A rectangular prism has perpendicular edge lengths x,2x and 2xcm, where x>0. Its space diagonal is 12cm. Find its volume, in cm3.
Rectangular prismDiagram not to scaleShow worked answer
Answer: 256
Pythagoras in three dimensions gives (1+4+4)x2=122.
Thus x=4.
The volume is (1×4)(2×4)(2×4)=256cm3.
Example 2
A box has edges 12,24 and 27cm. A cable must go over its surface between opposite vertices, without entering the box. The three possible unfolded rectangles are shown. What is the shortest cable length, in cm?
Three unfolded surface routesDiagram not to scaleShow worked answer
Answer: 45
Use Pythagoras for each unfolding.
Squared lengths are 2025,2097,2745cm2.
The smallest is 2025
the shortest length is √2025=45cm.
Example 3
A rectangular prism has perpendicular edge lengths 3x,4x and 12xcm, where x>0. Its space diagonal is 52cm. Find its volume, in cm3.
Rectangular prismDiagram not to scaleShow worked answer
Answer: 9216
Pythagoras in three dimensions gives (9+16+144)x2=522.
Thus x=4.
The volume is (3×4)(4×4)(12×4)=9216cm3.
Use right triangles in three dimensions and unfolded surfaces practice | Year 10 | Ascent Arcade