Lesson 6: Solve Geometry Applications: Volume and Surface Area

New Concept

Master calculating the space inside (volume) and area outside (surface area) of 3D shapes. You'll use specific formulas and a problem-solving strategy for rectangular solids, cubes, spheres, cylinders, and cones to tackle geometry applications.

What’s next

You're ready to start calculating! Next, you’ll solve interactive practice problems for finding the volume and surface area of each 3D shape.

Property

For a rectangular solid with length $L$, width $W$, and height $H$:

Volume: $V = LWH$

Surface Area: $S = 2LH + 2LW + 2WH$

Property

A cube is a rectangular solid whose length, width, and height are equal. For any cube with sides of length $s$:

Volume: $V = s^3$

Surface Area: $S = 6s^2$

Property

A sphere is a three-dimensional circle, and its size is determined by its radius. For a sphere with radius $r$:

Volume: $V = \frac{4}{3}\pi r^3$

Surface Area: $S = 4\pi r^2$

Property

A cylinder is a solid figure with two parallel circular bases of the same size. For a cylinder with radius $r$ and height $h$:

Volume: $V = \pi r^2 h$ or $V = Bh$

Surface Area: $S = 2\pi r^2 + 2\pi rh$

Property

A cone is a solid figure with one circular base and a vertex. For a cone with radius $r$ and height $h$:

Volume: $V = \frac{1}{3}\pi r^2 h$

Examples

• To find the volume of a cone with a height of 6 inches and a base radius of 2 inches, use $V = \frac{1}{3}\pi r^2 h$. With $\pi \approx 3.14$, the volume is $V \approx \frac{1}{3}(3.14)(2^2)(6) \approx 25.12$ cubic inches.