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

$ABCDEFGH$is a cuboid.

$$\begin{gathered} AB=7.3\mathrm{cm} \\ CH=8.1\mathrm{cm} \\ \mathrm{Angle}BCA=48° \end{gathered}$$

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

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 correct to 3 significant figures.

Find the size of angle $DAB$.

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

Angle $ABC=140°$
$AB=BC=8cm$
$TB=10cm$
$M$ 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.

The diagram shows cuboid 

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

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.

..............................................${\mathrm{cm}}^3$

Here is a cube $ABCDEFGH$.

$M$ is the midpoint of the edge $GH$.

Find the size of the angle between the line $MA$ and the plane $ABCD$.
Give your answer correct to $1$ 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.

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.

$\begin{array}{rcl}DE& =& CH=10\mathrm{cm}\\ AD& =& BC=15\mathrm{cm}\end{array}$

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

$ABCDEF$ 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 .
You must show your working.

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

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

Right-angled triangle$ABC$is the cross section of a prism.

$AB=5\mathrm{cm}BC=12\mathrm{cm}$
$M$ is the midpoint of $CF$.
$AM=16\mathrm{cm}$

Work out the volume of the prism.

......................................cm³

The diagram shows a cuboid ABCDEFGH.

Calculate angle BHF.

The length of the longest diagonal of a cube is 25cm.

Calculate the total surface area of the cube.

................................................... cm²

The diagram shows a right-angled triangular prism ABCDEF.

Calculate angle AFB.

$MNOPQRST$ is a cuboid.

$MP=11$cm

$RP=17$cm

Angle $STR=54°$

Work out the size of Angle $RMT$.

Give your answer correct to the nearest degree.

A square-based pyramid has base $ABCD$ where $AB=12\mathrm{cm}$.

The point $X$ is the apex of the pyramid perpendicular to the centre of the square base.

The lengths $AX$ and $AB$ are related by $2AX=3AB$.

Find the size of the angle $XAC$ to the nearest degree.

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.

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 $MAJ=y°$
Given that            $\tan y°=T$
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 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 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 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.

$VABCD$ 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

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.

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 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.