Simplify exact expressions involving pi

Width of the frame

\(\displaystyle 2\text{ m}\)

Straight side length

\(\displaystyle 1.5\text{ m}\)

A window frame has a rectangular part 2 m wide with 1.5 m straight sides and a semicircular top. What is the exact length of frame needed?

Move a and b and compare the two marked values. The second folds the constant into the π term

For every position of the sliders, when did the two marked values land on the same spot?

Exact expressions with π

An exact expression keeps π as a symbol until a decimal is asked for. Numbers multiplying π combine into one coefficient, and a constant stays separate from the π term

1. 1Multiply numerical factors Numbers multiplying π combine into a single coefficient \(\displaystyle 3\times4\pi=12\pi\)
2. 2Keep unlike terms separate A constant and a π term cannot be added into one term \(\displaystyle 5+\pi\)
3. 3Simplify fractional coefficients Multiply the fraction by the number before simplifying the π term \(\displaystyle \frac{1}{3}\times12\pi=4\pi\)
4. 4Approximate only at the end Keep π exact through the working. Round a supplied decimal value, or use the π key when a calculator is allowed \(\displaystyle 8+4\pi\approx20.57\)

Radius

\(\displaystyle 7\text{ m}\)

Angle at the centre

\(\displaystyle 90°\)

A quarter-circle patio of radius 7 m is to be edged all the way round. Find the coefficient of π in its exact perimeter

Find the curved part as a fraction of the full circumference, then read off the number multiplying π

Radius

\(\displaystyle 7\text{ m}\)

Angle at the centre

\(\displaystyle 90°\)

Exact perimeter

\(\displaystyle 14+3.5\pi\text{ m}\)

For the same patio, use the exact perimeter given above. Its decimal value begins 24.995574… m. Round that value to 2 decimal places

Use the supplied decimal value and look at its third decimal place. Write exactly 2 decimal places, including any final zeroes

Keep the sign of the π term

In the displayed form, a is the ordinary constant and b includes its sign. A product containing only π has no constant part

1. 1Product Multiply the numerical factors \(\displaystyle \begin{gathered}3\times 4\pi=12\pi\\a=0,\quad b=12\end{gathered}\)
2. 2Difference The minus sign belongs to the π coefficient \(\displaystyle \begin{gathered}8-2\times3\pi=8-6\pi\\a=8,\quad b=-6\end{gathered}\)

Which is the first incorrect line in this simplification?