O volume of the sphereis calculated based on the measurement of its radius. A sphere is a geometric shape that has three dimensions. The main elements of a sphere are its radius and diameter. The volume of the sphere is calculated using a specific formula that will be presented below. In addition to the volume, we can calculate the surface area of the sphere.
Read too: How to calculate the volume of a cylinder
Summary of sphere volume
- Several objects in our daily lives have a spherical shape, such as a soccer ball.
- The main elements of the sphere are its radius and diameter.
- To calculate the volume of the sphere, we use the formula:
\(V=\frac{4\pi R^3}{3}\)
- There are other important formulas, such as the formula for the area of a sphere: \(A=4\pi r^2\).
Video lesson on sphere volume
What is a sphere?
A sphere is a single three-dimensional shape, defined as a three-dimensional figure whose points are equally distant from its center. It is one of the most symmetrical shapes and is present in our world in many ways. We can perceive the presence of the sphere in nature, in the human body, in the study of the planets, among other situations in our daily lives.
A sphere is a geometric solid. The billiard, football and basketball ball are examples of spheres. It is made up of all points that are at a constant distance from a central point called the center of the sphere. And this constant distance is known as the radius of the sphere.
Sphere elements
The sphere has some interesting parts:
- Center: as the name suggests, it is the point that is in the center of the sphere.
- Diameter: straight line segment that connects two opposite points on the sphere, passing through the center.
- Ray: segment that goes from the center to any point on the surface.
- Surface: outer layer of the sphere.
- Inside: space inside the sphere.
How do you calculate the volume of the sphere?
The volume of the sphere is calculated by the formula:
\(V=\frac{4}{3}\pi R^3\)
- V: is the volume of the sphere.
- A: is the radius of the sphere.
- π: is a constant.
Oconstant value πmost commonly used is approximately 3.14, but we can consider π equal to approximately 3, or approximately 3.1, or even approximately 3.1415, depending on how many decimal places we want to consider, since the π is an irrational number, and irrational numbers have infinite decimal places.
- Example:
A sphere has a radius of 6 cm. What is the volume of this sphere, considering that π=3?
Resolution:
Calculating the volume of the sphere, we have:
\(V=\frac{4\pi R^3}{3}\)
\(V=\frac{4\cdot3\cdot6^3}{3}\)
\(V=\frac{12\cdot216}{3}\)
\(V=\frac{2592}{3}\)
\(V=864\ cm^3\)
So, the volume of this sphere is 864 cm³.
Another sphere formula
In addition to the formula presented to calculate the volume of the sphere, there is another important formula, which is the surface area formula. To calculate the surface area of the sphere, the formula is:
\(A=4\pi r^2\)
A surface of the sphere is nothing more than the region that surrounds the sphere. For example, in a plastic ball, the sphere is the entire ball, and the surface is the region of the plastic that is the contour of that ball.
- Example:
What is the surface measurement of a sphere that has a radius of 5 cm?
Resolution:
As the value of π, we will not replace it with any value, so:
\(A=4\cdot\pi\cdot5^2\)
\(A=4\cdot\pi\cdot25\)
\(A=100\pi\ cm²\)
The area of this sphere is in 100π cm2.
Know more: What is the difference between circumference, circle and sphere?
Solved exercises on sphere volume
Question 1
A spherical object has a radius of 6 cm. Then the volume of this object (using π=3,14) is approximately equal to:
A) 314.42 cm³
B) 288.00 cm³
C) 424.74 cm³
D) 602.38 cm³
E) 904.32 cm³
Resolution:
Alternative E
Substituting the values given in the statement into the formula \(V=\frac{4}{3}\pi R^3\), we have:
\(V=\frac{4}{3}\pi6^3\)
\(V=\frac{4}{3}\pi216\)
\(V=288\pi\approx288\cdot3,14=904.32{\cm}^3\)
Question 2
A container has a spherical shape. It is known that it has volume in 288π cm³. Knowing its volume, we can then state that the measurement of the radius of this container is:
A) 3 cm
B) 4cm
C) 5cm
D) 6 cm
E) 7 cm
Resolution:
Alternative D
We know that \(V=288\pi\).
Substituting the values given in the statement into the formula \(V=\frac{4}{3}\pi R^3\), we have \(288\pi=\frac{4}{3}\pi R^3\).
Canceling the π on both sides and cross-multiplying:
\({4R}^3=864\)
\(R^3=216\)
\(R=\sqrt[3]{216}\)
\(R=\sqrt[3]{6^3}\)
\(R=6\ cm\)
Sources
DOLCE, Osvaldo; POMPEO, José Nicolau. Fundamentals of Elementary Mathematics: Spatial Geometry, vol. 10, 6. ed. São Paulo: Current, 2005.
LIMA, E. et. al. High school mathematics. volume 2. Rio de Janeiro: SBM, 1998.
