Geometry and measures · KS3

3D Shapes, Plans and Elevations

A cereal box, a tin of beans, a Toblerone bar, a football: all solid shapes you can pick up. This page shows how to describe them (faces, edges and vertices), give them their proper names, unfold them into flat nets, work out how much card covers them, draw them on isometric paper, and read the flat “views” that architects and engineers use to describe them.

National curriculum KS3: use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D; calculate surface area of cuboids and other right prisms; draw and interpret 3-D shapes, nets, plans and elevations

1 · Flat or solid? 2D and 3D

A 2D shape (two-dimensional) is flat. It has a length and a width and nothing else, like a square drawn on paper or the shadow of your hand. You can draw it, but you can't pick it up.

A 3D shape (three-dimensional) has a third measurement as well: depth (you might also call it thickness or height). That's what lets you hold it. A 3D shape is often called a solid, even if it's hollow inside like a cereal box.

The quickest way to feel the difference: take a flat shape and pull it out of the page. Slide the depth up from 0. Drag the picture to turn it round.

flat shape depth

Notice two things. A square pulled out to the same depth as its side becomes a cube. And pulling a shape straight out always gives a solid that looks the same all the way through, like a stick of rock. Hold on to that idea: it's exactly what a prism is (section 3).

2 · Faces, edges and vertices

To describe a 3D shape we count three things. Think of a dice:

A face is a flat surface. It's a side the shape could rest on if you put it on the table. A dice has 6 faces.
An edge is a line where two faces meet. Run your finger along it. A dice has 12 edges.
A vertex is a corner, where edges meet. The plural is vertices. A dice has 8 vertices.

Some shapes also have a curved surface, like the side of a tin. Pick a shape, choose what to highlight, and drag to spin it. Dashed lines are edges hidden round the back.

shape

show hidden edges

Faces

Edges

Vertices

Common mistake: counting only what you can see. From any one direction you can see at most 3 faces of a cube, but it has 6. Look at the “facing you” numbers above: they're always smaller than the totals. Count the back too: spin the shape, or count the dashed hidden edges.

A pattern to discover

Every shape you pick above with only flat faces is added to this table. These shapes are called polyhedra (one is a polyhedron): solids with flat faces and straight edges. Try at least four, then press the button and look at the last column.

This is Euler's formula (say “oiler”): for every polyhedron without holes, F + V − E = 2. It's a brilliant way to check your counting. If you count a pentagonal prism as 7 faces, 10 vertices and 14 edges, then 7 + 10 − 14 = 3, not 2, so you've missed an edge somewhere. (It's 15.)

3 · Naming 3D shapes: prisms, pyramids and curved solids

Imagine slicing straight through a shape, like cutting a cucumber or a loaf of bread. The flat shape you see on the cut is called a cross-section.

A prism is a solid with the same cross-section all the way along (a constant cross-section). It has two identical end faces, joined by rectangles. A prism is named after its cross-section: a Toblerone is a triangular prism, an unsharpened pencil is a hexagonal prism. A cuboid is a rectangular prism, and a cube is a square prism.

A pyramid has a flat base and triangles that all meet at one point at the top, the apex. A pyramid is named after its base: the ones in Egypt are square-based pyramids. Slide the knife through each shape and watch the cross-section.

shape slice position

Name (example)	F	E	V
Cube dice, sugar cube	6	12	8
Cuboid cereal box, brick	6	12	8
Triangular prism Toblerone, tent	5	9	6
Hexagonal prism pencil, some nuts and bolts	8	18	12
Square-based pyramid Egyptian pyramids	5	8	5
Triangular-based pyramid (tetrahedron) some tea bags, a four-sided dice	4	6	4
Cylinder tin of beans, candle	2 flat + 1 curved	2	0
Cone ice-cream cone, traffic cone	1 flat + 1 curved	1	1
Sphere football, marble	1 curved	0	0

F = faces, E = edges, V = vertices. Shortcut for prisms: if the cross-section has n sides, the prism has n + 2 faces, 3n edges and 2n vertices. For a pyramid with an n-sided base: n + 1 faces, 2n edges, n + 1 vertices. The edges of curved shapes are where a curved surface meets a flat one, so a cylinder's 2 edges are the two circles.

