# 3D Shapes and Volume

A three-dimensional (3D) shape has three dimensions: length, width, and height. A 2D shape has only length and width.

There are several 3D shapes, several of which can be put in one of these categories: prism or pyramid:

• Prism: a prism has two congruent shapes at opposite ends, parallel to each other, and connected by rectangular faces
• Pyramid: a pyramid has a basic shape on the base, with triangular faces that come to a point

Depending on the shape at the ends/base of these two basic 3D shapes, we can generate a whole bunch of prisms and pyramids.

Note how the name is determined by the shape of the base: if the base is a square and the sides are triangles that come to a point, it is a square pyramid. If the two shapes at the ends are squares, parallel to each other and connected by rectangles, it is a square prism.

There are two special 3D shapes whose names don’t include the word “prism” even though they technically are prisms:

• Cube: a cube is essentially a prism (two squares opposite and parallel to each other, connected by rectangles), except that the connecting rectangles are actually squares. Cubes have six faces, all squares.
• Cylinder: a cylinder is a prism because it has two circles opposite and parallel to each other, which are connected by a rectangle. It’s not as obvious as other prisms, but think of a can of peas: if you take the label off, it is the shape of a rectangle.

3D shapes have volume: the amount of cubic space inside of them.

To find volume, you basically need the three dimensions: length, width, and height.

For prisms, the formulas are derived by taking the area of the shape at the end, and multiplying that times the figure’s height.

• Rectangular Prism: length x width x height
• Cube: length x width x height OR side x side x side (since they are all the same)
• Triangular Prism: (base x height ÷ 2) x height*
• *this height is the height of the prism
• Cylinder: (π x r x r) x height

For pyramids, the formulas are almost the same as for prisms, only they are divided by 3.

• Square or Rectangular Pyramid: (length x width x height) ÷ 3
• Triangular Pyramid: ((base x height ÷ 2) x height*) ÷ 3
• *this height is the height of the prism
• Cone: (π x r x r x height) ÷ 3

Examples math problems related to volume:

Example math problem related to Prisms:

1. Find the volume of this rectangular prism.
• Recall the formula: length x width x height
• (It won’t change things if you’re not sure which part is which, but usually length is the long side, width is the short side, and height is how tall it is.)
• Plug in the numbers to the formula:
• L x W x H = 12 x 4 x 6 = 288
• To label volume, we use cubic units.
• This rectangular prism has a volume of 288 cm3.
2. Find the volume of this cube.
• Recall the formula: length x width x height OR side x side x side
• Notice that only one side is labeled. This is okay since all the sides are the same. We know that length, width, and height are all 5 cm.
• Plug the numbers into the formula:
• L x W x H (or s x s x s) = 5 x 5 x 5 = 125
• This cube has a volume of 125 cm3.
3. Find the volume of this triangular prism.
• Recall the formula: (base x height ÷ 2) x height
• First, let’s identify each part.
• The base is the “bottom” of one of the triangles: 7
• The height is the height of one of the triangles: 5*
• (If you get the 7 and 5 backwards, that’s okay; just make sure you use numbers that are perpendicular to each other. Note that the other side of the triangle is not labeled. This side is not needed to find volume.)
• The second height is the height of the prism: 6
• Now plug in the numbers:
• (base x height ÷ 2) x height = (7 x 5 ÷ 2) x 6 = 105
• The volume of this triangular prism is 105 ft3.
4. Find the volume of this cylinder.
• Recall the formula: (π x r x r) x height
• We only need to know two things: radius (from the center of the circle to the edge) and the height of the cylinder. Remember that π = 3.14.
• Plug in the numbers:
• (π x r x r) x height = (3.14 x 3 x 3) x 10 = 282.6
• The volume of this cylinder is 282.6 ft3.

Example math problems related to pyramids:

1. Find the volume of this square pyramid.
• Recall the formula: (length x width x height) ÷ 3
• Remember, for a square, length, width, and height are all the same.
• Plug in the numbers:
• (length x width x height) ÷ 3 = (2 x 2 x 9) ÷ 3 = 12
• The volume of this square pyramid is 12 cm3.
2. Find the volume of this triangular pyramid.
• Recall the formula: ((base x height ÷ 2) x height*) ÷ 3
• Remember that the first height is the height of the triangle base, and the second height is the height of the pyramid, perpendicular to the base.
• Plug in the numbers:
• ((base x height ÷ 2) x height*) ÷ 3 = ((7 x 8 ÷ 2) x 9) = 252
• The volume of this triangular pyramid is 252 cm3.
3. Find the volume of this cone.
• Recall the formula: (π x r x r x height) ÷ 3
• A cone has a circular base with a pointy top. All we need is the radius and the height of the cone to find its volume.
• Plug in the numbers:
• (π x r x r x height) ÷ 3 = (3.14 x 4 x 4 x 17) ÷ 3 = 284.69
• The volume of this cone is 284.69 in3.