Plans and Elevations

A plan or an elevation is a flat, 2D drawing of a 3D solid, such as a cuboid, a prism or a shape made from cubes. Each one shows what you see when you look at the solid from one direction.

In the exam you may be asked to draw the plan, the front elevation or the side elevation of a solid on a centimetre grid, for 2 or 3 marks. Other questions give you the views and ask you to sketch the solid, or to work out a length, a volume or a number of cubes from them.

On a centimetre grid, one square length is 1 cm. Count squares to read a length off a view, and use the same number of squares when you draw that length.

• The plan is the view from directly above.
• The front elevation is the view from the front.
• The side elevation is the view from the side.

Draw each view flat, as if you were looking straight at the solid from far away. A view shows no depth, so it never looks 3D. The lengths you draw are the real widths, depths and heights of the solid, unless the question gives a scale.

Each view of a cuboid is a rectangle that shows two of the cuboid's three measurements. The depth is how far back the cuboid goes from the front.

• Plan: the length and the depth.
• Front elevation: the length and the height.
• Side elevation: the depth and the height.

When a solid is made from centimetre cubes, each view is made of 1 cm squares. Draw one square for each place where you can see at least one cube from that direction.

Cubes in a line, one behind another, show as just one square. So a view usually has fewer squares than the solid has cubes.

In a 3D drawing some cubes are hidden. A cube on top must have a cube under it, because cubes cannot float, so count the hidden ones too.

A useful way to describe a solid is to write, in each square of the plan, how many cubes are stacked on it. The total of the numbers is the number of cubes. Look along each column of the plan, from front to back. The biggest number is how tall the front elevation is there.

A view is more than an outline. Where two faces of the solid meet at an edge you can see, draw a solid line inside the view. For example, the plan of a step has a line where the low top ends and the high top starts.

An edge that is there but hidden behind the solid from that direction is drawn as a dashed line. Use dashes only for hidden edges.

A sloping face looks shorter in a view. Look straight at a ramp whose sloping face is 13 cm long but rises only 5 cm, and the slope looks just 5 cm high. A height in a view is always the vertical height, never a sloping length.

To work out the solid from its views, look at the shapes first. A circle for the plan and rectangles for both elevations make a cylinder. A triangle for the front elevation, with rectangles for the plan and the side elevation, makes a triangular prism.

Then match up the lengths. The plan and the front elevation have the same length, the front and side elevations have the same height, and the plan and the side elevation have the same depth.

Draw a 3D sketch and write the measurements on it. The sketch does not need to be accurate, but a sketch with no measurements loses marks.

Each view shows two measurements of a solid, so two views together usually give all three. When you are given only one view, the question gives you something else, such as the volume.

For a cuboid:

V=l\times w\times h

V is the volume, l the length, w the width (or depth) and h the height.

If you know the volume and two of the lengths, multiply the two lengths, then divide the volume by your answer. A cuboid with volume 70\text{ cm}^3, length 7 cm and depth 2 cm has height 70\div14=5 cm.

For a cuboid, or a cylinder standing on one end, the area of the plan times the height gives the volume. Never multiply the area of the plan by the area of an elevation: area times area is not a volume.

• Mixing up the plan and the side elevation. The plan is the view from above, and the side elevation is the view from the side, so it shows the height.
• Using a sloping length as a height. A height in a view is the vertical height, not the length of a sloping edge.
• Drawing a 3D picture instead of a view. Each view is a flat 2D shape, drawn on its own.
• Leaving out the lines inside a view. Draw a solid line for each edge you can see and a dashed line for each hidden edge.
• Sketching a solid with no measurements. Write the lengths from the views on your sketch.
• Multiplying the area of the plan by the area of an elevation. For the volume of a cuboid or an upright cylinder, multiply the area of the plan by the height.