Cross-sections of solids and drawing in space

The plane and space, grade 12 (3 units) Lesson 13 of 15Learning

Cross-sections of solids and drawing in space

A cross-section of a solid

A cross-section of a solid is the shape one gets by cutting it with a plane – like a slice of bread, or cheese cut with a knife. The shape depends on the direction of the cut.

Cross-sections of a cube whose edge is $\displaystyle a$ :

Parallel to a face – a square of $\displaystyle a$ by $\displaystyle a$ .
Through two opposite edges – a rectangle: one side is the edge $\displaystyle a$ , and the other a face diagonal, $\displaystyle a\sqrt{2}$ .
Cutting off a corner – a triangle.

Cross-sections of round solids

A cylinder cut along its axis (the line between the centres of its bases) – a rectangle of $\displaystyle 2r$ by $\displaystyle h$ . Parallel to the base – a disc of radius $\displaystyle r$ .
A cone cut through its apex and axis – an isosceles triangle: the base $\displaystyle 2r$ , the height $\displaystyle h$ , and the equal sides are slant heights.
A cone cut parallel to its base – a disc, smaller the closer the cut is to the apex.
A sphere – every cross-section is a disc. The cut through the centre is the largest, and its radius is $\displaystyle r$ .

Exercise 1

What is the shape of the cross-section?

A salami (a cylinder) cut parallel to its base
A square pyramid cut parallel to its base
A watermelon (a sphere) cut in two
A milk carton (a box) cut from an edge of the base to the opposite edge of the top
An ice-cream cone (a cone) cut lengthwise, through the sharp end

Exercise 2

Find the area of the cross-section. Round to two decimal places where needed.

A cylinder whose radius is 3 cm and height 10 cm, cut along its axis

/ – 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

A cube whose edge is 4 cm, cut through two opposite edges

/ – 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

A watermelon whose radius is 15 cm, cut through the centre

/ – 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

A cone whose radius is 6 cm and height 8 cm, cut through its apex and axis

/ – 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

Drawing a box in space

A drawing in space shows a solid as one sees it from the front, a little from the right and from above. On squared paper:

Draw the front in its real size: a rectangle.
From each vertex of the front draw a line going back, along the diagonal of the squares. To make the solid look natural, these lines are half the real depth.
Join their ends: this is the back face. An edge the solid hides is drawn as a dashed line.

In a drawing in space, edges that are parallel in the solid stay parallel, but not every angle is right: the angles of the top face and of the side face look acute or obtuse.

Exercise 3

Make a drawing in space of a box: 4 squares long, 4 squares deep and 3 squares high. Mark the hidden edges with dashed lines.

Exercise 4

This is the building map of a building of cubes. Draw the building in space: start with the cubes of the back row, then add what is in front of them.

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

Open as a page

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 $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$