Three flat views
A drawing of a solid on flat paper has to leave something out. One way to draw it exactly is to draw it three times, each time looking straight at it from one direction: from directly above, from straight in front, and from the side. Each of these views is flat, so every length in it can be measured at its true size.
This solid is made of four unit cubes: three in a row along the floor, and one more stacked on the cube at the left end.
The solid of four cubes, then its three flat views: from above, from the front, and from the side.
The plan
The view from directly above is called the plan. Looking straight down, you see the top of every column of cubes and nothing of its height: a column one cube tall and a column two cubes tall each show as one square.
This solid has three columns, so its plan is 3 squares in a row, even though the solid has 4 cubes. The stacked cube hides the cube under it.
The plan is 3 squares: one for each column, whatever its height.
The elevations
The view from straight in front is the front elevation. It shows widths and heights, and depth disappears: a cube standing behind another is hidden by it. This solid’s front elevation shows all 4 cubes, the row of 3 along the bottom and the stacked one above the left end, in an L-shape.
The view from the side is the side elevation, here seen from the left. It shows depths and heights. Looking along the row, the three floor cubes stand one behind another and show as a single square, and the stacked cube shows above it: 2 squares, one on the other.
The front elevation shows the row of 3 cubes and the one stacked on its left end: 4 squares in an L.
Each view keeps two lengths
A cuboid 4 cm long, 2 cm wide and 3 cm high has a plan that is a 4 by 2 rectangle, a front elevation that is 4 by 3, and a side elevation that is 2 by 3. Each view loses the one measurement that points straight at you, and keeps the other two at their true size.
So every measurement appears in two views. The length, 4 cm, is in both the plan and the front elevation, which is a check that the views belong to one solid. A light shining straight at a face casts a shadow of the same size and shape as the view from that direction.
Worked example: The Shadow of a Block
Question A rectangular block is 4 cm long, 3 cm wide and 5 cm tall. Light shining straight down casts its shadow on the table; light shining horizontally at its 4 cm by 5 cm face casts a shadow on the wall behind. Find the area of each shadow.
1.From above the light sees the top face, 4 cm by 3 cm: shadow = 12 cm2.
A block 4 by 3 by 5. 2.From the front it sees the 4 cm by 5 cm face: shadow = 20 cm2.
Lit from above, the shadow is the top face: 4 × 3 = 12 cm². 3.The third face, 3 by 5, would be the shadow from the side: 15 cm2, not asked.
Lit from the front, the shadow is the 4 by 5 face: 20 cm².
Answer: 12 cm2; 20 cm2
Common mistakes
- Using the volume or a mix of all three dimensions for a shadow.
- Casting the front-lit shadow with the top face.
One view is never enough
Different solids can share a view. A cube and a cylinder as wide as it is tall have the same front elevation, a square; only the plan tells them apart, since one plan is a square and the other a circle. Every view loses one direction, so one view alone cannot fix a solid.
The three views together usually can. Each view is 3 squares for this stack of 4 cubes, and there is only one way to place the cubes so that all three views come out right.
A stack of 4 cubes whose plan, front elevation and side elevation are 3 squares each. Only this stack fits all three.
Working back from the views
Sometimes the three views allow more than one solid, and the question is how many cubes there can be. The plan says which positions have at least one cube. The front elevation gives the height of the tallest column in each column of positions, from left to right, and the side elevation gives the tallest in each row, from front to back.
A column can be no taller than either of those two limits allows. So the most cubes the solid can have puts the smaller of its two limits at every position. The fewest puts one cube on every position in the plan, then raises as few columns as the elevations need: one column can reach a front limit and a side limit at the same time when the two are equal.
Worked example: Fewest and Most Cubes for Three Views
Question A stack of cubes covers a 3 by 2 grid of positions seen from above. Seen from the front, the three columns of the silhouette are 3, 1 and 2 cubes high. Seen from the right side, the two columns are 3 and 2 high. What is the greatest number of cubes the stack can have? What is the least?
1.Most: at each position the height is the smaller of its front limit and its side limit. Front limits 3, 1, 2 across; side limits 3 (back row) and 2 (front row).
Most: each column as tall as both its views allow. 2.Back row: 3, 1, 2; front row: 2, 1, 2. Most = 3 + 1 + 2 + 2 + 1 + 2 = 11.
Back row 3, 1, 2 and front row 2, 1, 2: 11. 3.Fewest: every position has at least 1 cube (the top view). The front's 3 and the back row's 3 can be one column of 3 at the back-left; the front's 2 and the front row's 2 can be one column of 2 at the front-right.
Fewest: one cube everywhere, plus a 3 at the back-left and a 2 at the front-right. 4.Fewest = 3 + 2 + 1 + 1 + 1 + 1 = 9.
3 + 2 + 1 + 1 + 1 + 1 = 9.
Answer: 11; 9
Common mistakes
- Adding the two views' heights, 6 + 5, as if every view showed different cubes.
- Letting a column exceed one of its views: a 3 in the front row would show as 3 from the side.