Exact calculation: fractions, surds and π worksheet — GCSE Higher

1.A student works out the exact area of a circle with radius 4 cm by squaring the radius but forgetting to multiply by π. Work out the correct exact area of the circle, in terms of π.

2.Rationalise the denominator of 10/(4 − √6), giving your answer in its simplest form.

3.Rationalise the denominator of 6/√3, giving your answer in its simplest form.

4.A pizza is cut into 12 equal slices. Ben eats 5 slices and Mia eats 3 slices. What fraction of the pizza is left, giving your answer in its simplest form?

5.A cake recipe needs 3/4 of a kilogram of sugar. Aisha wants to make half the recipe. Work out how much sugar she needs, giving your answer as a fraction of a kilogram in its simplest form.

6.Work out 3/4 − 5/12 exactly, giving your answer in its simplest form.

7.A square tile has an area of 72 cm². Work out the exact perimeter of the tile, giving your answer in the form k√2 cm.

8.Chloe wants to work out 3 1/4 − 1 2/3. She converts both mixed numbers to twelfths, then subtracts the whole numbers and the fraction parts separately, without checking whether she needs to exchange first. Work out the correct value of 3 1/4 − 1 2/3, giving your answer as a mixed number in its simplest form.

9.A circle has a circumference of 8π cm. Work out the exact area of the circle in terms of π.

10.Work out 5/6 × 2/9, giving your answer in its simplest form.

11.A semicircle has a diameter of 8 cm. Work out the exact area of the semicircle, in terms of π.

12.Simplify √45.

13.3/5 of a 5/6 litre bottle of juice is poured out. Work out the exact volume poured out, in litres.

14.The two shorter sides of a right-angled triangle are √12 cm and √24 cm. Work out the exact length of the hypotenuse.

15.A quarter-circle has a radius of 6 cm. Work out the exact perimeter of the quarter-circle, giving your answer in terms of π.