Missing dimensions and roots
Reverse area and volume calculations and interpret square and cube roots as lengths.
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Undo the multiplication that made the volume
A cuboid’s volume is length × width × height. Each factor supplies one length direction. To find a missing dimension, divide the volume by the product of the other two dimensions.
A cuboid has volume 420 cm³, base length 12 cm and base width 5 cm. One layer of height 1 cm holds 12 × 5 = 60 cm³. Therefore
Dividing by only 12 would give 35 cm²: that is the area of a face perpendicular to the 12 cm edge, not the missing height. The units help you check which quantity you have found.
Use base area when it is given
The same cuboid can be described by its volume, 420 cm³, and base area, 60 cm². Its height is 420÷60 = 7 cm. The area already combines two directions; do not multiply by another base length before dividing.
Check your understanding 1 Distinguish a length from an area
A cuboid has volume 240 cm³. Its length is 8 cm and its width is 5 cm. Find its height. A second cuboid has volume 240 cm³ and base area 30 cm². Find its height too.
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The first height is 240÷(8 × 5) = 6 cm. The second is 240÷30 = 8 cm. Both calculations divide volume by base area, but only the first needs you to calculate that base area from two lengths.
Find a face area from the perpendicular dimension
Imagine slicing a cuboid into layers parallel to the face you want. Multiplying that face area by the dimension perpendicular to the face gives the volume. Divide by that perpendicular dimension to recover the face area.
A cuboid has volume 420 cm³ and height 7 cm. Its base area is 420÷7 = 60 cm². You cannot determine its base length and width separately from this information: a 12 cm by 5 cm base and a 10 cm by 6 cm base both have area 60 cm².
Check your understanding 2 Choose the perpendicular dimension
A cuboid’s volume is 330 cm³. Its length is 11 cm. Find the area of an end face perpendicular to that length. Can you also determine both side lengths of that face?
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The end-face area is 330÷11 = 30 cm². This gives the product of its two side lengths, not each length separately. For example, 5 cm by 6 cm and 3 cm by 10 cm are both possible.
A cube has three equal dimensions
For a cube, the same edge length occurs three times. A cube of volume 125 cm³ has edge cube root of 125 = 5 cm because 5 × 5 × 5 = 125. The cube-root sign asks which number, multiplied by itself three times, gives the number inside. Dividing by 3 does not undo cubing.
A square face of area 81 cm² has edge square root of 81 = 9 cm because 9 × 9 = 81. A square root undoes squaring; a cube root undoes cubing. For a physical length, take the non-negative root. Squaring or cubing a length also changes the unit: cm² for area and cm³ for volume.
Check your understanding 3 A volume does not give an edge by division
A cube has volume 64 cm³. Find its edge length and the area of one face. Explain why dividing the volume by 3 or by 6 is not the rule for either answer.
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The edge is cube root of 64 = 4 cm. One face has area 4² = 16 cm². Dividing by 3 does not reverse the product of three equal lengths. Dividing by the six faces does not turn a cubic measure into the area of one face.
Round a calculator result only when asked
Some roots do not have a terminating decimal. For example, square root of 2 ≈ 1.41421356…, so to two decimal places it is 1.41. Keep the calculator digits until the final rounding step. A task asking for three decimal places needs three places, including a final zero when necessary.
A cube with volume 2 cm³ has edge cube root of 2 ≈ 1.259921… cm. To three decimal places, write 1.260 cm. Check by cubing the unrounded calculator value, rather than expecting the rounded length to reproduce the volume exactly.
Check your understanding 4 Keep the two inverse operations distinct
Find square root of 7 to two decimal places. Then find the edge of a cube with volume 10 cm³ to three decimal places.
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square root of 7 ≈ 2.645751…, so it rounds to 2.65. The cube edge is cube root of 10 ≈ 2.154434… cm, so it rounds to 2.154 cm. The first calculation reverses a square; the second reverses a cube.
Transfer to a tank
A tank’s internal base area is 250 cm². Pouring in 1.5 ℓ raises the water depth by how much, assuming no overflow or objects occupy the tank?
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1.5 ℓ is 1500 cm³. The depth increase is 1500÷250 = 6 cm. Dividing volume by base area gives a length, just as it did for the solid cuboid.
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