Geometry

Surface Area and Volume

Big Picture

Surface area and volume are two very fundamental properties of 3-dimensional shapes. Often times in geometry we will be asked to find the surface area or volume of a shape. The more simple shapes can be solved by using a general formula. More complex shapes will require us to apply our knowledge of one or several 2-dimensional shapes.

Key Terms

Surface Area: The sum of the areas of the faces.

Lateral Area: The sum of the areas of the lateral faces only.

Lateral Face: A face that is not the base.

Volume: The measure of how much space a 3-dimensional figure occupies.

Surface Area and Volume

Surface Area

Surface area can be calculated by constructing a net of the 3-dimensional figure and using the Area Addition Postulate.

      Area Addition Postulate: The surface area of a 3-dimensional figure is the sum of the areas of all its non-overlapping parts.

The lateral area is the surface area of the 3-dimensional figure minus the area of the base(s).

Volume

Volume is measured in the cubic units. Two postulates that help us find the volume are:

  • Volume Congruence Postulate: If two polyhedrons are congruent, then their volumes are congruent.
  • Volume Addition Postulate: The volume of a solid is the sum of the volume of all of its non-overlapping parts.

Volume can be found by “counting boxes,” which is a method where we count how many units, like building blocks, create our figure.

Volume can be found by “counting boxes,” which is a method where we count how many units, like building blocks, create our figure.

cubic units

The volume of the counting boxes can be found using the Volume of a Cube Postulate:

  • Volume of a Cube Postulate: The volume of a cube is the cube of the length of its side, or \( s^3\).

For oblique figures where the lateral edges are not perpendicular to the base, the volume can be found by applying Cavalieri’s Principle.

  • Cavalieri’s Principle: If two solids have the same height and the same cross-sectional area at every level, then the two solids have the same volume.
counting boxes

Think of Cavalieri’s Principle as a stack of books. Each book in the stack still has the same volume regardless of whether the stack is leaning or not.

Geometry

Surface Area and Volume Cont.

Composite Solids

Sometimes we will be asked to find the volume and surface area of a hollowed out figure. An example of this would be a pipe.

Volume of a Composite Solid

  • Find the volume of the entire figure as if it had no hole.
  • Find the volume of the hole.
  • Subtract the volume of the hole from the volume of the entire cylinder.

Surface Area of a Composite Solid

  • Find the lateral area of the larger figure.
  • Find the lateral area of the hole.
  • Find the area of the bases of the larger figure.
  • Find the area of the bases of the hole.
  • Subtract the area of hole’s bases from the larger figure’s bases.
  • Add to the lateral area of the larger figure and the lateral area of the hole.
Volume of a Composite Solid

Similar Solids

Similar solids are two solids of the same type with equal ratios of corresponding linear measures (for example, heights and radii). All ratios for corresponding measures must be the same.

Characteristics of Similar Figures

Recall that when two polygons are similar, the ratio relating any two corresponding lengths is equal to the scale factor.

  • The ratio of the areas of two similar figures is then equal to the square of the ratio between the corresponding linear sides.
  • For example, if we doubled the sides of a cube, the surface area would quadruple.
  • The ratio of the volumes of two similar figures is equal to the cube of the ratio between the corresponding linear sides.

Scale Factor for Similar Figures

Ratios
Units
Scale Factor
\(\frac{a}{b}\)  \(in, ft, cm, m, etc.\)
Ratio of the Surface Areas
\((\frac{a}{b})^2\)  \(in^2, ft^2, cm^2, m^2, etc.\)
Ratio of the Volumes
\((\frac{a}{b})^3\)  \(in^3, ft^3, cm^3, m^3, etc.\)

Notes