Solid Geometry - Surface Area and Volume
Triangles, polygons, and circles are all considered two‐dimensional objects and you’ve learned how to find the perimeter and area of these objects. You will also need to know how to analyze the measurements of their three‐dimensional counterparts, or solid geometry. For solid geometry, you will need to know the volume and the surface area of a three‐dimensional object. The surface area of a solid figure is like the perimeter of a two‐dimensional object. The volume of a solid figure is like the area of a two‐dimensional object.
Later, you will learn how to calculate the surface area and volume of three-dimensional objects such as prisms, cylinders, cones, and spheres. This lesson will only cover cubes and rectangular solids as those are the most relevant three-dimensional shapes at this point.
To find the volume of a cube, you must cube the length of the side. For side length s, you calculate s × s × s. To calculate surface area, you calculate the area of one side (s × s) and then multiply that by 6, since there are 6 equivalent sides.
To find the volume of any rectangular solid, you simply multiply the length by the width by the height.
To calculate surface area, you will see that there are 3 sets of 2 equivalent sides:
- length × width
- length × height
- width × height
Three dimensional objects can be difficult to visualize, but always remember that each side has a pair on the opposite side:
Therefore, you calculate the area of each combination and then multiply each one by 2.
Practice Problems
- Find the surface area and volume of a rectangular solid with the following measurements: l = 3, w = 4, h = 5.
- Find the surface are and volume of a cube with side length s = 6.
- Find the surface area and volume of a rectangular solid with the following measurements: l = 8, w = 6, h = 4.
- Find the surface area and volume of a cube with side length s = 10.
Answers to Practice Problems
- SA=90, V=60
- SA=216, V=216
- SA=208, V=192
- SA=600, V=1000
