Similar Shapes

    OCR
    GCSE

    Similarity in geometry extends beyond linear dimensions to govern the proportional relationships of area and volume in mathematically similar figures. When two shapes share a linear scale factor k, the ratio of their surface areas is defined by k² and the ratio of their volumes by k³. Mastery requires the bidirectional manipulation of these factors: squaring or cubing to find higher-order ratios, and extracting square or cube roots to derive linear dimensions from area or volume data. This framework is essential for solving complex problems involving frustums, composite solids, and physical properties such as mass and weight in objects of constant density.

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    Objectives
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    Exam Tips
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    Pitfalls
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    Key Terms
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    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for correctly establishing the linear scale factor (LSF) by calculating the square root of the area ratio or cube root of the volume ratio
    • Award M1 for cubing the derived linear scale factor to obtain the volume scale factor (VSF)
    • Award M1 for a complete method to find the volume of a frustum: Volume of large cone − Volume of small cone (subtraction method)
    • Award A1 for the correct final value, accurately rounded to the specified degree of accuracy
    • Credit responses that explicitly link mass ratio to volume ratio when density is constant

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You have correctly identified the linear scale factor; remember to cube this value when dealing with volume"
    • "Your method for the frustum is incomplete—you must calculate the volume of the 'removed' cone and subtract it from the total"
    • "Excellent use of the square root to find the linear ratio from the area information"
    • "Be careful with units—you applied the volume scale factor correctly but forgot to convert the final answer to litres as requested"

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for correctly establishing the linear scale factor (LSF) by calculating the square root of the area ratio or cube root of the volume ratio
    • Award M1 for cubing the derived linear scale factor to obtain the volume scale factor (VSF)
    • Award M1 for a complete method to find the volume of a frustum: Volume of large cone − Volume of small cone (subtraction method)
    • Award A1 for the correct final value, accurately rounded to the specified degree of accuracy
    • Credit responses that explicitly link mass ratio to volume ratio when density is constant

    Examiner Tips

    Expert advice for maximising your marks

    • 💡Immediately write down 'LSF = $k$, ASF = $k^2$, VSF = $k^3$' at the start of the question to prevent using the wrong factor under pressure
    • 💡When solving frustum problems, always sketch the full cone and use similar triangles to find the height of the missing smaller cone before calculating volumes
    • 💡If a question asks for the ratio of masses for similar objects made of the same material, treat this exactly as a volume ratio question ($k^3$)
    • 💡Check whether the question gives the ratio of 'A to B' or 'B to A'—inverting the fraction is a frequent cause of lost accuracy marks

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Applying the linear scale factor ($k$) directly to area or volume calculations instead of squaring ($k^2$) or cubing ($k^3$) it
    • Dividing the area ratio by 2 or the volume ratio by 3 when working backwards, rather than taking the square or cube root
    • Attempting to calculate frustum volume by applying a scale factor to a single dimension rather than subtracting the volume of the removed smaller cone
    • Confusing surface area ratio with volume ratio when dealing with problems involving painting or coating solids

    Study Guide Available

    Comprehensive revision notes & examples

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    Key Terminology

    Essential terms to know

    Relationship between Linear (k), Area (k²), and Volume (k³) Scale Factors
    Derivation of scale factors using roots (√ASF and ∛VSF)
    Application to mass and weight via density assumptions
    Decomposition of frustums into similar cones or pyramids

    Likely Command Words

    How questions on this topic are typically asked

    Calculate
    Show that
    Determine
    Work out
    Explain

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