Two boards, two views (the dihedral system)
One picture of a solid hides its depth. So we use two flat boards that meet at a right angle. The floor is the horizontal plane (HP). The wall is the vertical plane (VP). They meet at the XY line. This set-up is called the dihedral system.
- From the front, the solid makes the front view on VP. It shows height and width.
- From above, it makes the top view on HP. It shows width and depth.
- A projector is a thin line from a corner that meets a board at a right angle.
Folding the floor: rabatment
Paper is flat, so we cannot draw on two boards at right angles. We fold the floor down about the XY line until it lies in the same plane as the wall. This turning of a plane into another plane is called rabatment.
After folding, the top view sits below the front view. A corner shows in both views, and its two points lie on one upright line. This is the main rule of drawing: the front and top view of a point are joined by a vertical line.
True length and true size
A line is seen in its true length only on a board that it is parallel to. If it is slanted to the board, the view is shorter (foreshortened). For an axis of length L that makes an angle θ with HP and is parallel to VP:
- front view length = L (true)
- top view length = L cos θ (shorter)
In the same way a flat shape shows its true size only on a board it is parallel to. For a line, if you know its top-view length d and the difference in height h of its ends, the true length is √(d² + h²).
Rotations and plane changes
To see a solid in a tilted position you have two choices. They give the same drawing.
- Rotate the solid: keep the boards fixed and move the solid. First draw it in a simple position. Then redraw one view tilted by the given angle (same shape, new position). Project the other view from it. Tilt first, then turn about the vertical line, so the second move keeps the first angle.
- Change the plane: keep the solid fixed and put a new board (an auxiliary plane) parallel to the part you want to see in true shape. Project onto it with new projectors.
The solid never changes. Only the views change.
Prisms, pyramids and sections
A prism has two equal flat ends joined by rectangles. A pyramid has one flat base and sloping triangles that meet at the apex. Start with the base lying on HP and the axis upright. The top view then shows the true shape of the base: a square prism gives a square. A square pyramid gives a square crossed by two diagonals, which are the four sloping edges seen from above.
- Edges you cannot see from the viewing side are drawn dashed (hidden lines).
- When a flat knife cuts the solid, the cut face is a section. Its true shape needs a plane parallel to the knife. See the lesson Sections of solids.
Cylinders, cones and regular polyhedra
With the axis upright, a cylinder gives a circle in the top view and a rectangle in the front view. A cone gives a circle with a dot at the centre (the apex) in the top view and a triangle in the front view. Tilt them and the circles become ellipses, joined by two straight lines that just touch them.
Regular polyhedra (the cube, tetrahedron, octahedron and others) are built from equal regular faces. Draw one face or one edge first, find the height of the remaining corners, then project. For a cube with a body diagonal upright, the top view is a regular hexagon.
Key formulas and definitions
- True length L; angle θ with HP and parallel to VP: front view = L, top view = L cos θ
- True length from a top view d and end-height difference h: L = √(d² + h²)
- Front view: on VP, shows height and width. Top view: on HP, shows width and depth
- Fold the floor down about XY: top view lies below front view, joined by upright lines
Worked examples
1. An axis of length 8 cm is parallel to VP and makes 30° with HP. Find its front-view and top-view lengths.
It is parallel to VP, so the front view is true: 8 cm. Top view = 8 × cos 30° = 8 × 0.866 = 6.93 cm.
2. A square prism stands with its axis upright. What does its top view look like? And its front view?
The top view is a square, the true shape of the base. The front view is a rectangle, with width equal to the side of the square and height equal to the length of the axis.
3. A line has a top view 6 cm long. Its two ends are at heights 2 cm and 10 cm above HP. Find its true length.
Height difference h = 10 − 2 = 8 cm. L = √(6² + 8²) = √100 = 10 cm.
4. A cone with its axis upright has base diameter 4 cm and height 5 cm. Describe its two views.
Top view: a circle of diameter 4 cm with a dot at the centre (the apex). Front view: an isosceles triangle with base 4 cm and height 5 cm.
5. An axis 10 cm long is parallel to VP and its top view is 6 cm. Find the angle it makes with HP.
Top view = L cos θ, so cos θ = 6 / 10 = 0.6. θ = cos⁻¹ 0.6 ≈ 53.1°.
6. A square pyramid (base 4 cm, base on HP) is tilted so one base edge rests on HP and the base makes 30° with HP. Give the order of steps to draw it.
1) Draw the top and front views with the base flat on HP. 2) Redraw the front view with the base tilted 30° about the resting edge, same size and shape. 3) Project upward from this front view and across from the first top view to get the new top view. 4) Join corners; draw hidden edges dashed.
Common mistakes
- Placing the top view above the front view. After folding the floor down, the top view is below.
- Thinking a tilted solid changes shape. Only its views change; the solid is the same.
- Reading a length from a view where the line is slanted. It is shorter than the real length.
- Drawing every edge as a full line. Edges hidden behind the solid are dashed.