Pythagoras' theorem and trigonometry in 3D

A pyramid with a square base $ABCD$ is shown in the diagram. Every sloping edge of the pyramid has length 20 cm, and $AC = 17$ cm. The figure is marked NOT TO SCALE. The sketch shows vertex $V$ at the top, the base points $A$, $B$, $C$, $D$, one sloping edge labelled 20 cm, and the diagonal $AC$ labelled 17 cm.

Feb/March 2017

The diagram depicts a pyramid $VABCD$ with a rectangular base. $V$ is directly above $M$, where the diagonals $AC$ and $BD$ intersect. $AB = 12\,\text{cm}$, $BC = 10\,\text{cm}$ and $VC = 14\,\text{cm}$.

Feb/March 2021

The prism ABCDQP has a length of 7 cm and is triangular. Its cross-section is triangle PAB, with PA = 4 cm, AB = 5 cm and angle PAB = 90^{\circ}.

Feb/March 2023

Calculate the angle of elevation to the top of the tower.

May/June 2015

The diagram depicts a cuboid. $HD=3$ cm, $EH=5$ cm and $EF=7$ cm. It is labelled NOT TO SCALE.

May/June 2016

The diagram depicts a cube $ABCDEFGH$ with side length 26 cm. It is not drawn to scale.

May/June 2017

The diagram depicts a pyramid standing on rectangular base $ABCD$. $AC$ and $BD$ meet at $M$, and $P$ is vertically above $M$. $AB = 8$ cm, $BC = 6$ cm and $PM = 9$ cm.

May/June 2017

For pentagon $ABCDE$, angle $ACB$ is $90^\circ$ and angle $AED$ is $90^\circ$. Triangle $ACD$ is equilateral with side length 12 cm. Also, $DE = BC = 6$ cm.

May/June 2018

The figure is a pyramid with a square base $ABCD$, where each side measures 8 cm. The diagonals of the square, $AC$ and $BD$, meet at $M$. $V$ is vertically above $M$ and $VM = 10\text{ cm}$. The diagram is labelled NOT TO SCALE.

May/June 2019

A cuboid $ABCDEFGH$ is displayed, and it is NOT TO SCALE. $AB = 18\text{ cm}$, $BC = 7\text{ cm}$ and $CG = 12\text{ cm}$.

May/June 2019

A cuboid’s length is $L$ cm, its width is $W$ cm and its height is $H$ cm. The diagram gives the net of this cuboid. The ratio $W:L = 1:2$.

May/June 2021

A cuboid has dimensions of 24 cm, 12 cm and 8 cm.

May/June 2022

The sketch represents a solid triangular prism $ABCDEF$ with length 15 cm. $AB = 6.4$ cm, $EB = 5.7$ cm and the prism has volume 145 cm$^3$. The sketch is labelled NOT TO SCALE.

May/June 2023

Calculate $x$.

May/June 2023

Calculate the area of the shape.

May/June 2023

The diagram depicts a cuboid. $HD = 4$ cm, $EH = 6.5$ cm and $EF = 9.1$ cm.

May/June 2024

The figure represents a triangular prism whose cross-section is triangle $BCV$. Here, angle $BCV = 90^\circ$, $BC = 5\text{ cm}$, $CV = 4\text{ cm}$ and $AB = 15\text{ cm}$. This sketch is not drawn to scale.

May/June 2024

Calculate the volume occupied by the water in the tank. Give your answer correct to the nearest $10\,\text{cm}^3$.

May/June 2024

The diagram represents a pyramid $OABCD$. Its base is a square, $ABCD$, whose sides are $10.5\text{ cm}$. The vertex $O$ lies directly above the centre of the base, $M$. The pyramid has a height of $24\text{ cm}$.

May/June 2025

The diagram depicts tent ABCD. The tent's front face is an isosceles triangle ABC, with $AB = AC$. The two side faces are congruent triangles ABD and ACD. The diagram is labelled NOT TO SCALE.

Oct/Nov 2015

The diagram depicts a cube with edge length $8$ cm.

