Surds and exact values – Year 10 GCSE maths worksheet with answers to download as PDF

Leave each answer in terms of \(\pi\), then give a decimal to 1 decimal place.

1. the circumference of a circle with radius 5 cm
2. the area of a circle with diameter 12 cm
3. the area of a semicircle with radius 8 cm
4. the volume of a cylinder with radius 3 cm and height 10 cm

A surd is a root that cannot be written as a fraction (it is irrational). Decide whether each number is a surd.

1. \(\sqrt{16}\)
2. \(\sqrt{12}\)
3. \(\sqrt{\tfrac94}\)
4. \(\sqrt7\) and \(\pi\)

Simplify each surd by finding the largest square factor. Higher tier.

1. \(\sqrt{50}\)
2. \(\sqrt{72}\)
3. \(\sqrt{48}\)
4. \(\sqrt{200}\)

Simplify each surd first and then collect like terms. Higher tier.

1. \(3\sqrt2+5\sqrt2\)
2. \(\sqrt{12}+\sqrt{27}\)
3. \(\sqrt{50}-\sqrt8\)
4. \(2\sqrt{45}-\sqrt{20}\)

Simplify or expand. Higher tier.

1. \(\sqrt3\times\sqrt{12}\)
2. \((2+\sqrt3)(2-\sqrt3)\)
3. \((1+\sqrt5)^2\)
4. \((3-\sqrt2)(3+\sqrt2)\)

Rationalise the denominator of each fraction. Higher tier.

1. \(\dfrac{6}{\sqrt3}\)
2. \(\dfrac{10}{\sqrt5}\)
3. \(\dfrac{1}{1+\sqrt2}\)
4. \(\dfrac{3}{2-\sqrt3}\)

A circular pond has a radius of 4 m. A path of width 1 m goes all the way round it. Give exact answers in terms of \(\pi\).

1. Find the area of the pond.
2. Find the outer radius of the path and the area of the whole circle (pond and path).
3. Find the area of the path.
4. Find the circumference of the pond and the circumference of the outer edge of the path.

A gate frame is a rectangle 3 m wide and 5 m tall. A diagonal brace goes from one corner to the opposite corner. Give exact answers.

1. Find the length of the brace as a surd in its simplest form.
2. A square gate has side 5 m. Find the length of its diagonal brace.
3. Write the answer to (b) to 1 decimal place.
4. Which brace is longer, and by how much? Give your answer to 2 decimal places.

A sail is a right-angled triangle with the two shorter sides 4 m and 6 m. Give exact answers.

1. Find the length of the longest side as a surd in its simplest form.
2. Find the area of the sail.
3. Find the perimeter as an exact value.
4. The sail is made of cloth that costs £12 per m². Work out the cost of the cloth.

A roof panel is an equilateral triangle with side 6 m. Give exact answers.

1. Use Pythagoras to find the height of the triangle. Write it in simplest surd form.
2. Find the exact area of the panel.
3. Find the area to 1 decimal place.
4. Four of these panels are used. Find the total area in exact form.

A cylindrical can has radius 3 cm and height 10 cm. A cone has radius 3 cm and vertical height 4 cm. Give exact answers in terms of \(\pi\).

1. Find the volume of the can.
2. Find the volume of the cone.
3. Find the slant height of the cone.
4. Find the curved surface area of the cone.

A cube has side 4 cm. Give exact answers.

1. Find the diagonal of a face, as a surd in simplest form.
2. Find the length of the space diagonal (corner to opposite corner through the cube).
3. Write your answer to (b) to 2 decimal places.
4. Show that the space diagonal is \(\sqrt3\) times the side.

A beach ball has radius 6 cm. A hemisphere bowl has the same radius. Leave answers in terms of \(\pi\).

1. Find the volume of the ball.
2. Find the surface area of the ball.
3. Find the volume of the bowl.
4. Find the volume of the ball as a decimal to the nearest cm³.

A wheelchair ramp rises 1 m over a horizontal distance of 12 m. Give answers to 2 decimal places.

1. Find the length of the ramp using Pythagoras, as a surd.
2. Write this length to 2 decimal places.
3. Find the gradient as a fraction and as a percentage.
4. The guidance says the gradient should not be steeper than 1 in 12. Does this ramp meet the guidance?

Higher tier. Show that each statement is true, with working that shows every step.

1. \((\sqrt8+\sqrt2)^2=18\)
2. \(\sqrt{18}+\sqrt{8}=5\sqrt2\)
3. \(\dfrac{4+\sqrt{12}}{2}=2+\sqrt3\)
4. \(\dfrac{\sqrt2+1}{\sqrt2-1}=3+2\sqrt2\)

Higher tier. A rectangle has sides \((3+\sqrt5)\) cm and \((3-\sqrt5)\) cm.

1. Show that the area is 4 cm².
2. Find the perimeter.
3. Find the length of the diagonal, in simplest surd form.
4. Check the diagonal to 2 d.p. using the decimal side lengths.

Higher tier. Solve each equation, leaving answers in simplest surd form.

1. \(x^2=20\)
2. \((x+3)^2=7\)
3. \(2x^2=50\) (give both integer solutions)
4. \(x^2-6x+2=0\) using the formula

Higher tier. A triangle has sides \(\sqrt{12}\) cm, \(\sqrt{28}\) cm and \(2\sqrt{10}\) cm.

1. Write each side in simplest surd form.
2. Show that the triangle is right-angled.
3. Find the area of the triangle in exact form.
4. Find the area to 2 d.p.

Higher tier. Use the exact values in this table.

1. Find \(\sin30^\circ\times\cos60^\circ\) exactly.
2. Show that \(\sin60^\circ\cos30^\circ+\cos60^\circ\sin30^\circ=1\).
3. A ladder of length 6 m makes an angle of \(60^\circ\) with the ground. Find exactly how far up the wall it reaches.
4. Rationalise \(\tan30^\circ=\dfrac1{\sqrt3}\).

Higher tier. You want to find \(a\) and \(b\) (whole numbers) such that \((a+b\sqrt2)^2=17+12\sqrt2\).

1. Expand \((a+b\sqrt2)^2\).
2. Compare the rational parts and the surd parts to form two equations.
3. Solve to find \(a\) and \(b\) (try \(ab=6\)).
4. Hence write down \(\sqrt{17+12\sqrt2}\).