O cone volume is calculated when we multiply the base area and height and divide by three. This is one of the calculations that can be done in relation to this geometric solid, classified as a round body because it is formed by a circular base or because it is formed through the rotation of a triangle.
Read too: What are the volume measurements?
Cone volume summary
To calculate the volume of the cone, it is necessary to know the measurements of the base radius and height.
The volume of cone is calculated by the formula:
\(V=\frac{\pi r^2\cdot h}{3}\)
Since the base of the cone is a circle, we use the circle area formula to calculate the area of the cone's base, i.e. \(A_b=\pi r^2\).
Video lesson on cone volume
What are the elements of the cone?
The cone is known as a round body or solid body of revolution because it has a base formed by a circle. This geometric solid is quite common in our daily lives, used, for example, in traffic to signal an area where cars cannot pass. The cone has three important elements: the height, the base and the apex.
What is the formula for the volume of the cone?
The volume of a cone is calculated by product between the area of the base and the height divided by three, that is, it can be calculated by the formula:
\(V=\frac{A_b\cdot h}{3}\)
V: volume
AB: base area
h: cone height
It turns out that The area of the base is not always known. In this case, as the base of a cone is formed by a circle, we can use the circle area formula to calculate the area of the base. In other words, in a cone the area of the base is calculated by \(A_b=\pi r^2\), which allows us to calculate its volume using the formula:
\(V=\frac{\pi r^2\cdot h}{3}\)
V: cone volume
r: base radius
h: cone height
How is the volume of the cone calculated?
To calculate the volume of the cone, It is necessary to find the values of its height and radius. Knowing this data, simply substitute the values in the cone volume formula and perform the necessary calculations.
Example 1:
Calculate the volume of the cone that has a radius of 5 cm and a height of 12 cm.
Resolution:
We know that:
r = 5cm
h = 12 cm
Substituting into the formula:
\(V=\frac{\pi r^2\cdot h}{3}\)
\(V=\frac{\pi\cdot5^2\cdot12}{3}\)
\(V=\frac{\pi\cdot25\cdot12}{3}\)
\(V=\frac{300\pi}{3}\)
\(V=100\pi cm^3\)
Example 2:
Calculate the volume of the following cone, using 3.1 as an approximation for the value of π.
Resolution:
The data is:
r = 6 cm
h = 12 cm
π = 3,1
Calculating the volume of the cone:
\(V=\frac{\pi r^2\cdot h}{3}\)
\(V=\frac{3,1\cdot6^2\cdot12}{3}\)
See too: How is the volume of the cylinder calculated?
Solved exercises on cone volume
Question 1
A reservoir was built in the shape of a cone. Knowing that it has a base diameter of 8 meters and a height of 5 meters, with π = 3, the volume of this reservoir is:
A) 12 m³
B) 15 m³
C) 18 m³
D) 20 m³
E) 22 m³
Resolution:
Alternative D.
Considering that the diameter of the base is 8 meters and that the radius is half the diameter:
r = 8: 2 = 4 m
The other information is that h = 5 and π = 3.
Calculating the volume of the cone:
\(V=\frac{\pi r\cdot h}{3}\)
\(V=\frac{3\cdot4\cdot5}{3}\)
\(V=4\cdot5\)
\(V=20\ m^3\)
Question 2
A cone-shaped package must be 310 m³. Since the height of this package is 12 cm, its radius must be: (Use 3.1 as an approximation of π)
A) 3 cm
B) 4cm
C) 5cm
D) 6 cm
E) 7 cm
Resolution:
Alternative C
The data are that V = 310, h = 12 and π = 3.1.
Substituting the known values into the volume formula:
\(V=\frac{\pi r^2\cdot h}{3}\)
\(310=\frac{3,1\cdot r^2\cdot12}{3}\)
\(310\cdot3=3,1\cdot r^2\cdot12\)
\(930=37.2r^2\)
\(\frac{930}{37,2}=r^2\)
\(25=r^2\)
\(r=\sqrt{25}\)
\(r=5\ cm\)
Therefore, the radius must be 5 cm.
