Summary: Solving Problems Using Volume and Surface Area

Key Concepts

  • Volume and Surface Area of a Rectangular Solid
    • [latex]V=LWH[/latex]
    • [latex]S=2LH+2LW+2WH[/latex]
  • Volume and Surface Area of a Cube
    • [latex]V={s}^{3}[/latex]
    • [latex]S=6{s}^{2}[/latex]
  • Volume and Surface Area of a Sphere
    • [latex]V=\Large\frac{4}{3}\normalsize\pi {r}^{3}[/latex]
    • [latex]S=4\pi {r}^{2}[/latex]
  • Volume and Surface Area of a Cylinder
    • [latex]V=\pi {r}^{2}h[/latex]
    • [latex]S=2\pi {r}^{2}+2\pi rh[/latex]
  • Volume of a Cone
    • For a cone with radius [latex]r[/latex] and height [latex]h[/latex] :Volume: [latex]V=\Large\frac{1}{3}\normalsize\pi {r}^{2}h[/latex]

Glossary

cone
A cone is a solid figure with one circular base and a vertex.
cube
A cube is a rectangular solid whose length, width, and height are equal.
cylinder
A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.

 

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

A table is shown that summarizes all of the formulas in the chapter. The first cell is for Supplementary and Complementary Angles, and says that the measure of angle A plus the measure of angle B equals 180 degrees for supplementary angles A and B and the measure of angle C plus the measure of angle D equals 90 degrees for complementary angles C and D. There is an image of two angles A and B that together form a straight line and two angles C and D that together form a right angle. The next cell says Rectangular Solid and shows the formulas Volume equals LWH and Surface Area equals 2LH plus 2LW plus 2WH. An image of a rectangular solid with sides L, W, and H is shown. The next cell says Triangle. An image of a triangle is shown with sides a, b, and c, vertices A, B, and C, and height h. It says, “For triangle ABC, angle measures measure of angle A plus measure of angle B plus measure of angle C equal 180 degrees. Below this is Perimeter, P equals a plus b plus c. Below this is Area, A equals one-half bh. The next cell says Cube and shows an image of a cube with sides s. It says Volume V equals s cubed and Surface Area S equals 6 times s squared. The next cell says Similar Triangles. It shows two similar triangles ABC and XYZ. It says if triangle ABC is similar to triangle XYZ, then measure of angle A equals measure of angle X, measure of angle B equals measure of angle Y, and measure of angle C equals measure of angle Z. It then says a over x equals b over y equal c over z. The next cell says Sphere and shows an image of a sphere with radius r. It says volume V equals four-thirds times pi times r and Surface Area S equals 4 times pi times r squared. The next cell says Circle. There is an image with two radii labeled r and the diameter labeled d. It says Circumference C equals 2 pi times r and C equals pi times d. It says Area equals pi times r squared. The next cell says Cylinder and shows an image of a cylinder with height h and radius of the base r. It says Volume V equals pi times r squared times h. Below this is V equals Bh. Below that is Surface Area S equals 2 times pi times r squared plus 2 times pi times rh. The next cell says Rectangle and shows an image of a rectangle with sides W and L. It says Perimeter P equals 2L plus 2W, then Area A equals LW. The next cell says Cone and shows an image of a cone with height h and radius of the base r. It says Volume V equals one-third times pi times r squared times h. The last cell says Trapezoid and shows an image of a trapezoid with bases little b and capital B, and height h. It says Area A equals one-half times h times parentheses little b plus capital B.

This image shows a row with three columns. The first column says Rectangular solid with the formula below that says volume: V equals LWH. Under this, it says Surface Area: S equals 2LH plus 2LW plus 2WH. An image shows an image of a rectangular solid with the sides labeled L , W and H. The middle column says Rectangle. Under this it says Perimeter P equals 2L plus 2W, then Area A equals LW. An image of a rectangle with sides W and L. The right column says Cube. Under this it says “Volume: V equals s to the third power.” Under this is says “Surface area: S equals 6 times s squared. Below it is an image of a cube with three sides labeled “s”.