3D Pythagoras & Trigonometry (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

The diagram shows cuboid $ABCDEFGH$ of length 20 cm and width 5.5 cm.

The volume of the cuboid is 495 cm³ .

Find the angle between the line $AG$ and the base of the cuboid $ABCD.$

$ABCDEFGH$ is a cuboid.

$AB=18\text{cm}$, $BC=7\text{cm}$ and $CG=12\text{cm}$.

Calculate the angle that the diagonal $AG$ makes with the base $ABCD$. 

The diagram shows a pyramid with a square base ABCD.

The diagonals AC and BD intersect at M.

The vertex V is vertically above M.

AB = 11 cm and AV = 18.6 cm.  

Calculate the angle that AV makes with the base.

The diagram shows a cuboid with dimensions 5.5 cm, 8 cm and 16.2 cm.

Calculate the angle between the line AB and the horizontal base of the cuboid.

The diagram shows a triangular prism.

AB = 12 cm, BC = 6 cm, PC = 4 cm, angle BCP = 90° and angle QDC = 90°.  

Calculate the angle between AP and the rectangular base ABCD.

The diagram shows a prism ABCDEF.

The cross-section is a right-angled triangle BCD.

BD = 10 cm, BC = 5.2 cm and ED = 18 cm.

Calculate angle BEC.

Angle BEC = ...................................................

The point G lies on the line ED and GD = 7 cm.

Work out angle BGE.

Angle BGE = ................................................

The diagram shows a field, ABCD, on horizontal ground.

There is a vertical post at C.

From B, the angle of elevation of the top of the post is 19°.

Find the height of the post.

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

The diagram shows an open box $ABCDEFGH$ in the shape of a cuboid.

$AB=20$ cm, $BC=18$ cm and $AE=16$ cm.

A thin rod $AGX$ rests partly in the box as shown.

The rod is $40$ cm long.

Calculate $GX$, the length of the rod which is outside the box.

$GX=$ ............................................ cm

Calculate the angle the rod makes with the base of the box.

The diagram shows a cuboid $ABCDEFGH$.

$AB=7\mathrm{cm},AF=5\mathrm{cm}\mathrm{and}FC=15\mathrm{cm}.$

Calculate the volume of the cuboid.

Give your answer correct to $3$ significant figures.

The diagram shows a pyramid.

$BCDE$is a square with sides of length 10 cm.

The other faces of the pyramid are equilateral triangles with sides of length 10 cm.

Calculate the volume of the pyramid.

Give your answer in the form $\frac{a}{b}\sqrt{2}{\mathrm{cm}}^3$ where $a$ and $b$ are integers to be found.

Find the size of angle $DAB$.

The diagram shows a cuboid.

$\begin{array}{rcl}AB& =& 5\mathrm{cm}\end{array}$ and $\begin{array}{rcl}AH& =& 4\mathrm{cm}\end{array}$.

The size of the angle between $CH$ and the plane $ABCD$ is $35°$

Calculate the volume of the cuboid.

Give your answer correct to $3$ significant figures.

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.

The diagram shows a triangular prism.

The base, $ABCD$, of the prism is a square of side length 15 cm. Angle $ABE$and angle $CBE$are right angles.

Angle $EAB$= 35°

$M$is the point on $DA$such that

$DM:MA=2:3$

Calculate the size of the angle between $EM$and the base of the prism. Give your answer correct to 1 decimal place.

$VABCD$is a solid pyramid.

$ABCD$is a square of side 20 cm.

The angle between any sloping edge and the plane $ABCD$is 55°

Calculate the surface area of the pyramid.

Give your answer correct to 2 significant figures.

$VABCD$ is a square-based pyramid.

The horizontal base $ABCD$ has side length 10 cm and centre $M$.

Angle $VMA$ is 90°

Angle $VAM$is 68°

Work out the volume of the pyramid.

....................cm³

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.

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

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.

 [1]

ii) Calculate AX.

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

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

[2]

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 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.

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 solid pyramid $ABCDE$ with a horizontal base.

The base, $BCDE$ , of the pyramid is a square of side 10 $\mathrm{cm}$.

The vertex $A$ of the pyramid is vertically above the centre $O$ of the base so that  $AB=AC=AD=AE$

The total surface area of the pyramid is 360 ${\mathrm{cm}}^2$

Work out the size of the angle between $AC$ and the base $BCDE$. 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.