3D Pythagoras & Trigonometry (Edexcel IGCSE Maths): Exam Questions

The diagram shows the prism $ABCDEF$ with cross section triangle $ABC$.

Angle $BEC$= 40° and angle $ACB$is obtuse. $AC$ = 6 cm and $CE$= 13cm

The area of triangle $ABC$is 22 $\mathrm{cm}^{2}$

Calculate the length of $AB$. Give your answer correct to one decimal place.

..................$\mathrm{cm}$

$ABC$ is an isosceles triangle in a horizontal plane. The point $T$ is vertically above $B$.

Angle $ABC=140^{\circ}$ $AB=BC=8cm$ $TB=10cm$ $M$ is the midpoint of $AC$.

Calculate the size of the angle between $MT$ and the horizontal plane $ABC$. Give your answer correct to one decimal place.

ABCDEFGH is a cuboid.

AB = 7.3 cm CH = 8.1 cm Angle BCA = 48°

Find the size of the angle between AH and the plane ABCD. Give your answer correct to 1 decimal place.

The diagram shows a solid prism $ABCDEFGH$.

The trapezium $ABCD$, in which $AD$ is parallel to $BC$, is a cross section of the prism. The base $ADEH$ of the prism is a horizontal plane. $ADEH$ and $BCFG$ are rectangles. The midpoint of $BC$ is vertically above the midpoint of $AD$ so that $BA=CD$.

$AD=37\mathrm{cm}$ $GF=28\mathrm{cm}$ $DE=24\mathrm{cm}$

The perpendicular distance between edges $AD$ and $BC$ is 20 $\mathrm{cm}$.

Calculate the size of the angle between $AF$ and the plane $ADEH$. Give your answer correct to one decimal place.

The diagram shows a triangular prism $ABCDEF$ with a horizontal base $ABEF$.

$AC=BC=FD=ED=12\mathrm{cm}$ $AB=10\mathrm{cm}$ $BE=15\mathrm{cm}$

Calculate the size of the angle between $AD$ and the base $ABEF$. Give your answer correct to 3 significant figures.

$ABCD$ is a horizontal rectangular field.

A vertical pole, $AE$, is placed at the corner A of the field.

$AE=12m$ $AB=18m$ $AD=8m$

Calculate the size of the angle between $EC$ and the plane $ABCD$ Give your answer correct to one decimal place.

The diagram shows the prism $ABCDEFGHJK$ with horizontal base $AEFG$

$ABCDE$ is a cross section of the prism where

$ABDE$ is a square $BCD$ is an equilateral triangle

$EF=2\times AE$
$M$is the midpoint of $GF$ so that $JM$ is vertical.

Angle $MAJ=y^{\circ}$ Given that $\mathrm{tan}y^{\circ}=T$ find the value of$T$, giving your answer in the form $\frac{\sqrt{p}+\sqrt{q}}{17}$ where $p$ and $q$ are integers.

$T$ = ...........................

The diagram shows a prism $ABCDEFGH$ with a horizontal base.

The base of the prism, $EFGH$, is a square of side $12\mathrm{cm}.$ Trapezium $ADEF$ is a cross section of the prism where $AF$ and $DE$ are vertical edges.

$\begin{gathered} D E & = C H = 10 \text{cm} \\ A D & = B C = 15 \text{cm} \end{gathered}$

Work out the size of the angle between $CF$ and the base $EFGH.$ Give your answer correct to one decimal place.

Work out the length of $BE.$ Give your answer correct to one decimal place.

.............................................. cm

The diagram shows a cuboid $ABCDEFGH$.

$EH=9\mathrm{cm},HG=5\mathrm{cm}$ and $GB=6\mathrm{cm}.$

Work out the size of the angle between $AH$ and the plane $EFGH$. Give your answer correct to 3 significant figures.

Alvin has a crate in the shape of a cuboid. The crate is open at the top. The internal dimensions of the crate are 46cm long by 46cm wide by 55cm high.

Alvin has a stick of length 95cm. Alvin places the stick in the crate so that the shortest possible length extends out above the top of the crate.

Calculate the length of the stick that extends out of the crate.

..................................................... cm

Calculate the angle the stick makes with the base of the crate.

$ABCDEF$ is a triangular prism which represents part of a hill.

$ABCF$ is the horizontal rectangular base. $D$ is vertically above $C$.

Work out the height $CD$.

..........................m

Jamil walks in a straight line from $A$ to $D$.

Work out the size of angle $DAC$. You must show your working.

........................degrees

The diagram shows a triangular prism.

$AF=10\mathrm{cm},AB=24\mathrm{cm}$ and$BC=8\mathrm{cm}$. Angle$FAB$ = angle $ADC$ = angle $BCD=90^{\circ}$

Work out the size of the angle between the line $BE$ and the plane $ABCD$. Give your answer correct to 1 decimal place.

The diagram shows a pyramid with a square base ABCD of side length 8cm. The diagonals of the square, AC and BD, intersect at M. V is vertically above M and VM = 10 cm.

Calculate the angle between VA and the base.

Rectangle $ABCD$ is the horizontal base of a triangular prism $ABCDEF.$

$AE=BE$

$E$ is vertically above M, the midpoint of $AB$.

$AB=16$ cm $AE=17$cm $BC=30$ cm

Show that $EM=15$ cm

Work out the size of angle $ECM.$

...........................degrees

Here is a cube with edge length $x$cm

One diagonal is shown.

Circle the length, in centimetres, of the diagonal.

The diagram shows a prism with a rectangular base, ABFE. The cross-section, ABCD, is a trapezium with AD = BC. AB = 8 cm, GH = 5 cm, BF = 12 cm and angle ABC = 70°.

The perpendicular from G onto EF meets EF at X.

i) Show that *EX *= 6.5 cm.

ii) Calculate AX.

AX = ........................................... cm [2]

iii) Calculate the angle between the diagonal AG and the base ABFE.

The diagram shows a cuboid ABCDEFGH.

Calculate angle BHF.

The diagram shows a right-angled triangular prism ABCDEF.

Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm.

Calculate the exact length of AF.

..................................................... cm

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

Ivana has a pencil of length 13cm.

Does this pencil fit completely inside the cuboid? Show how you decide.

The diagram shows a right-angled triangular prism ABCDEF.

Calculate angle AFB.

The diagram shows a cuboid $PQRSTUVW$. $PV=17.2\mathrm{cm}$ The angle between the line $PV$ and the base $TUVW$ of the cuboid is $43^{\circ}$.

Calculate $PT$.

$PT=$ ........................................... cm

The diagram shows a pyramid with a square base ABCD. DB = 8 cm. P is vertically above the centre, X, of the base and PX = 5 cm.

Calculate the angle between PB and the base ABCD.

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

i) Calculate angle CAB.

Angle* CAB* = ................................................ [2]

ii) Calculate angle GAC.

Angle* GAC *= ................................................ [2]

The diagram shows a cuboid. $AB$ = 8 cm, $BC$ = 4 cm and $CR$ = 5 cm.

Calculate the angle between the diagonal $AR$ and the plane $BCRQ$.