?

A: A sector of a circle is a part of a circle enclosed by two radii and an arc. It is a portion of the circle that is bounded by two radii and a portion of the circle's circumference.

Q: How is a sector of a circle used to form a cone?

A: A sector of a circle is used to form a cone by rolling the sector into a cone shape. The radii of the sector become the slant height of the cone, and the arc of the sector becomes the curved surface of the cone.

Q: What is the formula for calculating the base radius of a cone formed by a sector of a circle?

A: The formula for calculating the base radius of a cone formed by a sector of a circle is:

r = (θ / 360) × R

where r is the base radius of the cone, θ is the angle of the sector in degrees, and R is the radius of the circle.

Q: What is the formula for calculating the height of a cone formed by a sector of a circle?

A: The formula for calculating the height of a cone formed by a sector of a circle is:

h = R - r

where h is the height of the cone, R is the radius of the circle, and r is the base radius of the cone.

Q: What is the formula for calculating the total surface area of a cone formed by a sector of a circle?

A: The formula for calculating the total surface area of a cone formed by a sector of a circle is:

A = πr^2 + πrl

where A is the total surface area of the cone, π is a mathematical constant approximately equal to 3.14, r is the base radius of the cone, and l is the slant height of the cone.

Q: What is the formula for calculating the volume of a cone formed by a sector of a circle?

A: The formula for calculating the volume of a cone formed by a sector of a circle is:

V = (1/3)πr^2h

where V is the volume of the cone, π is a mathematical constant approximately equal to 3.14, r is the base radius of the cone, and h is the height of the cone.

Q: What is the formula for calculating the vertical angle of a cone formed by a sector of a circle?

A: The formula for calculating the vertical angle of a cone formed by a sector of a circle is:

θ = arcsin(h / l)

where θ is the vertical angle of the cone, h is the height of the cone, and l is the slant height of the cone.

Q: What are some real-world applications of a sector of a circle?

A: A sector of a circle has many real-world applications, including:

Architecture: Sectors of circles are used to design and build domes, arches, and other curved structures.
Engineering: Sectors of circles are used to design and build pipes, tubes, and other cylindrical structures.
Art: Sectors of circles are used to create circular patterns and designs.
Science: Sectors of circles are used to model and analyze circular motion and other circular phenomena.

Q: What are some common mistakes to avoid when working with sectors of circles?

A: Some common mistakes to avoid when working with sectors of circles include:

Not accounting for the angle of the sector when calculating the base radius and height of the cone.
Not using the correct formula for calculating the total surface area and volume of the cone.
Not considering the slant height of the cone when calculating the vertical angle.
Not using the correct units of measurement when working with sectors of circles.

Conclusion

In this article, we have answered some common questions about sectors of circles and their applications in mathematics and real-world scenarios. We have also provided formulas and examples to help you understand and work with sectors of circles.