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Getting ready for solid geometry

Practicing finding area of 2D figures, volume of 3D figures, and comparing proportional relationships will help us apply what we know from simpler figures to more complicated solid figures.
Let’s refresh some concepts that will come in handy as you start the solid geometry unit of the high school geometry course. You’ll see a summary of each concept, along with a sample item, links for more practice, and some info about why you will need the concept for the unit ahead.
This article only includes concepts from earlier courses. There are also concepts within this high school geometry course that are important to understanding solid geometry. If you have not yet mastered the Transformations properties and proofs unit, it may be helpful for you to review that before going farther into the unit ahead.

Area of 2D figures

What is this, and why do we need it?

Area is the amount of space enclosed in a 2D figure. We'll find the area of bases and cross-sections of 3D figures as a first step toward finding their volume. Also, understanding the area of cross-sections will help us relate volumes of tricky figures to the volumes of more familiar figures.

Practice

Problem 1.1
What is the area of the following semicircle?
Either enter an exact answer in terms of pi or use 3, point, 14 for pi and enter your answer as a decimal.
A semi-circle with a diameter of ten units.
Area equals
start text, c, m, end text, squared

For more practice, go to Area of a circle and Area of triangles.

Where will we use this?

Here are a few of the exercises where reviewing area might be helpful:

Volume of solid figures

What is this, and why do we need it?

Volume is the amount of space enclosed in a 3D figure. We'll use the volumes of familiar solids, like right rectangular prisms, to help us find the volumes of more unusual figures, like prisms that lean to one side, all sorts of pyramids, and figures with different shapes of bases.

Practice

Problem 2.1
Find the volume of the cylinder.
Either enter an exact answer in terms of pi or use 3, point, 14 for pi.
A cylinder with a radius of six units and a height of four units.
unitscubed

Where will we use this?

Here are a few of the exercises where reviewing the volume of solid figures might be helpful:
We'll also explore why several of these volume formulas work. Come back at the end of the unit to compare and contrast these examples and see how we could solve them with fewer different formulas.

Comparing proportional relationships

What is this, and why do we need it?

Proportional relationships are two quantities where the ratio between the two quantities always stays the same.
Density is a kind of proportional relationship that relates some quantity (such as mass or number of people) to the volume or area of a region. When we apply density, we often need to compare the density below some value, such as keeping the mass of a loaded ship less than 1, comma, 000, start text, space, k, g, end text per cubic meter, so that the ship will float.

Practice

Problem 3
The recipe for Jack's Famous Chili uses 2 teaspoons left parenthesis, start text, t, end text, right parenthesis of red hot
chili powder
for each cup left parenthesis, start text, c, end text, right parenthesis of tomato sauce.
His rival, Marsha, claims her chili is spicier. She makes chili in large quantities, and uses the chart below.
Whose chili is spicier?
Chili powder left parenthesis, start text, t, end text, right parenthesisTomato sauce left parenthesis, start text, c, end text, right parenthesis
43, point, 5
87, point, 0
1210, point, 5
Choose 1 answer:
Choose 1 answer:

For more practice, go to Rates & proportional relationships.

Where will we use this?

Here is an exercise where reviewing how to compare proportional relationships might be helpful:

Want to join the conversation?

  • female robot ada style avatar for user brain fog
    I didn't remember any of the formulas for volume in 2D shapes. Do we have to have these memorized? (ex. volume of sphere)
    (1 vote)
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    • aqualine ultimate style avatar for user Shall Win
      It depends on what grade your in. If your in/going to be in Geometry, I find it helpful to memorize, because it doesn't take that long to learn all of it, and sometimes it might pop-up on a test in your math class. You usually don't need to go really deep with complex shapes, but mastering the fundamentals are vital.

      Here are a few things that I'd recommend having memorized.

      - Area of all 2D Shapes
      - Volume of 3D Shapes (Spheres, Cones, etc)
      - Surface area of 3D Shapes (Spheres, Cones, etc).

      To help with memorization I'd recommend watching videos on it, taking notes, doing problems including them once in a while, until when you see it on a test, you know the formula and can do it.

      Hope this helps.
      (3 votes)
  • blobby green style avatar for user ibarraj
    Why is this no new to me??
    (1 vote)
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  • blobby green style avatar for user shaunevers25
    will these help me in life
    (1 vote)
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  • blobby green style avatar for user shaunevers25
    how do i get better at this
    (1 vote)
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  • primosaur seed style avatar for user Sky
    Why does Jack’s tomato sauce is spicier? I still don’t get it.
    (0 votes)
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    • duskpin ultimate style avatar for user Alex Samuelson
      Jack's chili is spicier than Marsha's chili because he uses a more concentrated amount of the chili powder for each cup of tomato sauce. Consider this:
      The problem indicates that for every 1 cup (c) of tomato sauce Jack uses he adds 2 teaspoons (t) of chili powder. This gives him a "2 t of chili powder"-to-"1 c of tomato sauce" ratio in his chili dish.
      The problem also indicates that Marsha adds 1 1/7 t of chili powder for every 1 c of tomato sauce. This means she has a "1 1/7 t of chili powder"-to-"1 c of tomato sauce ratio" in her chili dish.
      Jack adds 6/7 t of chili powder more to his chili than Marsha for every c of tomato sauce that is in the dish. This makes his chili more spicy because he has a higher ratio of chili powder to tomato sauce.

      The solution to this is shown in more detail in the "Explain" tab under Problem 3.
      (2 votes)
  • blobby green style avatar for user Fatima Zaragoza
    What are proportional relationships used for?
    (0 votes)
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  • blobby green style avatar for user robert.nelson
    At Pm where is the sun
    (0 votes)
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