Pythagoras’ theorem and trigonometry in 3D
Section: Trigonometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Finding Lengths in 3D Solids Using Pythagoras
Extended Only
Pythagoras' theorem can be applied twice in succession to find a diagonal that runs through the inside of a 3D solid, such as the space diagonal of a cuboid.
- First find the diagonal of a 2D face using Pythagoras, treating it as a right-angled triangle
- Then use this face diagonal as one side of a second right-angled triangle, together with the remaining edge, to find the full 3D (space) diagonal
- Sketching each right-angled triangle separately, clearly labelling the known lengths, makes a 3D problem much easier to manage
Worked Example: Finding the Space Diagonal of a Cuboid
- Question: A cuboid has length 8 cm, width 6 cm and height 5 cm. Find the length of its space diagonal (from one corner to the opposite corner), correct to 3 significant figures.
- Step 1: Find the diagonal of the base (length 8, width 6): √(8^2+6^2)=√(64+36)=√(100)=10
- Step 2: Use this base diagonal (10) with the height (5) to find the space diagonal: √(10^2+5^2)=√(100+25)=√(125)11.2
- Answer: 11.2 cm (3 s.f.)
Common Mistakes
MistakeTrying to find the space diagonal directly in one step, without first finding a face diagonal
Fix3D length problems almost always need Pythagoras applied twice - once for a 2D face, and again to reach the full 3D diagonal
MistakeUsing the wrong pair of edges when calculating the base diagonal, e.g. mixing up which two edges are on the same face
Fixsketch the base rectangle separately first, clearly labelling only the two edges that actually form its diagonal
Finding Angles in 3D Solids Using Trigonometry
Extended Only
The angle between a sloping line and a flat plane (such as a base) can be found by identifying the right-angled triangle formed by the line, its projection onto the plane, and a vertical height.
- The angle between a line and a plane is measured between the line and its projection (shadow) onto that plane
- To find this angle, identify (or construct) the right-angled triangle containing the sloping line as the hypotenuse, the projection as the adjacent side, and the vertical height as the opposite side
- Once the right-angled triangle is identified, use ordinary right-angle trigonometry (SOH CAH TOA) to find the angle
Worked Example: Finding the Angle Between a Diagonal and the Base of a Cuboid
- Question: A cuboid has a base diagonal of 10 cm and a height of 5 cm. Find the angle between the space diagonal and the base, correct to 1 decimal place.
- Step 1: The right-angled triangle has the base diagonal (10 cm) as the side adjacent to the angle, and the height (5 cm) as the side opposite
- Step 2: tan(angle)=(5/10)=0.5
- Step 3: angle=tan^-1(0.5)26.6°
- Answer: 26.6° (1 d.p.)
Common Mistakes
MistakeUsing the full space diagonal as one of the two shorter sides in the tangent ratio, instead of correctly identifying it as the hypotenuse
Fixthe space diagonal is always the hypotenuse of the angle-finding triangle - the base diagonal and the height are the two shorter sides
MistakeMeasuring the angle between the line and the wrong reference line, rather than its projection onto the specified plane
Fixthe angle between a line and a plane is always measured to the line's projection (shadow) on that plane - sketch this projection clearly before setting up the triangle
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