Grade 7-12 Math Review · Grade 11-12 · Solid Mensuration
Naming Solids, Nets, and Cross-Sections
Every surface-area formula in this unit is really the area of a flat net. Learn to unfold a solid and you stop memorising.
Which solid is this?
A solid has 5 faces, 9 edges and 6 vertices.
A triangular prism has two triangular bases and three rectangles: 2 + 3 = 5 faces. Its two triangles hold 3 + 3 = 6 vertices, joined by 3 + 3 edges around the triangles and 3 more running between them, so 9 edges. A square pyramid also has 5 faces, which is the trap — but it has only 5 vertices and 8 edges, so the counts settle it. Check with F + V − E: 5 + 6 − 9 = 2 ✓.
1
A solid is named by its base, not by how it looks.
Say the base first and the family second. A prism whose base is a hexagon is a
hexagonal prism; a pyramid whose base is a square is a square pyramid.
Prismtwo identical basesjoined by rectangles; the same all the way up
Pyramidone base, one apexjoined by triangles; narrows to a point
Cylindera prism with a circular baseno edges, no vertices
Conea pyramid with a circular baseone vertex, one curved surface
2
A net is the solid unfolded, and that is what surface area means.
Cut along enough edges to lay a solid flat and you get its net. The surface area of the
solid is simply the area of that flat shape — nothing more. Every formula later in
this unit is a shortcut for adding up one particular net.
Three rectangles — one for each side of the triangle. That is why the lateral area of
any prism turns out to be the base perimeter times the height: the rectangles sit
side by side and their widths add up to the perimeter.
3
A cross-section is what the knife leaves behind.
Cylinder, cut parallel to the base → a circle
the same circle, at any height
Cylinder, cut straight down through the axis → a rectangle
width 2r, height h
Cone, cut parallel to the base → a smaller circle
it shrinks as you climb
Cone, cut straight down through the apex → a triangle
base 2r, height h
Sphere, cut anywhere → a circle
largest through the centre
A prism is exactly the solid whose parallel cross-section never changes. That single
sentence is why its volume is base area × height: identical layers stacked h deep.
4
Euler’s formula checks a count you were unsure of.
F + V − E = 2
True for every polyhedron with flat faces and no holes. It will not give you a count on
its own, but it will tell you within seconds whether the count you made is possible.
8 + 12 − E = 2
substitute what you have
E = 8 + 12 − 2 = 18
a hexagonal prism fits
An edge is a line; a vertex is a corner
Counting a cube, it is easy to answer “12 corners” because 12 is the number you
remember. A cube has 8 vertices and 12 edges — and the check
6 + 8 − 12 = 2 confirms it. Swap them and you get 6 + 12 − 8 = 10, which is
not 2, so the error announces itself.
5
The four counts you will keep needing.
Solid
Faces
Vertices
Edges
Cube
6
8
12
Triangular prism
5
6
9
Square pyramid
5
5
8
Triangular pyramid
4
4
6
Every row satisfies F + V − E = 2. A prism on an n-sided base always has
n + 2 faces, 2n vertices and 3n edges; a pyramid on one has n + 1 faces, n + 1 vertices
and 2n edges.
6
Variations: the missing count, the base from the edges, an impossible solid.
The same two tools, F + V − E = 2 and the n-rules, used from a different end.
Vertices missing: a polyhedron has 9 faces and 16 edges. How many vertices?
9 + V − 16 = 2
substitute the two counts we have
V = 2 − 9 + 16 = 9
an octagonal pyramid fits: 8 + 1 faces, 2(8) edges
Move the two known counts across; the sign of each one flips.
Backwards: a prism has 21 edges. What is its base?
3n = 21, so n = 7
a prism has 3n edges
F = 7 + 2 = 9, V = 2(7) = 14
a heptagonal prism
9 + 14 − 21 = 2 ✓
Euler agrees
Given a count, run the n-rule in reverse to find the base, then rebuild the other two counts.
No such solid: can a pyramid have 15 edges?
2n = 15, so n = 7.5
a pyramid has 2n edges
A base cannot have 7.5 sides: no
every pyramid has an even number of edges
When the n-rule gives a fraction, the solid described does not exist.
On the exam
Periodical exams and entrance tests ask this as a count: give two of faces, vertices and edges and ask for the third, or name a solid from its counts or its net. The fastest route is the n-rule for the family (prism n + 2, 2n, 3n; pyramid n + 1, n + 1, 2n). The 10-second check is F + V − E = 2 on your final three numbers.
Now you try
Practice problem
Count the edges
A pentagonal prism is a prism whose base is a five-sided polygon. How many edges does it have?
Five edges round the top pentagon, five round the bottom, and five uprights joining them: 5 + 5 + 5 = 15, which is the 3n rule. A counts only the two pentagons and forgets the uprights. Check it: the prism has 5 + 2 = 7 faces and 2(5) = 10 vertices, and 7 + 10 − 15 = 2 ✓.
Hint: build the count in three pieces — the top base, the bottom
base, and the edges running between them — then check the total with
F + V − E = 2 before you commit.
