A composite 3D figure is a three dimensional figure made up of basic three dimensional figures such as cubes, prisms, pyramids, cylinders, cones, etc.

To find the surface area of a composite 3D figure, add the areas of each geometric figure making up the composite 3D figure.

To find the volume of a composite 3D figure, draw any necessary planes to view the figure as basic three dimensional figures, then:

• add basic figure volumes belonging to the composite shape
• subtract basic figure volumes NOT belonging to the composite figure

PROBLEM

Find the surface area and volume of the composite 3D figure below.

View the composite 3D figure as a right rectangular prism together with a cube, then find any missing lengths in order to calculate the surface area and volume.

SURFACE AREA SOLUTION

Note that there are more elegant solutions to calculate the surface area than the one shown below, but the solution shown below illustrates separating a composite 3D figure into basic 3D figures.

S: Surface area of composite 3D figure.

A: Area.

S = A_FRONT + A_BACK + A_LEFT + A_RIGHT + A_TOP + A_BOTTOM

S = [(15 cm)(5 cm) + (5 cm)(5 cm)] + (20 cm)(5 cm) + (10 cm)(5 cm) + [(5 cm)(5 cm) + (5 cm)(5 cm)] + [(15 cm)(10 cm) + (5 cm)(5 cm)] + [(15 cm)(10 cm) + (5 cm)(5 cm)]

S = 650 cm²

The surface area of the given composite 3D figure is 650 cm²

VOLUME SOLUTION ONE

V: Volume.

V_{composite 3D figure} = V_{rectangular right prism} + V_cube

V_{composite 3D figure} = (15 cm)(10 cm)(5 cm) + (5 cm)³

V_{composite 3D figure} = 750 cm³ + 125 cm³

V_{composite 3D figure} = 875 cm³

The volume of the given composite 3D figure is 875 cm³

AREA SOLUTION TWO

Note that there are other similar ways to calculate the volume of the given composite 3D figure. For example rather than viewing the composite 3D figure as a rectangular right prsim together with a cube, it can be viewed as rectangular right prism less a cube.

V: Volume.

V_{composite 3D figure} = V_{rectangular right prism} - V_cube

V_{composite 3D figure} = (20 cm)(10 cm)(5 cm) - (5 cm)³

V_{composite 3D figure} = 1000 cm³ - 125 cm³

V_{composite 3D figure} = 875 cm³

The volume of the given composite 3D figure is 875 cm³