The diagram shows a cuboid with length 8 cm, width 6 cm and height 4 cm. The vertices are labelled \(A\) to \(H\). 8 cm6 cm4 cmABCDEFGH© EAGLE BEACON GLOBAL...
Question 1 Report
The diagram shows a cuboid with length 8 cm, width 6 cm and height 4 cm. The vertices are labelled \(A\) to \(H\).
Calculate the length of the space diagonal \(AG\). Give your answer correct to 3 significant figures. (5)
Answer Details
This question tests finding the space diagonal of a cuboid by applying Pythagoras' theorem twice: once across the base, then again up through the solid.
First find the base diagonal \(AC\), using the length \(AB = 8\) cm and width \(BC = 6\) cm: [2]
\[AC^2 = 8^2 + 6^2 = 64 + 36 = 100\] \[AC = \sqrt{100} = 10\text{ cm}\]The space diagonal \(AG\) is the hypotenuse of a second right-angled triangle formed by the base diagonal \(AC\) and the vertical edge \(CG = 4\) cm: [3]
\[AG^2 = AC^2 + CG^2 = 10^2 + 4^2 = 100 + 16 = 116\] \[AG = \sqrt{116} = 10.8\text{ cm (3 s.f.)}\]Finding the space diagonal of a cuboid always works this way: find a face (or base) diagonal first with Pythagoras, then use that diagonal as one side of a second right-angled triangle with the remaining edge.
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