Surface area of cuboids

A closed cuboid is 8 cm long, 5 cm wide and 3 cm high. Find the total area of its six faces. The drawing is a schematic, not to scale

This net folds into a box: the four rectangles in the horizontal strip are its sides, the upper tab is its top and the lower tab is its base. The top and base are each 4 cm by 3 cm. All diagram lengths are in cm. Change the height h and compare the closed-box and open-top area readouts

What stays the same as height changes?

Count faces, not edges

Surface area is the sum of the included face areas. A closed cuboid has three equal opposite-face pairs. An open-top box is missing only its top face

1. 1Use square units Each rectangular face area is the product of two perpendicular lengths
2. 2Check what is included For an open top, include one length-by-width base and both pairs of sides \(\displaystyle SA_{\text{open}}=lw+2lh+2wh\)
3. 3Convert before multiplying Use a common length unit first. Do not mix centimetres and metres in one product

Length

\(\displaystyle 1.2\text{ m}\)

Width and height

\(\displaystyle 80\text{ cm},\;50\text{ cm}\)

A cuboid container is 1.2 m long, 80 cm wide and 50 cm high. The top is open. Find the area of the base and four sides

There is one base, two long sides and two short sides

Length, width and height

\(\displaystyle 7\text{ cm},\;4\text{ cm},\;2\text{ cm}\)

A closed cuboid is 7 cm long, 4 cm wide and 2 cm high. Find its total surface area

Group the six faces into three equal pairs

Joined solids

When two solids are attached along a face, the join is inside the combined object. That face was included in each separate surface area, so remove the shared area twice

1. 1Find the actual contact area Only the touching portion becomes internal; any exposed remainder stays in the surface area
2. 2Do not remove it once Adding the two separate surface areas counted the contact region from both sides

Lower cuboid dimensions

\(\displaystyle 8\times6\times2\text{ cm}\)

Upper cuboid dimensions

\(\displaystyle 4\times3\times3\text{ cm}\)

Contact rectangle

\(\displaystyle 4\text{ cm by }3\text{ cm}\)

A cuboid 4 cm long, 3 cm wide and 3 cm high is attached on top of a cuboid 8 cm long, 6 cm wide and 2 cm high. Its entire 4 cm by 3 cm base touches the larger cuboid. Find the exposed surface area of the combined object, including its underside

Add the separate surface areas, then subtract the contact area twice

A closed cuboid is 6 cm long, 4 cm wide and 3 cm high. Which is the first incorrect line?