The math sections on the SAT and ACT will require you to answer geometry related questions.
If you’re taking the test during your junior or senior year, chances are you may be a little rusty when it comes to geometry concepts you learned years before. Before test day, you’ll want to brush up on key formulas for calculating the area, circumference, and volume of different shapes, including spheres.
While you will likely only encounter a question or two about spheres on the SAT or ACT, it will be important for you to be able to answer these questions if you want to earn a score in the 99th percentile.
This guide will walk you through everything you’ll need to know about spheres and calculating their volume. I’ll also give you some tips and tricks for memorizing formulas and mastering the math sections on the SAT and ACT.
What is a sphere?
A sphere is a round, three-dimensional shape that consists of a set of points that are a given, equal distance from its center. It is the three-dimensional version of a two-dimensional circle.
The points that make up a sphere are a specific distance, also known as r or the radius of the sphere.
What is the volume of a sphere
The volume of a sphere or any shape is the amount of space that the shape can occupy. If you were to fill up the sphere with water or some other substance, the sphere’s volume would be the amount of water it could hold.
This is why when you hold up a fully inflated basketball it does not weigh the same as a fully inflated soccer ball. In addition to being made out of different materials, these two spheres hold different volumes.
Volume of a sphere formula
Fortunately, mathematicians don’t need to cut spheres in half and carry around buckets of water to determine their volume.
There is a simple formula that they can use as long as they know the radius of the sphere (which is half of the diameter).
The formula used to calculate the volume of a sphere is V= 43πr3
In this formula, V represents volume and r represents the radius of the sphere.
Calculating the volume of a sphere
To calculate the volume of a sphere, you only need to complete a few steps.
First, make sure you know the formula used to calculate this volume: V= 43πr3.
Once you have the formula, you need to find the radius of the sphere. On a math test, you may be given the radius, which will make this step easy. However, you may be given the diameter instead.
If that is the case, you need to divide the diameter by 2 in order to get the radius. Once you have the radius, you can plug it into the equation.
After you have the radius, you need to cube this number. For example, if the radius of the sphere was 2, you would find 23, which is 8. You need to cube this number because a sphere is a three-dimensional shape.
Your next step will be to multiply this number by π, which is roughly 3.1415926.
Finally, multiply the number you get in the previous step by 43. This will give you the volume of the sphere.
Ultimately, to calculate the volume of a sphere, you need to do the following steps:
- Find the volume formula
- Find the radius of the sphere
- Cube the radius
- Multiply your answer by π
- Multiply your answer by 43
When you finish these steps, you will have your answer in cubic units.
Practicing calculating the volume of a sphere
The more you practice these steps, the easier it will be for you to calculate the volume of a sphere on assignments and tests for your math classes.
Here are a few “practice problems” you can use to get started:
1.) Given a sphere with a radius of 3 in, calculate its volume.
2.) Calculate the volume of the sphere pictured below:
3.) Determine the volume of a sphere with a diameter of 4 cm.
Pause and take the time to answer the questions above. When you are finished, check your answers below!
Answer #1: When given a sphere with a radius of 3 in, the volume is ≈113.1 cubic inches.
Start with V= 43πr3
Plug in the radius, which is 3 for this sphere, to get V= 43π33
Cube the radius to get V= 43π27
When you complete this multiplication, you should get roughly 113.1.
Answer #2: The sphere in the diagram for question #2 has a radius of 10 and a volume of ≈4188.7902 cubic inches.
Start with V= 43πr3
Plug in the radius, which is 10 for this sphere, to get V= 43π103
Cube the radius to get V= 43π1000
When you complete this multiplication, you should get roughly 4188.7902.
Answer #3: Given a sphere with a diameter of 4 cm, the volume is ≈33.51032 cubic centimeters.
Start with V= 43πr3
To determine the radius, you must divide the diameter by 2. This will give you a radius of 2.
Plug in the radius, which is 2 for this sphere, to get V= 43π23
Cube the radius to get V= 43π8
When you complete this multiplication, you should get roughly 33.51032.
If you answered all three of these questions correctly, congratulations! You’re doing great. If you missed some of these questions, don’t worry. With extra practice, you’ll have this concept down in no time.
Learning additional math tips
Once you’ve mastered the steps required to calculate the volume of a sphere, you can move on to calculating the circumference and surface area of spheres as well as the volume of other three-dimensional shapes.
You can learn the formulas you’ll need for these math concepts as well as additional tips to help you do well in your math courses when you sign up for private math tutoring through Prep Expert.
Earning a high grade in math is essential for students who want to get into a prestigious college or go into a STEM field in the future.
Working with a private tutor will ensure that you are on track to earn good grades in your math courses and help you develop the foundational math skills needed to thrive in STEM. You will be able to learn at your own pace, reviewing math topics where you need extra practice and moving forward through math concepts that you’ve already mastered.
At Prep Expert, we hire high quality math tutors who know how to teach complex math concepts in a way that makes them simple to understand. Learn more or sign up for private math tutoring through Prep Expert today when you visit our website.