HP, VP and the two views
Imagine two flat screens that meet at a line. HP is the horizontal screen (the floor). VP is the vertical screen (the wall). In first-angle projection the solid sits above HP and in front of VP.
- Front view (FV): what you see looking at VP from the front. It is drawn on VP. It shows height and width.
- Top view (TV): what you see looking down on HP. It shows width and depth.
A right regular solid has a regular shape as its base (square, hexagon, circle) and its axis (the line joining the centre of the base to the top or apex) is at right angles to the base.
Axis perpendicular to HP
The solid stands on its base. The base is parallel to HP, so the top view shows the base in true shape.
- Cube or prism: TV is the base shape (a square, hexagon...). FV is a rectangle.
- Pyramid: TV is the base with lines from the corners to the centre (the apex). FV is a triangle.
- Cone: TV is a circle with a dot at the centre. FV is a triangle.
- Cylinder: TV is a circle. FV is a rectangle.
- Sphere: both views are circles of the same size.
This is the easiest position, and the first step of every harder problem.
Axis perpendicular to VP, or parallel to both
Axis perpendicular to VP
Now the base is parallel to VP, so the front view shows the base in true shape and the top view is the 'side' shape. A cone pointing at the wall has a circle as FV and a triangle as TV.
Axis parallel to HP and to VP
The solid lies with its axis running left to right. A cylinder gives two rectangles. A cone gives two triangles. A square pyramid gives two triangles.
Rule: where the axis is perpendicular to a plane, the view on that plane is the true base. Where the axis is parallel to a plane, the view on that plane is a side shape (a triangle or rectangle).
Frustums and hemispheres
A frustum is a cone or pyramid with its top cut off by a plane parallel to the base. Its views are those of the full solid with the top removed: FV is a trapezium (axis upright), TV is two concentric circles (frustum of a cone) or two nested squares joined at the corners (frustum of a square pyramid).
A hemisphere is half a sphere. With its flat face down, TV is a circle and FV is a semicircle. With its flat face towards VP, FV is a circle and TV is a semicircle.
Axis inclined to one plane: the tilting method
When the axis is tilted at an angle φ to HP (and still parallel to VP), use two stages.
- Stage 1: draw the solid in the simple position (axis perpendicular to HP). Draw FV and TV.
- Stage 2: redraw the FV tilted by φ. Its shape is unchanged. Then draw the new TV by projecting upwards from the tilted FV and across from the first TV (the points keep their distance from VP).
Because the axis is parallel to VP, the FV shows the true length of the axis and the true angle φ. In the TV the axis looks shorter: top-view length = L cos φ. If the axis is instead inclined to VP, swap the roles of the views.
Key formulas and definitions
- True length of axis = L (seen in the view where the axis is parallel to the plane)
- Top-view length of an axis inclined at φ to HP = L cos φ
- Front-view length of an axis inclined at θ to VP = L cos θ
- Axis perpendicular to a plane → view on that plane is the true base
- Key term: frustum = solid with its top cut off by a plane parallel to the base
Worked examples
1. Name the front view and top view of a cylinder standing on its base (axis perpendicular to HP).
The base is parallel to HP, so the top view is a circle. The side is straight, so the front view is a rectangle of width equal to the diameter and height equal to the axis.
2. A square pyramid has base side 40 mm and axis 60 mm. Its axis is perpendicular to HP. Describe both views.
TV: a square of 40 mm with lines from each corner to the centre. FV: a triangle with base 40 mm and height 60 mm, apex above the middle of the base.
3. A cone with axis 70 mm lies with its axis parallel to HP and perpendicular to VP, apex towards VP. What are the views?
Axis perpendicular to VP means FV shows the base in true shape: a circle. TV shows the side: a triangle with height equal to the axis, 70 mm, pointing towards VP.
4. A pyramid with axis L = 60 mm is tilted so its axis makes 30° with HP and is parallel to VP. How long is the axis in the top view?
The axis is parallel to VP, so the front view shows 60 mm at 30°. Top-view length = L cos φ = 60 × cos 30° = 60 × 0.866 = about 52 mm.
5. A cylinder (diameter 40 mm, axis 70 mm) has its axis inclined at 45° to HP and parallel to VP. Give the steps to draw both views.
Stage 1: draw the upright cylinder: TV circle of 40 mm, FV rectangle 40 × 70. Stage 2: redraw the FV rectangle tilted so its axis makes 45° with the ground line. Project the TV upward from the tilted FV and across from the first TV. The TV is an elongated shape made of the two end circles (now ellipses) joined by tangent lines.
6. A cone has base diameter 50 mm and axis 60 mm. Its axis is inclined at 60° to HP, parallel to VP. What is the angle shown in the front view, and how long is the axis in the top view?
The front view shows the true angle, 60°, and the true axis length, 60 mm. Top-view axis = 60 × cos 60° = 30 mm.
Common mistakes
- Drawing the tilted solid with the wrong shape. In Stage 2 the front view keeps its exact shape; only its position turns.
- Mixing up which view is true. The axis is true length only in the view where it is parallel to that plane.
- Forgetting the lines from the corners to the apex in the top view of a pyramid.
- Lowering the solid below HP or behind VP. In first angle the solid stays above HP and in front of VP.