Checkpoint 8 Maths · Geometry and Measure · 8Gg.05 · 8Gg.07

3D Shapes: Euler's Formula and Plans and Elevations

Count faces, vertices and edges, use Euler's formula F + V − E = 2 to find missing values or spot impossible shapes, and draw plans and elevations of 3D shapes to scale.

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In the curriculum

Understand and use Euler's formula, which connects the number of faces (F), vertices (V) and edges (E) of a 3D shape: F + V − E = 2. (8Gg.05)
Draw the front view (front elevation), side view (side elevation) and top view (plan) of 3D shapes to scale. (8Gg.07)

1. Key words

Word / phrase	Meaning / use
face (F)	وجه — a flat surface of a 3D shape
edge (E)	حرف — a line where two faces meet
vertex, vertices (V)	رأس، رؤوس — a corner where edges meet
polyhedron (plural: polyhedra)	متعدد السطوح — a 3D shape whose faces are all flat polygons
prism	منشور — the same cross-section all the way through; two identical ends joined by rectangles
pyramid	هرم — a polygon base with triangles meeting at one point (the apex)
plan (top view)	المسقط الأفقي — what you see looking straight down from above
front elevation	المسقط الرأسي الأمامي — what you see looking straight at the front
side elevation	المسقط الجانبي — what you see looking straight at the side
to scale	بمقياس رسم — every length is multiplied or divided by the same number
isometric drawing	رسم أيزومتري — a 3D drawing on triangle-dot paper

2. Faces, vertices and edges

Four polyhedra with their numbers of faces (F), vertices (V) and edges (E). Dashed lines are hidden edges: count them too.

Count carefully and in an organised way: faces — count the front and back, then the faces around the sides; vertices — count the corners on the top layer, then the bottom layer (or the base, then the apex); edges — count the edges round each end, then the edges joining the ends.

3D shape	F	V	E	F + V − E
cube / cuboid	6	8	12	6 + 8 − 12 = 2
triangular prism	5	6	9	5 + 6 − 9 = 2
square-based pyramid	5	5	8	5 + 5 − 8 = 2
tetrahedron (triangle-based pyramid)	4	4	6	4 + 4 − 6 = 2
pentagonal prism	7	10	15	7 + 10 − 15 = 2
hexagonal prism	8	12	18	8 + 12 − 18 = 2
octahedron (8 triangles)	8	6	12	8 + 6 − 12 = 2

3. Euler's formula

Look at the last column of the table: it is always 2. This pattern was discovered by the mathematician Leonhard Euler and is called Euler's formula.

Euler's formula

F + V − E = 2 (faces + vertices − edges = 2). It can also be written F + V = E + 2. It works for polyhedra: shapes with flat faces, straight edges and no holes through them.

Rearranging the formula lets you find a missing number:

E = F + V − 2
F = E + 2 − V
V = E + 2 − F

Prisms and pyramids in general (generalising). If the end of a prism (or the base of a pyramid) has n sides:

Shape	F	V	E	Check F + V − E
prism with an n-sided end	n + 2	2n	3n	(n + 2) + 2n − 3n = 2 ✓
pyramid with an n-sided base	n + 1	n + 1	2n	(n + 1) + (n + 1) − 2n = 2 ✓

Curved shapes do not follow the formula. A cylinder has 3 faces (2 flat, 1 curved), 2 curved edges and 0 vertices: 3 + 0 − 2 = 1, not 2. So use Euler's formula only for shapes with flat faces.

Compound shapes also follow the formula, if they have flat faces and no holes. The tower below is a cube with a square-based pyramid on top. The square where they join is inside, so it is not a face.

Cube + square-based pyramid: F = 5 + 4 = 9, V = 8 + 1 = 9, E = 12 + 4 = 16, and 9 + 9 − 16 = 2.

قانون أويلر بالعربي

عدد الأوجه + عدد الرؤوس − عدد الأحرف = 2 دايمًا (F + V − E = 2) لأي مجسم أوجهه كلها مستوية ومفيهوش خرم. لو ناقصك رقم، عوّض في القانون وهات المجهول: مثلاً الأحرف E = F + V − 2. والقانون ده مش بيشتغل مع الأسطوانة أو المخروط لأن فيهم أسطح منحنية. ولو حد قالك مجسم مجموعه مش بيدي 2، يبقى المجسم ده مستحيل.

4. Plans and elevations

A 3D shape can be shown by three flat (2D) drawings, each looking straight at one side:

Plan (top view): looking straight down from above.
Front elevation (front view): looking straight at the front, from the direction marked F.
Side elevation (side view): looking straight at the side, from the direction marked S.

In each view you see only the outline and the edges facing you, with no depth. You draw a line wherever there is an edge: where the height or the depth of the shape changes. A flat surface made of several cubes is drawn as one region, with no lines across it.

A solid made from 9 cubes. F shows the front direction, S the side direction.

Its three views. Front: the left column is 3 cubes high. Side: the back column (on the right, as S sees it) is 3 high. Plan: the lines show where the height changes.

Find the width, height and depth of the shape (count the cubes, or read the measurements).
Front elevation: stand at F. For each column across, how high is it? Draw that outline. Add a line wherever a nearer part stops and a part further back continues.
Side elevation: stand at S. Your left and right are now different! Check which end is on your left.
Plan: look from above with the front at the bottom of your drawing. Draw the outline of the whole 'footprint', then lines where the height changes.
To scale: multiply or divide every real length by the scale (for example 1 cm represents 2 m means divide metres by 2). Use a ruler and a sharp pencil; label each view.

