Intuition The one core ideaA net is a 3D shape's skin peeled flat : cut some edges of a solid, unfold it onto a table, and you get a 2D pattern that folds back into the solid. Because the flat pattern keeps every face at its true size, the area of the flat net equals the surface area of the solid — that is the single idea every formula on the parent page is built from.
Before you can read the parent note you must own a small toolbox of pictures and symbols. This page builds each one from nothing, in the order they lean on each other. Never skip forward: every symbol below is earned before it is used.
Before any letters, look at the solid itself. Every flat-sided 3D shape is made of just three kinds of thing.
Definition Face, edge, vertex
A face is one flat panel of the solid — a whole wall of the box. In the picture it is a coloured patch.
An edge is a straight line where two faces meet — the crease you would cut or fold along.
A vertex is a corner point where edges meet (plural: vertices ).
Unfolding a net is exactly the act of cutting some edges and rotating faces flat about the un-cut edges . So to understand a net you must first see the solid as faces held together at edges. That is the whole game.
Face-and-edge counting A cube has 6 faces, 12 edges, 8 vertices.
Every net formula measures distances. A length is just "how long", measured with a ruler in some unit (cm, m). We give lengths short letters so we can talk about them without redrawing.
Definition The length letters used on the parent page
s slant height of a cone/pyramid (defined in §7).l , w , h a base of a pyramid.r
Intuition Why letters and not numbersA formula like 6 s 2 every cube at once. Put in s = 3 s = 10
s s slant height elsewhere. Always check the shape first. When two ideas share a letter, the picture decides which one is meant.
Area is how much flat surface a shape covers — how many unit squares (e.g. 1 cm by 1 cm) fit inside it. Its picture is a tiled floor.
The symbol × multiply — repeated adding. 4 × 3
The little raised 2 superscript , read "squared") means multiply a thing by itself once :
s 2 = s × s .
Intuition Why "squared" literally means a squareA square of side s s s s × s = s 2 squared is a memory of this picture: the area of a square is the side squared . This is why surface area — a sum of areas — is measured in "square units" like cm 2
See Triangles and Area for how this rectangle rule gives the triangle rule we need in §7.
Circles appear the moment we meet cylinders and cones, so we need the symbol π
π π 3.14159 how many diameters fit around a circle's rim . It never changes, whatever the circle's size.
Intuition Why circumference is thewidth of the unrolled tin
When you unroll the curved wall of a cylinder into a flat rectangle, the side that used to hug the circle straightens out. Its length is exactly the distance around that circle — the circumference 2 π r 2 π r Circles and Circumference .
Circumference of a rim with r = 3 2 π × 3 = 6 π ≈ 18.85
The cone net (parent §4) needs one more circle idea: measuring an angle by the arc it cuts.
Definition Radian and the symbolθ θ θ radians means: the angle equals the arc length it cuts, divided by the radius .
Intuition Why radians and not degrees hereRadians make arc = r θ no extra conversion factor . When we unroll a cone, we set the flat sector's arc equal to the base rim's circumference, and radians let us equate them cleanly:
flat sector arc l θ = base rim 2 π r ⇒ θ = l 2 π r . 180 π l
The pyramid and cone use 2 1 2 1 split into two equal parts, take one — half.
Intuition Why the half is thereAny triangle is exactly half of the rectangle you get by boxing it in (base wide, height tall). Two copies of the triangle fill that rectangle, so one triangle is half of base × height 2 1 2 1 a s
Half of a rectangle 6 × 5 2 1 × 6 × 5 = 15
The slant height is the distance measured along the sloping surface , from the middle of a base edge (pyramid) or base rim (cone) up to the apex. It is not the vertical height inside the solid.
Common mistake Vertical height vs slant heightThe vertical height h inside . The slant height s l outside face . In a net you always see the outside faces laid flat, so the triangle heights are slant heights. Using h
They are linked by Pythagoras Theorem : for a cone, l = r 2 + h 2
Which height appears in a pyramid net's triangles The slant height (along the sloping face), not the vertical height.
Definition Equals and plus= +
Intuition Why every surface-area formula is a sumThe net breaks the skin into separate flat pieces with no overlap. Surface area is just the total paper, so you add each piece's area. That is the meaning of, e.g.,
cuboid area = top+bottom 2 l w + front+back 2 l h + left+right 2 w h . Surface Area of 3D Shapes .
Length letters s l w h a r
Radians and arc = r theta
Slant height and Pythagoras
Surface Area of 3D Shapes
Related uses of these tools: Volume of 3D Shapes , Platonic Solids , Unfolding and Folding Problems , Tessellations .
Test yourself — cover the right side.
I can point to a face, an edge, and a vertex on a box A face is a flat wall, an edge is a fold-line between two faces, a vertex is a corner point.
I know what s 2 s 2 = s × s s
I can state the rectangle and triangle area rules Rectangle = base × height = 2 1 × base × height
I know what π π ≈ 3.14159 = 2 π r = π r 2
I understand why the unrolled cylinder wall is 2 π r Straightening the rim lays out its full circumference 2 π r
I can measure an angle in radians and use arc = r θ A full turn is 2 π
I know slant height is not vertical height Slant runs along the outside face (hypotenuse); vertical height goes straight up inside.
I know why surface area is always a sum The net's pieces don't overlap, so total paper = add every face's area.
Recall Quick self-quizWhy can one letter like s s r θ l θ = 2 π r
