3-D shapes

 





Three‐dimensional objects are the solid shapes you see every day, like boxes, balls, coffee cups, and cans.
Here are some helpful vocabulary terms for solids:

• Base: the bottom surface of a solid object.
• Edge: the intersection of two faces on a solid object. This is a line.
• Face: a flat side of a 3‐dimensional object.
• Prism: a solid object with two congruent and parallel faces.
• Pyramid: a solid object with a polygon for a base and triangles for sides.

Say Hello to the 3-D Shapes

Name	Properties	Picture
Rectangular Prism	A prism with rectangular bases. There are six faces. All faces meet at 90 degrees. Opposite faces are parallel.
Cube	A prism with six congruent faces. All faces meet at 90 degree angles. Opposite faces are parallel. It is a form of rectangular prism.
Triangular Prism	A prism with triangular bases. Only the bases are parallel.
Octagonal Prism	A prism with octagons for bases. Opposite faces are parallel.
Triangular Pyramid, aka Tetrahedron	A pyramid with a triangular base. A tetrahedron made up of four equilateral triangles is called a regular tetrahedron.
Square Pyramid	A pyramid with a square base.
Cylinder	A solid with two parallel circular bases. If you "unwrap" the middle section and lay it flat, it is a rectangle.
Cone	A solid with a circular base and a curved side that ends in one point It has one vertex.
Sphere	A solid figure where all points are an equal distance from the center point. It's a ball.

The volume of a solid is the amount of space inside the object. It's how much water fits inside a bathtub, how much sand fills a bucket, or how much soda your friend can chug and hold in his stomach.
Take a look at this rectangular prism:

If we consider the front face (Rectangle ABCD) to be the base we can see that it has an area of 12 square units. There are four rows of this face, each with 12 cubes in it.

So, if we multiply the area of the face (12) by the 4 rows, we find that there are 48 cubes, or a volume of 48 cubic units!
Let's look at another type of prism: a cube! Here's a sample of the surface area and volume of a cube.

Now let's look at a cylinder:

If the area of the circular base is equal to 16π square units, and there are five rows of these circular bases, then the volume would be 16π × 5 = 80π cubic units, or approximately 251.2 cubic units.
Look Out: volume is always cubic units (units³). This is because we are dealing with the three-dimensional objects now. You're in the big time!
This is pretty much all you have to do to find the volume of any prism or cylinder: find the area of the base and multiply it by the height.
Volume of a Prism or Cylinder = area of the Base x height
Volume = Bh
Look Out: note the difference between small "b" and large "B". In the examples above (and often in geometry in general), small "b" is the length of the base of a 2D shape. Large "B" is the area of the base of a 3D solid.
The formula for the volume of pyramids and cones tells you how much space is inside each object.
For these two solid shapes, the volume formula is the same: it's one third of the area of the base times the height.

Volume of Pyramids or Cones = ⅓ Base × height

Why? Here it is in a nutshell. The volume of three pyramids is equal to the volume of one prism with the same base and height. Similarly, the volume of three cones is equal to the volume of one cylinder with the same circular base and height.

The volume of each cone is equal to ⅓Bh = ⅓(28.3 × 10) = 94 ⅓ cm³. All three cones combined equals 283 cm³. The volume of the cylinder is equal to Bh = 28.3 × 10 = 283 cm³, ta da!

The volume of each pyramid is equal to ⅓Bh = ⅓(18 × 8) = 48 cm³. All three pyramids combined equals 144 cm³. The volume of the prism is equal to Bh = 18 × 8 = 144 cm³.
To find the volume of a sphere, follow this simple formula (which took a brilliant ancient Greek mathematician named Archimedes years to derive):

Volume of a Sphere = 4/3π x radius cubed = 4/3πr³

The surface area of a solid is the area of each surface added together.
There are few formulas to memorize (w00t!). The keys to success: make sure that you don't forget a surface and that you have the correct measurements.
Surface area is often used in construction. If you need to paint any 3‐D object you need to know how much paint to buy.

Surface Area of a Rectangular Prism

If we "unfold" the box, we get something that is called – in the geometry world – a "net".

Using the net we can see that there are six rectangular surfaces.

Side 1	4 x 8	32 cm2
Side 2	8 x 6	48 cm2
Side 3	4 x 8	32 cm2
Side 4	8 x 6	48 cm2
Side 5	4 x 6	24 cm2
Side 6	4 x 6	24 cm2
	TOTAL	208 cm2

If we study the table we will see that there are two of each surface. That's because the top and bottom of a rectangular prism are congruent, as are the two sides, and the front and back.

Surface Area of a Triangular Prism

If we break down our triangular prism into a net, it looks like this:

In a triangular prism there are five sides, two triangles and three rectangles.

Side 1	½(9 × 4)	18 cm2
Side 2	½(9 × 4)	18 cm2
Side 3	4.5 x 8.1	36.45 cm2
Side 4	9 x 8.1	72.9 cm2
Side 5	7.2 x 8.1	58.32 cm2
	TOTAL	203.67 cm2