Angles in solids: trigonometry in space

The plane and space, grade 12 (3 units) Lesson 9 of 15Learning

Angles in solids: trigonometry in space

Recap: trigonometry in a right-angled triangle

In grade 11 you learnt trigonometry in a right-angled triangle. $\displaystyle \alpha$ is one of the acute angles: $\displaystyle a$ is the leg opposite $\displaystyle \alpha$ , $\displaystyle b$ the leg next to $\displaystyle \alpha$ , and $\displaystyle c$ the hypotenuse.

\displaystyle \sin\alpha

\displaystyle =

\displaystyle \frac{a}{c}

\displaystyle \cos\alpha

\displaystyle =

\displaystyle \frac{b}{c}

\displaystyle \tan\alpha

\displaystyle =

\displaystyle \frac{a}{b}

When two sides are known, the angle is found with the calculator’s inverse key: $\displaystyle {\alpha}={\tan^{-1}\left(\frac{a}{b}\right)}$ , and likewise $\displaystyle \sin^{-1}$ and $\displaystyle \cos^{-1}$ .
When an angle and a side are known, another side is found: for example $\displaystyle {a}={b\cdot\tan\alpha}$ .
The calculator is in degrees; a rounded angle is written after $\displaystyle \approx$ .

An angle inside a solid

To find an angle in a solid, look for a right-angled triangle holding the angle:

The right angle is where a line perpendicular to the base – an upright edge, the height of a pyramid or a cone – meets the base.
Find the missing sides of the triangle by Pythagoras’ theorem.
Choose sine, cosine or tangent by the sides that are known.

The angle between a box’s diagonal and its base

The angle $\displaystyle \alpha$ between the box’s diagonal and its base is in the right-angled triangle with sides $\displaystyle e$ , $\displaystyle c$ and $\displaystyle d$ : the leg opposite $\displaystyle \alpha$ is the height $\displaystyle c$ , and the leg next to it is the base’s diagonal $\displaystyle e$ .

Example. In a room of 5 m by 4 m, 3 m high, the cable from a corner of the floor to the far corner of the ceiling:

$\displaystyle {e}={\sqrt{5^2+4^2}}={\sqrt{41}}$ .
$\displaystyle {\alpha}={\tan^{-1}\left(\frac{3}{\sqrt{41}}\right)\approx25.10^{\circ}}$ .

The angle between a pyramid’s face and its base

The angle $\displaystyle \beta$ between a side face and the base – how steep the roof is – is at the point $\displaystyle M$ , in the triangle $\displaystyle h$ , $\displaystyle OM$ , $\displaystyle k$ :

$\displaystyle {\tan\beta}={\frac{h}{OM}}$ , and $\displaystyle {OM}={\frac{a}{2}}$ .

In a cone the angle $\displaystyle \gamma$ between the slant height and the base is in the triangle $\displaystyle h$ , $\displaystyle r$ , $\displaystyle l$ : $\displaystyle {\tan\gamma}={\frac{h}{r}}$ .

Exercise 1

A box’s dimensions are 8 cm, 6 cm, and the height 5 cm.
What is the angle between the box’s diagonal and its base? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The glass pyramid at the Louvre: the side of the base is 35 m, and the height 21.6 m.
What is the angle between a side face and the base? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A pile of sand is a cone: the radius is 2 m, and the height 1.5 m.
At what angle does the side of the pile slope to the ground? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A roof is a square pyramid: the side of the base is 10 m, and each face slopes to the base at an angle of $\displaystyle 30^{\circ}$ .
Round each result to two decimal places.

a

The height of the roof

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The height of a face

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

A cube’s edge is 1 m.
What is the angle between the cube’s diagonal and its base? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 12 (3 units)

The box and the cube

The volume of a box – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle V$ $\displaystyle =$ $\displaystyle abc$
The surface area of a box – three pairs of equal faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2(ab+ac+bc)$
The volume of a cube – $\displaystyle a$ the edge
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$
The surface area of a cube – six square faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 6a^2$

The prism

The volume of a prism – $\displaystyle B$ the base area, $\displaystyle h$ the height (the distance between the bases)
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$
The lateral area – $\displaystyle P$ the perimeter of the base
$\displaystyle L$ $\displaystyle =$ $\displaystyle P\cdot h$
The surface area – the two bases and the lateral surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2B+L$

The cylinder

The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2h$
The curved surface area – a rectangle as long as the base’s circumference
$\displaystyle L$ $\displaystyle =$ $\displaystyle 2\pi rh$
The surface area – the two bases and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2\pi r^2+2\pi rh$

The pyramid and the cone

The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$
In a square pyramid – $\displaystyle k$ the height of a side face, $\displaystyle h$ the pyramid’s height, $\displaystyle a$ the side of the base
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$
The surface area of a square pyramid – the base and four triangles
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2+2ak$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2h}{3}$
The slant height $\displaystyle l$ – the hypotenuse of a triangle whose legs are the radius $\displaystyle r$ and the height $\displaystyle h$
$\displaystyle l^2$ $\displaystyle =$ $\displaystyle r^2+h^2$
The curved surface area of a cone
$\displaystyle L$ $\displaystyle =$ $\displaystyle \pi rl$
The surface area of a cone – the base and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2+\pi rl$

The sphere

The volume of a sphere – $\displaystyle r$ the radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4\pi r^3}{3}$
The surface area of a sphere
$\displaystyle A$ $\displaystyle =$ $\displaystyle 4\pi r^2$

Diagonals and distances

The diagonal of a box $\displaystyle d$ – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle d^2$ $\displaystyle =$ $\displaystyle a^2+b^2+c^2$
The diagonal of a cube – $\displaystyle a$ the edge
$\displaystyle d$ $\displaystyle =$ $\displaystyle a\sqrt{3}$
In a square pyramid – $\displaystyle s$ a side edge, $\displaystyle h$ the height, $\displaystyle a$ the side of the base
$\displaystyle s^2$ $\displaystyle =$ $\displaystyle h^2+\frac{a^2}{2}$

Trigonometry in a right-angled triangle

$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle, when both legs are known
$\displaystyle \alpha=\tan^{-1}\left(\frac{a}{b}\right)$

The shortest path on the surface of a box

A path from a vertex to the vertex farthest from it, across two faces: unfolded, the diagonal of a rectangle with sides $\displaystyle a+b$ and $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$