Analytical Applications of Derivatives: Background You’ll Need 3

  • Find the volume and surface area of different shapes

Finding the Volume and Surface Area of Rectangular Solids

When we explore three-dimensional shapes, understanding how to calculate the volume and surface area is crucial. Volume measures the space a shape occupies, while surface area describes the total area of all the surfaces of a three-dimensional object. For rectangular solids, which include cubes and rectangular prisms, these measurements are based on the object’s length, width, and height.

volume and surface area of a rectangular solid

For a rectangular solid with length [latex]L[/latex], width [latex]W[/latex], and height [latex]H[/latex]:

 

A rectangular solid, with sides labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.

For a rectangular solid with length [latex]14[/latex] cm, height [latex]17[/latex] cm, and width [latex]9[/latex] cm. Find the

  1. volume
  2. surface area

Finding the Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. Substituting, [latex]s[/latex] for the length, width, and height into the formulas for volume and surface area of a rectangular solid, we get:

[latex]\begin{array}{ccccc}V=LWH\hfill & & & & S=2LH+2LW+2WH\hfill \\ V=s\cdot s\cdot s\hfill & & & & S=2s\cdot s+2s\cdot s+2s\cdot s\hfill \\ V={s}^{3}\hfill & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\hfill \\ & & & & S=6{s}^{2}\hfill \end{array}[/latex]

So for a cube, the formulas for volume and surface area are [latex]V={s}^{3}[/latex] and [latex]S=6{s}^{2}[/latex].

volume and surface area of a cube

For any cube with sides of length [latex]s[/latex],

 

A cube. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.
A cube is [latex]2.5[/latex] inches on each side. Find the

  1. volume
  2. surface area

Finding the Volume and Surface Area of a Sphere

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below. Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course.

volume and surface area of a sphere

For a sphere with radius [latex]r\text{:}[/latex]

 

A sphere, with radius labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.

 

Note: We will approximate [latex]\pi[/latex] with [latex]3.14[/latex].

It is important to note that for both the volume and surface area of a sphere you use the radius of the sphere. Sometimes questions will only give you the diameter ([latex]d[/latex]). To find the radius when given the diameter [latex]r = \frac{d}{2}[/latex]
A sphere has a radius [latex]6[/latex] inches. Find its

  1. volume
  2. surface area

Finding the Volume and Surface Area of a Cylinder

If you have ever seen a can of vegetables, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height [latex]h[/latex] of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, [latex]h[/latex] , will be perpendicular to the bases.

A cylinder with an arrow pointing to the radius of the top labeling it r, radius. There is an arrow pointing to the height of the cylinder labeling it h, height.

 

volume and surface area of a cylinder

For a cylinder with radius [latex]r[/latex] and height [latex]h[/latex]:

 

A cylinder, with the height labeled h and the radius of the top labeled r. Beside it is Volume: V equals pi times r squared times h or V equals capital B times h. Below this is Surface Area: S equals 2 times pi times r squared plus 2 times pi times r times h.

 

For a cylinder, the area of the base, [latex]B[/latex], is the area of its circular base, [latex]\pi {r}^{2}[/latex]. This is different from the area of the base for rectangular solids.

A cylinder has height [latex]5[/latex] centimeters and radius [latex]3[/latex] centimeters. Find its

  1. volume
  2. surface area