3-Dimensional Figures

Volume

 

In a 3-dimensional shape, volume is the measurement of the amount of space is inside the shape.  It is measured in terms of distance cubed (cubic inches, cubic feet, cubic meters, etc.).  Formulas for volume can be found on your formula sheet.  Problems involving volume may be items like filling a tank or a can.

 

Surface Area

 

Surface area is a measure of the area of the outside surface area of a 3-dimensional shape.  Since it is a measure of area, it is measured in terms of distance squared.  Formulas for surface area can also be found on the formula sheet.  These problems may be things like painting the outside of a container of wrapping a present.

 

Right Prism/Rectangular Solid

 

A right prism or rectangular solid is a 3-dimensional rectangular shape.  This could be a box or a container with six flat sides.  A cube is a rectangular solid with all three sides the same length.  On your formula sheet, you may see both the rectangular solid and right prism volume formulas as V = lwh, which means volume equals length times width time height.  The formula sheet may also list the formula as V = Bh, or volume equals area of the base time the height.  The formula for surface area of a rectangular solid is also on your formula sheet:  SA = ph + 2B, or surface area equals the perimeter of the base times the height plus 2 times the area of the base.

 

Cylinder

 

A cylinder is a shape that has circles on the ends and straight sides, like a can.  To find the volume of a cylinder, use the formula V = πr2h, or volume equals pi times the radius of the end squared times the height.  Notice that this formula is the same as the formula for area of a circle times the height of the can.  For surface area of a cylinder, use the formula SA = 2πrs + 2πr2, or surface area equals 2 times pi times the radius times the height of a side plus 2 time pi times the radius squared.

 

 

 

 

Pyramid

 

A pyramid is a 3-dimensional shape with a square base and slanted sides that meet at a single point.  The formula for volume of a pyramid is V = Bh, or volume equals 1/3 times the area of the base times the height.  Remember that the height is the distance from the base to the top point, or the overall height of the pyramid.  The height is not the distance up one side; that is called the slant side.  The formula for surface area of a pyramid is SA = ps + B, or surface area equals one half times the perimeter of the base times the length of a slant side plus the area of the base.

 

 

 

Cone

 

A cone is a 3-dimensional shape with a circle on one end and sides that meet at a point.  The formula for volume of a cone is V = πr2h, or volume equals 1/3 times pi times the radius squared times the height.  The formula for surface area of a cone is SA = πrs + πr2, or surface area equals pi times the radius times the slant side plus pi times the radius squared.

 

 

 

 

 

 

Sphere

 

A sphere is a ball-like shape, or a three-dimensional circle.  The formula for volume of a sphere is V = πr3, or volume equals 4/3 times pi times the radius cubed.  The formula for surface area of a sphere is SA = 4πr2, or surface area equals 4 times pi times the radius squared.

 

 

 

 

 

 

 

Combined Figures

 

Sometimes you may have combined figures, which are shapes that don’t fit into any of the above shapes because they are multiple shapes stuck together.  For 2-dimensional combined figures, finding the area means dividing the shape into shapes you have the formula for, then finding the area of each and adding them together.  To find the perimeter, add up all of the outside edges.  You may have to do some addition or subtraction from opposite sides to find the length of missing sides.

 

For 3-dimensional shapes, you may have to find the volume.  Divide the shape into shapes you are familiar with, find the volume of each, and add them together.