3D Pythagoras & Trigonometry Flashcards

All 8 cards in this deck

What is the formula for the length of the space diagonal of a cuboid with edges $a$, $b$ and $c$?

e.g. a cuboid $2$ cm by $3$ cm by $6$ cm

$d=\sqrt{a^2+b^2+c^2}$

e.g. $d=\sqrt{4+9+36}=7$ cm

What are the steps to find the angle between a space diagonal of a cuboid and its base?

e.g. cuboid with base $3$ cm by $4$ cm and height $12$ cm

1. Pythagoras on the base to get the base diagonal: $\sqrt{3^2+4^2}=5$ cm
2. Sketch the right-angled triangle: base diagonal $5$, vertical edge $12$, hypotenuse the space diagonal
3. Use tan: $\tan\theta=\frac{12}{5}$, so $\theta=67.4^\circ$

In a pyramid with square base $ABCD$ and vertex $T$ vertically above the centre, which right-angled triangle gives the angle between edge $TA$ and the base?

e.g. angle $TAC$

Horizontal side $=\frac{1}{2}AC$ (half the base diagonal), vertical side $=$ height of the pyramid, hypotenuse $=TA$; so $\tan(\angle TAC)=\dfrac{\text{height}}{\frac{1}{2}AC}$

What equation links the radius $r$, vertical height $h$ and slant height $l$ of a cone?

e.g. $r=3$ cm, $h=4$ cm

$l^2=r^2+h^2$

e.g. $l=\sqrt{9+16}=5$ cm

True or false? The sine rule and cosine rule cannot be used on a triangle drawn inside a 3D solid because it is not flat.

False. Any triangle lies in a plane, so the sine and cosine rules work on triangles inside 3D solids just as in 2D.

To find the lower bound of a cube's edge from its measured space diagonal, which value of the diagonal do you use, and what equation do you solve?

e.g. diagonal $= 11.3$ cm correct to the nearest mm

The lower bound of the diagonal, $11.25$ cm; solve $a^2+a^2+a^2=11.25^2$ (i.e. $a=\frac{11.25}{\sqrt3}$)

When finding the lower bound for the edge of a cube from a measured space diagonal, which bound of the diagonal do you substitute?

e.g. diagonal $= 11.3$ cm correct to the nearest mm

The lower bound of the diagonal, $11.25$ cm.
The edge gets smaller as the diagonal gets smaller.

What are the steps to find the edges of a cuboid from its space diagonal and the ratio of its edges?

e.g. edges in the ratio $1:2:2$, space diagonal $12$ cm

1. Write the edges as multiples of $x$: $x$, $2x$, $2x$
2. Put them into 3D Pythagoras: $x^2+4x^2+4x^2=12^2$, so $9x^2=144$
3. Solve for $x$ and substitute back: $x=4$, edges $4$ cm, $8$ cm, $8$ cm