Is 3D Pythagoras on the Foundation paper?

No. Spec reference G23, which extends Pythagoras' theorem and the trigonometric ratios into three dimensional figures, is printed in bold in the 1MA1 specification, and bold marks Higher tier only content. Foundation candidates still meet Pythagoras and the sine, cosine and tangent ratios at G20, but only in flat figures.

Is the 3D version of Pythagoras' theorem on the Edexcel formula sheet?

No. The Exam Aid issued with every 1MA1 paper for the 2025 to 2027 series gives a^2 + b^2 = c^2 and the three right-angled ratios, and the Higher version adds the sine rule, the cosine rule and the area rule. The three dimensional form d = sqrt(a^2 + b^2 + c^2) is not on it. It is worth learning, with one caveat: it works for a cuboid, where the three edges meeting at a corner are at right angles to each other, and not for a general solid.

How do you find the angle between a line and a plane?

Find the point where the line meets the plane, then drop a perpendicular from the other end of the line down onto the plane and mark where it lands. Join that landing point back to the meeting point and you have a right-angled triangle; the angle you want is the one at the meeting point. In a cuboid ABCDEFGH with base ABCD and G above C, the angle between AG and the base is angle GAC.

How many marks is a 3D trigonometry question worth on Edexcel Higher?

Usually 4-5 marks, and it tends to appear once across a series rather than on every paper. The mark scheme normally splits into a method mark for the first triangle, a method mark for the second, and an accuracy mark for the final rounded value, so writing down a correct base diagonal is worth doing even if the rest of the question stalls.

3D Pythagoras and Trigonometry - Edexcel GCSE Higher Maths

Typically 4-6 marks, usually one longer question on Paper 2 or Paper 3 · Spec 1MA1

Spec reference G23 on Edexcel's 1MA1 runs to a single line: apply Pythagoras' theorem and the trigonometric ratios to find lengths and angles in three dimensional figures. It is printed in bold, which in this specification means Higher tier only, so a Foundation candidate never meets it. No new formula arrives with it. The theorem and the three ratios are the ones from G20; what is new is that the right-angled triangle you need is buried inside a solid, and you usually have to build it from a diagonal you calculate first.

A question in this style names its solid and letters every vertex, so it opens with something like ABCDEFGH is a cuboid, or VABCD is a pyramid with a horizontal square base, and then asks for a length such as AG or for the size of the angle between AG and the plane ABCD. Exact answers are rare, so these sit on Papers 2 and 3 where you have a calculator, generally in the last third of the paper for 4-5 marks, with the stem closing on correct to 1 decimal place or correct to 3 significant figures. Cone questions belong to the same spec point in practice: the radius, the perpendicular height and the slant height form a right-angled triangle, and pulling the missing one out of it is what lets you use the cone formulae printed alongside the question.

The marks split along the two stages. One method mark for the first triangle, which is almost always the diagonal of the base, one for the second triangle that uses that diagonal, then an accuracy mark for the rounded value. Two habits protect them. Keep the base diagonal squared rather than rooted, since $AC^2 = 65$ is exact where $AC = 8.1$ is not. And name the vertex directly beneath the far end of your line before deciding which side is adjacent, because the angle between AG and the base ABCD sits at A, not at G, and those two are complementary, so picking the wrong one loses the accuracy mark outright. One more thing worth knowing: $a^2 + b^2 = c^2$ is on the formulae sheet, but the three dimensional shortcut $\sqrt{a^2 + b^2 + c^2}$ is not.

Worked example

VABCD is a solid pyramid with a horizontal square base ABCD of side 14 cm. The diagonals AC and BD of the base cross at M, and the vertex V is vertically above M. Each of the sloping edges VA, VB, VC and VD is 25 cm. Work out the size of the angle between VA and the base ABCD. Give your answer correct to 1 decimal place.

[5 marks]

V is vertically above M, so VA meets the base at A and drops onto the base along AM. The angle asked for is angle VAM, in triangle VAM, which has its right angle at M.
M1: identifying the right-angled triangle VMA and placing the angle at A, where the line meets the plane
Pythagoras on the base square: $AC^2 = 14^2 + 14^2 = 392$ , so $AC = \sqrt{392} = 19.798\ldots$ cm.
M1: full diagonal of the base found first
M is the centre of the square, so $AM = 19.798\ldots \div 2 = 9.899\ldots$ cm.
M1: halving the diagonal, not the side
VA is the hypotenuse of triangle VAM and AM is adjacent to the angle at A, so $\cos(\angle VAM) = \frac{9.899\ldots}{25}$ .
M1: cosine chosen because the 25 cm edge is the hypotenuse
$\angle VAM = \cos^{-1}(0.39598\ldots) = 66.672\ldots$ , so the angle is $66.7^\circ$ correct to 1 decimal place.
A1: 66.7; a candidate who uses AM = 7 cm, half the side, gets 73.7 and scores no accuracy mark

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no diagrams, so every solid is set out in words: read which face is the base and which vertex sits above which before you sketch anything. Nearly all of these are calculator questions, so work in degree mode, hold the base diagonal unrounded between the two stages, and match the rounding the stem asks for.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [3 marks]

ABCDEFGH is a cuboid with horizontal rectangular base ABCD. E is vertically above A, F is vertically above B, G is vertically above C and H is vertically above D. The diagonal AC of the base is 12 cm and the vertical edge CG is 5 cm. Work out the size of the angle between the line AG and the base ABCD. Give your answer in degrees correct to 1 decimal place.

