Formula for calculating the lateral area of ​​a truncated cone, the total area of ​​a truncated cone, the volume of a truncated cone

We invite you to learn what a truncated cone is, the formula for calculating the volume of a truncated cone, the lateral area and the total area of ​​a truncated cone in the article below.

As learned from the previous article, a pyramid is formed when a right triangle is rotated around its axis (a right angle side) one revolution.

Truncated cone

• 1. Calculate the area of ​​a truncated cone
  • 1.1. Calculate the lateral area of ​​a truncated cone
  • 1.2. Calculate the total area of ​​a truncated cone
• 2. Calculate the volume of a truncated cone
• What is a truncated cone?

1. Calculate the area of ​​a truncated cone

The area of ​​a truncated cone is often referred to with two concepts: lateral area and total area.

1.1. Calculate the lateral area of ​​a truncated cone

In there:

• The perimeter is the area around the truncated cone.
• r1and r2is the radius of the two bases of the truncated cone.
• lis the length of the generator of the frustum cone.

The lateral area of ​​a truncated cone only includes the area of ​​the surrounding faces surrounding the cone, not including the area of ​​the two bases.

Formula for calculating lateral area: equal to the difference in lateral area of ​​the large cone and the small cone.

1.2. Calculate the total area of ​​a truncated cone

• r1and r2is the radius of the two bases of the truncated cone.
• lis the length of the generator of the frustum cone.

Hence:

In there:

• The area around a truncated cone is the area around it.
• Stoalphan is the total surface area of ​​a truncated cone.
• S2day is the area of ​​the 2 bottom surfaces

The total area is calculated as the magnitude of the entire space occupied by the figure, including the lateral area and the area of ​​the two circular bases.

Formula for calculating total area: equals lateral area plus area of ​​2 bases.

For example:

Given a truncated cone with base radii r1 and r2 equal to 5cm and 7cm respectively. The generator l connecting the top to the base of the cone is 6cm. What is the total area and lateral area of ​​this cone?

Prize:

Applying the formula to calculate the total area of ​​a truncated cone, we have r1 = 5cm, r2 = 7cm and the length of the generator l = 6cm. Therefore, the total area of ​​a truncated cone when applied according to the formula is as follows:

Stp = π.(5 + 7).4 + (π.52 + π.72) = π.12.4 + (π.25 + π.49) = 383.08 (cm2).

Thus the total area of ​​this truncated cone is approximately 383.08 cm2.

The lateral area of ​​a truncated cone is:

Sxq = π.(r1 + r2).l = π.(5 +7).6 ~ 226 cm2.

2. Calculate the volume of a truncated cone

The volume of a frustum is the amount of space the frustum occupies.

Formula for calculating the volume of a truncated cone: equal to the difference in volume of the large cone and the small cone.

In there:

• Vis the volume of the truncated cone.
• r1and r2is the radius of the two bases of the truncated cone.
• his the height of the truncated cone (distance between the 2 bases).

For example : Given a truncated cone with base radii r1 and r2 equal to 5cm and 9cm respectively. The height connecting the two base radii is 8cm long. What is the total surface area of ​​this cone?

Prize: 

Applying the formula for calculating the volume of a truncated cone, we have: r1 = 5cm, r2 = 9cm, h = 8cm.
V = 1/3π.8. (52 + 5.9 +92) = 1264.37 (cm3).

Thus the volume of this truncated cone is approximately 1264.367 cm3.

Example 2:

Given a truncated cone as shown in the figure.

Knowing that the radius of the small base is r = 3cm, the radius of the large base is R = 6cm, the length of AB = 4cm. Calculate the lateral area and volume of the truncated cone.

Prize:

The lateral area of ​​a truncated cone is:

Sxq = π(r + R)l = π(3 + 6).4 = 36π (cm2)

To calculate the height of a truncated cone, we have the following drawing:

Applying the Pythagorean theorem and the right triangle AHB at H, we have:

The volume of a truncated cone is:

• Formula for calculating the volume of a truncated pyramid, the lateral area and total area of ​​a truncated pyramid

What is a truncated cone?

When a cone is cut by a plane parallel to the base, the part of the plane lying inside the cone is a circle . The part of the shape lying between the above plane and the base is called a frustum cone.

It can be understood that a truncated cone is a shape with two bases which are two circles with different radii lying on two parallel planes with the line connecting the centers as the axis of symmetry.

You can easily see that we often encounter truncated cones in life such as buckets or lampshades... Hopefully through the above article, you have a better understanding of truncated cones and how to calculate the area and volume of truncated cones.