4 · Nets: unfolding a shape

Carefully open out a cereal box along its edges and lay it flat. The flat shape you get is called a net: a 2D pattern that folds up to make a 3D shape. Every face of the solid appears exactly once in the net. Slide to fold each net, and drag to look at it from different sides.

net of a flat folded

Look at what each net is made of. The prism net has two copies of its cross-section (the triangles) plus one rectangle for each side of the triangle. The pyramid nets have the base plus one triangle for each side of the base. That's a quick way to sketch a net for any prism or pyramid.

Will this net make a cube?

A cube needs 6 squares, but not every arrangement of 6 joined squares folds into a cube. There are exactly 11 different cube nets (not counting turning or flipping one over). Picture the folding in your head, then decide. After you answer, use the slider to fold it and see for yourself.

the net

fold it:

Two quick tests for cube nets. A net with a 2 × 2 block of squares can never be a cube net (four squares would meet at one corner, but a cube's corners only have three faces). And a net with 5 or more squares in a row can't work either: the 5th square wraps round and lands on top of the 1st.

5 · Surface area from a net

The surface area of a 3D shape is the total area of all its faces: how much card or paper would cover it exactly. Because the net is all the faces laid flat, surface area = area of the net. Area is measured in square units, such as cm².

For a cuboid the faces come in 3 matching pairs: top and bottom, front and back, left and right. Work out one of each pair, double, and add.

length width height

Common mistake: surface area vs volume. Surface area is the outside (wrapping paper, in cm²). Volume is the space inside (how much it holds, in cm³). For a 4 cm × 2 cm × 3 cm box the surface area is 52 cm² but the volume is 24 cm³. Different questions, different units.

Another one: counting each face only once. A cuboid has 6 faces, so your sum should have 6 areas in it (3 pairs).

6 · Drawing 3D shapes on isometric paper

Drawing a solid on squared paper is awkward. Isometric paper has dots (or lines) arranged in a pattern of triangles, which makes 3D drawings easy and accurate. Rules:

Hold the paper so the dots make vertical columns.
Vertical edges stay vertical.
Edges that go left–right or front–back are drawn sloping at 30°, along the dots, never horizontal.
One unit of length is the gap between two neighbouring dots, in any of these directions.

Step through the drawing of a cuboid.

length width height

step show hidden edges (dashed)

Common mistakes on isometric paper: drawing an edge horizontal (no edge is ever horizontal on isometric paper), turning the paper sideways so the dots make rows instead of columns, and drawing the hidden edges (on an isometric drawing you only draw what you can see).

7 · Plans and elevations

Architects and engineers can't hand a builder a 3D model, so they describe a shape with flat 2D views:

The plan is the view from directly above, like a bird looking down. It shows the “footprint”.
The front elevation is the view looking straight at the front.
The side elevation is the view looking straight at the side.

Each view is flat: you can't see depth in it at all. Lines inside a view show where there's a step, an edge between parts that are nearer and further away. Build a shape below: tap a square of the floor (or a cube) to add a cube there, switch to “Remove” to take the top cube off. The three views redraw as you build. The arrows show which way you look for the front (F) and the side (S).

show heights on the plan

Common mistakes with plans and elevations.

Mixing up front and side: check which way the arrow points, and imagine your eye at the end of it. The front elevation shows the faces coloured orange, the side elevation the plum ones, the plan the teal tops.

Trying to show hidden cubes: a view only shows what you'd see. A short column in front of a tall one doesn't make the view taller or add squares. It just adds a line where the step is.

Drawing a view back to front: the left of the front elevation is the left as you look along the F arrow.

Practice: match the view to the shape

8 · Building a shape from its plan and elevations

Now the reverse: you're given the three views and have to build the shape. A good method:

The plan tells you which floor squares have at least one cube.
The front elevation tells you the tallest column in each left-to-right position.
The side elevation tells you the tallest column in each front-to-back row.
Build each square up to the smaller of its two heights, then use the inner lines in the views to fix any steps.