Your answer (fractions like 1/2 are fine)
Question 2 Warm-up [3 marks]

VABCD is a solid pyramid with a horizontal square base ABCD. The diagonals AC and BD of the base cross at M, and the vertex V is vertically above M. AC = 16 cm and VM = 15 cm. Work out the length of the sloping edge VA, in cm.

Your answer (fractions like 1/2 are fine)
Question 3 Exam pace [3 marks]

A solid cone has a base diameter of 14 cm and a perpendicular height of 24 cm. Work out the slant height of the cone, in cm.

Your answer (fractions like 1/2 are fine)
Question 4 Exam pace [4 marks]

ABCDEFGH is a cuboid with horizontal square base ABCD. E is vertically above A, F is vertically above B, G is vertically above C and H is vertically above D. The base ABCD has sides of 9 cm and the diagonal AG is 21 cm. Work out the volume of the cuboid. Give your answer in cm $^3$ correct to the nearest cm $^3$ .

Your answer (fractions like 1/2 are fine)
Question 5 Exam pace [3 marks]

A swimming pool is a cuboid 25 m long and 10 m wide, with a uniform depth of 2 m. A rope is stretched in a straight line from a corner at the top of one end to the corner at the bottom of the other end, diagonally opposite. Work out the length of the rope, in metres.

Your answer (fractions like 1/2 are fine)
Question 6 Exam pace [3 marks]

A solid cone has a volume of $100\pi$ cm $^3$ and a base radius of 5 cm. Volume of a cone $= \frac{1}{3}\pi r^2 h$ . Work out the slant height of the cone, in cm.

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Using an edge of the base instead of the base diagonal when finding the angle a space diagonal makes with the base. In a cuboid with AB = 12 cm, BC = 5 cm and a height of 7 cm, $\tan\theta = \frac{7}{12}$ gives $30.3^\circ$ , where the correct triangle uses AC = 13 cm and gives $28.3^\circ$ .

Name the vertex directly below the far end of the line first. For AG in a cuboid that vertex is C, so the triangle is AGC and the adjacent side is AC, which needs Pythagoras before any trigonometry starts.
Rounding the base diagonal and carrying the rounded figure into stage two. With AB = 7 cm, BC = 4 cm and a height of 6 cm, writing AC = 8.1 cm instead of keeping $\sqrt{65} = 8.0622\ldots$ turns a correct $36.7^\circ$ into $36.5^\circ$ , outside the accepted range.

Keep the diagonal squared where you can: $AC^2 = 65$ is exact, and $\tan\theta = \frac{6}{\sqrt{65}}$ goes into the calculator in one press. Otherwise store the value and recall it.
Feeding a diameter into Pythagoras. For a cone of base diameter 16 cm and perpendicular height 15 cm, $\sqrt{16^2 + 15^2} = 21.9$ cm instead of the correct slant height $\sqrt{8^2 + 15^2} = 17$ cm.

Write r = 8 cm on the page before squaring anything. Then size-check: the slant height has to be the longest of the three, so a slant height shorter than the perpendicular height means a value has gone into the wrong slot.
Taking half the base side as the horizontal distance from the centre of a square base out to a corner. For a base of side 10 cm, the distance from the centre M to the corner A is half the diagonal, $5\sqrt{2} = 7.07$ cm, not 5 cm.

Decide which length the question needs. Half the diagonal goes with a sloping edge such as VA; half the side goes with the slant height of a triangular face, VN where N is the midpoint of AB. Both appear in surface area questions, one after the other.
Reading the angle at the top of the triangle rather than at the base. With AC = 13 cm and a height of 7 cm, angle AGC is $61.7^\circ$ and angle GAC is $28.3^\circ$ ; only the second is the angle between AG and the base.

The angle between a line and a plane always sits where the line meets the plane. Mark that point, here A, and read the angle there. If your two candidate answers add to $90^\circ$ , you have taken the wrong one.

Frequently asked questions

Is 3D Pythagoras on the Foundation paper?: No. Spec reference G23, which extends Pythagoras' theorem and the trigonometric ratios into three dimensional figures, is printed in bold in the 1MA1 specification, and bold marks Higher tier only content. Foundation candidates still meet Pythagoras and the sine, cosine and tangent ratios at G20, but only in flat figures.
Is the 3D version of Pythagoras' theorem on the Edexcel formula sheet?: No. The Exam Aid issued with every 1MA1 paper for the 2025 to 2027 series gives $a^2 + b^2 = c^2$ and the three right-angled ratios, and the Higher version adds the sine rule, the cosine rule and the area rule. The three dimensional form $d = \sqrt{a^2 + b^2 + c^2}$ is not on it. It is worth learning, with one caveat: it works for a cuboid, where the three edges meeting at a corner are at right angles to each other, and not for a general solid.
How do you find the angle between a line and a plane?: Find the point where the line meets the plane, then drop a perpendicular from the other end of the line down onto the plane and mark where it lands. Join that landing point back to the meeting point and you have a right-angled triangle; the angle you want is the one at the meeting point. In a cuboid ABCDEFGH with base ABCD and G above C, the angle between AG and the base is angle GAC.
How many marks is a 3D trigonometry question worth on Edexcel Higher?: Usually 4-5 marks, and it tends to appear once across a series rather than on every paper. The mark scheme normally splits into a method mark for the first triangle, a method mark for the second, and an accuracy mark for the final rounded value, so writing down a correct base diagonal is worth doing even if the rest of the question stalls.

3D Pythagoras and Trigonometry - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics