Direct proportion to the square

Last updated on May 25, 2026

Scaffolded direct proportion to the square worksheet

Direct proportion to the square is when one quantity increases or decreases in proportion to the square of another quantity. This relationship is common in areas like physics, where force and area are connected, or in geometry when calculating areas of shapes. Understanding this concept helps in solving problems where changes aren't linear but grow or shrink more rapidly. Jump to the questions

Practise now

Question

y is directly proportional to p²

y = when p =

Find the equation for y and p

Step 1 - describe the relationship using the correct symbol

y ∝ p²

Step 2 - replace the symbol with the constant of proportionality, k

y = p²

Step 3 - substitute the values from the question into the equation

= k

Step 4 - rearrange the equation to isolate k

÷ = k

Step 5 - determine the value of k

k =

Step 6 - write the final equation linking y and p²

y = p²

Find y when p =

Step 1 - restate the equation

y = p²

Step 2 - substitute the value you know

y = ×

Step 3 - determine the final value of y

y =

Find p when y =

Step 1 - restate the equation

y = p²

Step 2 - substitute the value you know

= × p²

Step 3 - rearrange the equation to isolate p²

÷ = p²

Step 4 - calculate the value of p²

p² =

Step 5 - calculate the value of p

p =

Topic guide

What this worksheet practises

This worksheet provides practice on direct proportion to the square of a number. This means that as one variable increases, the other variable increases at an accelerating rate. For example, if a car travels twice as fast, its braking distance doesn't just double; it quadruples (2²).

Key method

To calculate missing values, you must build an algebraic equation using a constant of proportionality, 'k'.

Write the proportionality statement: y ∝ x².
Convert this to an equation: y = kx².
Substitute the complete pair of given values into the equation to calculate 'k'.
Rewrite the equation with your newly found value for 'k'.
Substitute the final known value into this specific equation to find the missing answer.

Worked example

y is directly proportional to the square of x. When x = 4, y = 48. Find y when x = 6.

Step 1: Write the equation.

y = kx²

Step 2: Substitute the known values to find k.

48 = k × 4²

48 = k × 16

k = 48 ÷ 16 = 3.

Step 3: Write the full equation.

y = 3x²

Step 4: Answer the question by substituting x = 6.

y = 3 × 6²

y = 3 × 36 = 108.

Common mistakes to avoid

The most common mistake is confusing "y is proportional to the square of x" with "y is proportional to the square root of x". Read the wording very carefully. Another frequent error is multiplying by 'k' before squaring the 'x' value. The rules of BIDMAS dictate that indices (squares) must be calculated before multiplication.

Things to remember

When working with direct proportion to a square, scaling up is incredibly fast. If you multiply the 'x' value by 3, the 'y' value will be multiplied by 9 (which is 3²). This shortcut can often save you from doing long algebraic calculations if the numbers are simple multiples.