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Grade 5 (age 10-11)

grade 5 Rectangular Solid: Surface Area & Volume

A rectangular prism, or cuboid, is a box-shaped solid surrounded by six rectangles. It has 6 faces, 12 edges and 8 vertices, and opposite faces have the same shape and size, so the faces come in three pairs. Add up the areas of the six faces in its net, the box cut open and laid flat, and you get the surface area; the volume, the number of 1 cm³ cubes that fit inside, is width × depth × height. Change the width, depth and height and the box above and the net below change together, with the surface area and volume worked out again.

What is a rectangular prism?

📦 A box-shaped solid
①A solid surrounded by 6 rectangles is called a rectangular prism, or cuboid
②It has 6 faces, 12 edges and 8 vertices
③Opposite faces have the same shape and size, so the six faces make three pairs
④A rectangular prism whose edges are all the same length is a cube
4
2
3

Looking at the net

📋 Open it up: six rectangles
①Cutting along the edges of a box and laying it flat gives its net
②In the net, faces of the same color are opposite each other, so they have the same shape and size
③That is why the surface area is the three different face areas added and doubled. With the starting width 4, depth 2 and height 3, that is 2 × (12 + 6 + 8) = 52
④When the net is folded, the top goes above the front and the bottom below it, and the face below the bottom folds up to become the back

Summary

Surface area
SA = 2 × (W×H + H×D + W×D)
The three different face areas added together and doubled
Volume
V = W × D × H
Width × depth is the area of the base; multiply it by the height
This box
Surface area = 52 cm², volume = 24 cm³
width × depth × height = 4×2×3
🎯 At a glance
①Rectangular prism: a solid surrounded by 6 rectangles
②Surface area: the six face areas added together
③Volume: width × depth × height
④A cube is a rectangular prism with all edges the same length
⑤Area is measured in cm² and volume in cm³

Examples and Unit Test

Example 1
Find the volume of a rectangular prism 3 cm wide, 2 cm deep and 4 cm high.
1
The volume is width × depth × height.
2
3 × 2 × 4 = 24, so the volume is 24 cm³.
▸ 24 cm³
It is the same box as the starting one in the first picture (4 wide, 2 deep, 3 high) turned on its side, so the volume is also 24.
Example 2
Find the surface area of a rectangular prism 3 cm wide, 2 cm deep and 4 cm high.
1
Surface area = 2 × (width × depth + depth × height + width × height).
2
2 × (3 × 2 + 2 × 4 + 3 × 4) = 2 × (6 + 8 + 12) = 2 × 26 = 52, so the surface area is 52 cm².
▸ 52 cm²
Surface area adds up all three pairs of faces, and it is measured in cm².
Unit Test
What is the volume of a cube with edges of 2 cm?
①6 cm³
②8 cm³
③4 cm³
④12 cm³
▸ ② 8 cm³
1
A cube is a rectangular prism whose width, depth and height are all equal.
2
The volume is 2 × 2 × 2 = 8, so 8 cm³.
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