A rectangular box is 2 cm by 3 cm by 6 cm
- Base diagonal squared = 2² + 3² = 13.
- Space diagonal squared = 13 + 6² = 49.
- Space diagonal = √49 = 7 cm.
Multi-step problems & checking
Connect roots, geometry and a defensible final answer.
Plan the sequence of right triangles
Connect right triangles across several steps and check the final result against the context.
Some problems need two applications of Pythagoras. In a rectangular box, first find the base diagonal from length and width. Then use that base diagonal and the perpendicular height as legs of another right triangle. The space diagonal is the final hypotenuse.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
Carry square areas forward without rounding
For a box with length 3, width 4 and height 12, the base diagonal has square 3² + 4² = 25. The space diagonal has square 25 + 12² = 169, so its length is 13. In general d = √(length² + width² + height²) for a rectangular box. This is an extension built from two right triangles, not a rule for arbitrary three-dimensional shapes.
Check the size and communicate the reasoning
The space diagonal must exceed each box dimension and the base diagonal. Identify each right angle, give exact intermediate expressions, and round only the final result to the requested precision. If the answer is unreasonable, revisit the geometry before recalculating. A method is complete when the reader can see why the equation fits the situation.
Worked examples
A right triangle has equal legs and hypotenuse 10 m
- Let each leg have length x: x² + x² = 100.
- 2x² = 100, so x² = 50 and x = √50.
- Each leg is about 7.1 m, not 5 m.
Guided Practice
A box is 6 units by 8 units by 24 units. Find its space diagonal.
A first hint
The base diagonal is 10; use it with the height of 24.
Compare your reasoning
Space diagonal = √(100 + 576) = √676 = 26 units.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Connections
A square has diagonal 12 cm. Find its exact area without rounding a side length. Explain why the two equal leg-square areas together total 144 cm².
Videos & Explorations →