Build the mystery shape. Your three views are checked against the targets as you go.

Target views:

Your views:

Worked examples

Count the faces, edges and vertices of a pentagonal prism, and check with Euler's formula

Faces: 2 pentagon ends + 1 rectangle for each of the 5 sides of the pentagon = 2 + 5 = 7.

Edges: 5 round the front pentagon + 5 round the back pentagon + 5 running along the length joining them = 15.

Vertices: 5 corners at the front + 5 at the back = 10.

Check: F + V − E = 7 + 10 − 15 = 2 ✓. The shortcut agrees too: n = 5 gives n + 2 = 7, 3n = 15, 2n = 10.

Find the surface area of a cuboid 6 cm long, 4 cm wide and 3 cm high

Top and bottom: each 6 × 4 = 24 cm², so 2 × 24 = 48 cm².

Front and back: each 6 × 3 = 18 cm², so 2 × 18 = 36 cm².

The two ends: each 4 × 3 = 12 cm², so 2 × 12 = 24 cm².

Total: 48 + 36 + 24 = 108 cm². (Its volume is 6 × 4 × 3 = 72 cm³: a different measurement, in different units.)

Find the surface area of a triangular prism. The ends are right-angled triangles with sides 3 cm, 4 cm and 5 cm; the prism is 10 cm long.

Sketch the net: 2 triangles + 3 rectangles, one for each side of the triangle.

Triangles: the 3 cm and 4 cm sides are at right angles, so each triangle is ½ × 3 × 4 = 6 cm². Two of them: 12 cm².

Rectangles: 3 × 10 = 30 cm², 4 × 10 = 40 cm², 5 × 10 = 50 cm². Together: 120 cm².

Total: 12 + 120 = 132 cm². Check: 5 faces, 5 areas in the sum ✓.

Draw the plan, front elevation and side elevation of this cube stack

Read the shape: the front row has columns of 2, 1 and 1 cubes from left to right. Behind the left-hand column there's a column of 3.

Plan (from above): 4 squares in an L: three along the front, one behind on the left. There's a line between the front-left square and the one behind it, because their tops are at different heights (2 and 3). The front-left top is also higher than the one next to it (2 vs 1), so there's a line there too.

Front elevation (along F): the left column looks 3 tall (the back column shows above the front one), the other two are 1 tall. There's a line at height 2 in the left column, where the front column ends and the back one carries on.

Side elevation (along S, from the left): the back row is on your left and is 3 tall; the front row is on your right and is 2 tall. No inner lines: everything you see is the same flat side.

Build the shape: the plan is a 2 × 2 square, the front elevation has heights 2 then 1, the side elevation has heights 1 then 2

Plan: all 4 floor squares have at least one cube.

Front elevation: the left column is 2 tall, the right column is 1 tall. So both right-hand squares (front and back) have exactly 1 cube, since they can't be taller than 1 and the plan says they're not empty.

Side elevation (from the left): the back row is on the left and is 1 tall, the front row is on the right and is 2 tall. So the back-left square has 1 cube, and somewhere in the front row there's a column of 2. The front-right square is only 1, so the front-left square has 2.

Answer: front-left 2, everything else 1: 5 cubes. Check all three views ✓.

Quick check

What you've learned

2D shapes are flat (length and width); 3D shapes also have depth, so you can hold them.
A face is a flat surface, an edge is where two faces meet, a vertex is a corner. Count the hidden ones too.
For any polyhedron, F + V − E = 2 (Euler's formula). Use it to check your counts.
A prism has the same cross-section all the way along and is named after it; a pyramid has a base and triangles meeting at an apex and is named after its base. Cylinders, cones and spheres have curved surfaces.
A net is a flat pattern that folds into a solid. There are 11 cube nets; a 2 × 2 block or 5 in a row means it's not one.
Surface area = total area of all the faces = area of the net, in cm². It is not the volume (cm³).
On isometric paper, vertical edges stay vertical and the others slope at 30°; hidden edges aren't drawn.
The plan is the view from above, the front and side elevations are views from the front and side. Lines inside a view show steps. You can rebuild a shape from its three views.