Diagonals and distances in solids

The plane and space, grade 12 (3 units) Lesson 8 of 15Learning

Diagonals and distances in solids

The diagonal of a box

The diagonal of a box, $\displaystyle d$ , joins a vertex to the vertex farthest from it, through the inside of the box. It is found in two right-angled triangles:

The base’s diagonal, $\displaystyle e$ , is the hypotenuse of a triangle with legs $\displaystyle a$ and $\displaystyle b$ : $\displaystyle {e^2}={a^2+b^2}$ .
The edge $\displaystyle c$ is perpendicular to the base, so to $\displaystyle e$ too. The box’s diagonal is the hypotenuse of a triangle with legs $\displaystyle e$ and $\displaystyle c$ : $\displaystyle {d^2}={e^2+c^2}$ .

\displaystyle d^2

\displaystyle =

\displaystyle a^2+b^2+c^2

In a cube the three dimensions are equal, and $\displaystyle {d^2}={3a^2}$ :

\displaystyle d

\displaystyle =

\displaystyle a\sqrt{3}

Example: a cable in a room

A room’s floor is a rectangle of 5 m by 4 m, and the room is 3 m high.
A straight cable is stretched from a corner of the floor to the corner of the ceiling farthest from it.
How long is the cable?

d=\sqrt{5^2+4^2+3^2}=\sqrt{50}\approx7.07

The cable is about 7.07 m long.

A side edge of a pyramid

In a square pyramid too there is a right-angled triangle: the height $\displaystyle h$ , the segment from $\displaystyle O$ to a vertex of the base, and the side edge $\displaystyle s$ – the hypotenuse.

The segment from $\displaystyle O$ to the vertex is half the diagonal of the base. The diagonal of a square with side $\displaystyle a$ is $\displaystyle a\sqrt{2}$ , so the segment is $\displaystyle \frac{a\sqrt{2}}{2}$ , and its square is $\displaystyle \frac{a^2}{2}$ .

\displaystyle s^2

\displaystyle =

\displaystyle h^2+\frac{a^2}{2}

Exercise 1

A box is 30 cm by 20 cm, and 10 cm high.
Can a straight stick 35 cm long be put inside it, with the lid closed?

/ – 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

A lorry’s load space is a box: 4.2 m by 2 m, and 2.2 m high.
How long is the longest straight pipe that fits in it? 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 classroom’s floor is a rectangle of 6 m by 4 m, and the room is 3 m high.
A lamp hangs in the middle of the ceiling, exactly above the centre of the floor.
How far is the lamp from a corner of the floor? 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 tent is a square pyramid: the side of its floor is 6 m, and its height 4 m.
Along each side edge there is a pole.
Round each result to two decimal places.

a

The distance from the centre of the floor to one of its corners

/ – 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

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b

The length of a pole

/ – 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

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Exercise 5

A storage crate is a cube with an edge of 50 cm.
How long is the longest straight stick that fits in it? 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 $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$