Oct/Nov 2016

The figure represents a square-based pyramid $ABCDE$. The square’s diagonals intersect at $M$. $E$ is directly above $M$ on a vertical line. $AB = BC = 12\ \text{cm}$ and $EM = 9\ \text{cm}$. The figure is not drawn to scale.

Oct/Nov 2017

The prism has a length of 4 cm. Its cross section is a right-angled triangle. BC = 3 cm and CQ = 2 cm.

Oct/Nov 2017

A cuboid is illustrated with side lengths $5.5\text{ cm}$, $8\text{ cm}$ and $16.2\text{ cm}$. Line $AB$ is drawn on the cuboid, and the figure is labelled NOT TO SCALE.

Oct/Nov 2018

A triangular prism is drawn in the diagram. $AB = 12\text{ cm}$, $BC = 6\text{ cm}$, $PC = 4\text{ cm}$, angle $BCP = 90^\circ$ and angle $ODC = 90^\circ$. Its rectangular base is $ABCD$. This sketch is not drawn to scale. The labelled points are $A, B, C, D, O, P,$ and $Q$, and the lengths $12\text{ cm}$, $6\text{ cm}$ and $4\text{ cm}$ are shown.

Oct/Nov 2018

The sketch depicts a hemisphere with radius $6\text{ cm}$. Work out its volume. Include the units in your answer. [For a sphere of radius $r$, the volume is $V = \frac{4}{3}\pi r^3$.]

Oct/Nov 2019

The diagram depicts a cuboid $PQRSTUVVW$. $PV = 17.2 \text{ cm}$. The angle formed by the line $PV$ and the base $TUVW$ of the cuboid is $43^\circ$.

Oct/Nov 2020

ABCDEFGH is a cuboid with AB = 8 cm, BC = 5 cm and CG = 11 cm.

Oct/Nov 2020

The figure depicts a pyramid $ABCDE$. It has a square base $ABCD$ lying horizontally, with each side measuring 5 cm. The point $E$ is directly above the centre $O$ of the base. The pyramid’s perpendicular height $OE$ is 9 cm. The diagram is not drawn to scale.

Oct/Nov 2021

The diagram depicts a triangular prism. Angle $BPC = 90^{\circ}$. The base $AB$ measures 12 cm. The lengths $BC = 5$ cm and $CP = 4$ cm are indicated. The points $A, B, C, D, P, Q$ are labelled. The diagram is marked NOT TO SCALE.

Oct/Nov 2021

The sketch depicts triangle $ABC$ on level ground. $AC=15$ m, $BC=8$ m and $AB=20$ m. $BP$ and $CQ$ are vertical poles with unequal heights. $BP=3$ m and $CQ=4$ m. $AQ$ and $PQ$ are straight wires.

Oct/Nov 2022

Calculate the angle $QPR$.

Oct/Nov 2022

The diagram depicts a cuboid $ABCDEFGH$. $AB = 15.1\,$cm, $BC = 4.5\,$cm and $CG = 9.2\,$cm.

Oct/Nov 2023

A cuboid $ABCDEFGH$ is drawn in the diagram. $AB = 14$ cm, $BC = 5$ cm and $CG = 8$ cm. $M$ lies at the midpoint of $HG$. The diagram is NOT TO SCALE.

Oct/Nov 2023

The diagram depicts a cuboid with dimensions AB = 12 cm, BC = 5 cm and height = 4 cm. PB is a diagonal drawn inside the cuboid. A, B, C and D are the vertices of the base ABCD. The diagram is marked "NOT TO SCALE".

Oct/Nov 2024

ABCDEFGH is a solid cuboid. $AB = 10\text{ cm}$, $BC = 8\text{ cm}$ and $CG = 17\text{ cm}$. The diagram is not drawn to scale.

Oct/Nov 2024

The diagram represents a prism with length 15 cm. Its cross-section is a trapezium. Angle DAB = 90^{\circ} and angle ADC = 90^{\circ}. AB = 10 cm, AD = 5 cm and DC = 7 cm.

Oct/Nov 2025

The diagram represents a cuboid ABCDEFGH, with AE = 6.4 cm, EH = 5.1 cm and AG = 13 cm.

Oct/Nov 2025