Pythagoras' Theorem in 3D

OCR

GCSE

The extension of Pythagoras’ Theorem to three dimensions necessitates the visualization of orthogonal planes within solid geometry to determine spatial distances. Candidates must apply the relationship $d^2 = x^2 + y^2 + z^2$ (or iterative 2D applications) to calculate space diagonals in cuboids, pyramids, and prisms. Assessment frequently integrates the manipulation of surds to express exact values, moving beyond mere numerical approximation. Mastery requires identifying the correct internal right-angled triangles where the hypotenuse of a face becomes the leg of the internal triangle.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

What You Need to Demonstrate

Key skills and knowledge for this topic

Award M1 for a clear attempt to use Pythagoras in a valid right-angled triangle (e.g., identifying the face diagonal first)
Award M1 for correct substitution of values into the 3D Pythagoras formula or sequential 2D application
Award A1 for the correct final answer, strictly observing instructions regarding significant figures or exact surd form
In 'Show that' questions, credit is only given if the full substitution and intermediate surd values are explicitly shown before the final result

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You correctly identified the face diagonal, but you rounded it too early. Keep it as a square root to ensure accuracy."
"Draw the 2D triangle you are solving for out of the 3D shape to avoid confusing slant height with vertical height."
"Check the question for 'exact form' requirements—you gave a decimal which would lose the final mark."
"Excellent work on the calculation; to secure full marks on 'Show that' questions, write down the unsimplified surd before the final answer."

Marking Points

Key points examiners look for in your answers

Award M1 for a clear attempt to use Pythagoras in a valid right-angled triangle (e.g., identifying the face diagonal first)
Award M1 for correct substitution of values into the 3D Pythagoras formula or sequential 2D application
Award A1 for the correct final answer, strictly observing instructions regarding significant figures or exact surd form
In 'Show that' questions, credit is only given if the full substitution and intermediate surd values are explicitly shown before the final result

Examiner Tips

Expert advice for maximising your marks

💡Draw the 2D section containing the length you need to find; examiners reward clear diagrams that justify your method
💡If the question asks for an exact answer, leave your result as a surd (e.g., 5√2); converting to a decimal will lose the final accuracy mark
💡Look for 'hidden' right-angled triangles in pyramids, specifically formed by the vertical height, the apothem, and the base center

Common Mistakes

Pitfalls to avoid in your exam answers

Using the slant height instead of the vertical height when calculating dimensions within a pyramid
Rounding intermediate answers (e.g., writing 7.07 instead of √50), resulting in a final answer outside the allowed tolerance
Failing to identify the correct right angle in 3D space, often assuming a non-right-angled triangle is right-angled

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Visualisation of internal planes and diagonals

Iterative application of Pythagoras' Theorem

Distance between coordinates in 3D space

Algebraic manipulation of variable dimensions

Likely Command Words

How questions on this topic are typically asked

Calculate

Show that

Find

Determine

Give

Practical Links

Related required practicals

•{"code":"Construction","title":"Structural Integrity","relevance":"Calculating the length of diagonal cross-bracing in steel frameworks"}

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