Common solids. It helps to know these views:

Front elevation and plan of a square-based pyramid, a cylinder (standing up) and a cone. The dot in the plan of the cone is the apex.

Solid (standing on its base)	Front elevation	Side elevation	Plan
cube	square	square	square
cuboid	rectangle	rectangle	rectangle
square-based pyramid	triangle	triangle	square with both diagonals
cylinder	rectangle	rectangle	circle
cone	triangle	triangle	circle with a dot at the centre
sphere	circle	circle	circle

المساقط بالعربي

Plan = المسقط من فوق (كأنك طاير فوق المجسم). Front elevation = من قدام. Side elevation = من الجنب.
ارسم خط بس في الأماكن اللي فيها حرف — يعني الارتفاع أو العمق بيتغير. السطح المستوي ماترسمش جواه خطوط.
بمقياس رسم: كل الأطوال بتتقسم على نفس الرقم. مثلاً 1 cm بيمثل 2 m ← اقسم كل الأمتار على 2.

5. Worked examples

Worked example 1 [1 mark] A polyhedron has 8 faces and 12 vertices. Use Euler's formula to find the number of edges.

E = F + V − 2 = 8 + 12 − 2 = 18 edges. (This is a hexagonal prism: 3 × 6 = 18 ✓.)

Worked example 2 [2 marks] A polyhedron has 10 vertices and 15 edges. How many faces does it have? Which prism could it be?

F = E + 2 − V = 15 + 2 − 10 = 7 faces.
A prism with an n-sided end has 2n vertices: 2n = 10, so n = 5. It could be a pentagonal prism (7 faces, 10 vertices, 15 edges ✓).

Worked example 3 [2 marks] Samira says: 'I have made a polyhedron with 6 faces, 6 vertices and 11 edges.' Explain why she must have counted wrongly.

F + V − E = 6 + 6 − 11 = 1. For every polyhedron F + V − E = 2, and 1 ≠ 2, so these numbers are impossible. (1 mark for the calculation, 1 mark for comparing with 2.)

Worked example 4 [2 marks] A pyramid has 20 edges. How many faces and vertices does it have?

A pyramid with an n-sided base has 2n edges: 2n = 20, so n = 10 (a decagonal pyramid).
Faces = n + 1 = 11; vertices = n + 1 = 11.
Check with Euler: 11 + 11 − 20 = 2 ✓

Worked example 5 [3 marks] A tent is a triangular prism. The front is an isosceles triangle 3 m wide and 2 m high. The tent is 4 m long. Draw the front elevation, side elevation and plan to a scale of 1 cm to 1 m.

The tent: width 3 m, height 2 m, length 4 m.

The three views drawn to scale (1 cm represents 1 m).

Front elevation: a triangle with base 3 cm and height 2 cm (the top is above the middle of the base, 1.5 cm from each end).
Side elevation: a rectangle 4 cm long and 2 cm high (you see the sloping roof side-on).
Plan: a rectangle 3 cm by 4 cm with a line down the middle: the ridge (top edge) of the tent.
Common mistake: forgetting the ridge line in the plan, or drawing the side elevation with the sloping length instead of the height.

Worked example 6 [1 mark] The front elevation of a building is drawn to a scale of 1 cm to 5 m. On the drawing, the building is 7.2 cm wide. How wide is the real building? 7.2 × 5 = 36 m.

Common mistakes

Forgetting the hidden edges, faces and vertices at the back when counting.
Using Euler's formula for a cylinder, cone or sphere (curved surfaces).
Rearranging wrongly: E = F + V − 2, not F + V + 2.
Drawing a 3D (sloping) picture instead of a flat view. Each view is 2D.
Mixing up the side elevation and the front elevation, or swapping left and right in the side view.
Drawing lines across a flat surface in a plan or elevation, or missing the line where the height changes.
Not using the scale for every length (for example scaling the width but not the height).

How Checkpoint asks this

"Complete the table" of faces, vertices and edges, then "Show that Euler's formula works for this shape."
"A shape has 9 faces and 14 vertices. Work out the number of edges." Write the formula, substitute, answer.
"Explain why a polyhedron cannot have…" Show F + V − E and say it is not 2.
Isometric drawings of shapes made of cubes: "On the grid, draw the plan / front elevation / side elevation." Arrows on the drawing show the directions.
"Draw a ring around the plan of this shape" from several options.
Scale drawings: "Draw the front elevation to a scale of 1 cm to 2 m" or "How tall is the real object?" Use a ruler and label lengths.

Practice

Remember

Euler's formula: F + V − E = 2 for any polyhedron (flat faces, no holes). Rearrange it to find a missing value: E = F + V − 2.
Prism with an n-sided end: F = n + 2, V = 2n, E = 3n. Pyramid with an n-sided base: F = V = n + 1, E = 2n.
If F + V − E is not 2, the shape is impossible (or counted wrongly).
Plan = from above; front elevation = from the front; side elevation = from the side. Each is a flat 2D drawing.
Draw lines only where there is an edge (height or depth changes), and use the same scale for every length.

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