Exact Calculations with Pi

Remember that the radius ($r$) is the distance from the centre to the edge. The diameter ($d$) is the full distance across the circle through the centre, meaning $d = 2r$.

Rules for Calculating with $\pi$

When providing an exact multiple of $\pi$, your final answer should be written in the form $k\pi$, where $k$ is a number (for example, $25\pi$ or $\frac{4}{9}\pi$). You must apply BIDMAS correctly during your calculations.

In the area formula, you must always square the radius before multiplying by $\pi$. If the radius is a fraction, ensure you square both the numerator and the denominator. For instance, if $r = \frac{2}{3}$, then $r^2 = \frac{4}{9}$, making the exact area .

Worked Example: Area of a Circle

Calculate the exact area of a circle with a radius of $7\text{ cm}$. Give your answer in terms of $\pi$.

• Step 1: Write down the correct formula.

  • $A = \pi r^2$

• Step 2: Substitute the value of the radius.

  • $A = \pi \times 7^2$

Worked Example: Perimeter of a Semicircle

Find the exact perimeter of a semicircle with a diameter of $10\text{ cm}$.

• Step 1: Calculate the curved arc length.

  • The curved edge is exactly half of the full circumference.
  • Curved length = $\frac{1}{2} \times \pi d$
  • Curved length = $\frac{1}{2} \times \pi \times 10 = 5\pi \text{ cm}$

Higher Tier Applications: Sectors

Higher tier papers often apply exact $\pi$ calculations to a sector (a slice of a circle bounded by two radii). A sector is defined by its angle, $\theta$.

The arc length is simply a fraction of the full circumference: $L = \frac{\theta}{360} \times \pi d$. The sector area is a fraction of the full area: $A = \frac{\theta}{360} \times \pi r^2$.

To solve these without a calculator, simplify the fraction $\frac{\theta}{360}$ first. For example, if the angle is $60^\circ$, simplify $\frac{60}{360}$ to , then multiply this simplified fraction by your value.