Plans and Elevations

Think of the drawing an architect makes of a house as seen from above.

The plan is the view from directly above. Which two lengths can you see from above?

Imagine looking straight down on the cylinder from above.

Imagine standing where the arrow is and looking straight at the solid. How many cubes wide and how many cubes tall does it look?

The front elevation shows the length and the height. The plan shows the length and the depth.

Imagine standing at the Side arrow. Which two lengths of the cuboid can you see from there?

1. From the side you see the end face of the cuboid.
2. Its width is the depth of the cuboid, 4 cm, and its height is 3 cm.
3. So the side elevation is a rectangle 4 cm wide and 3 cm high. (7 cm by 3 cm is the front elevation.)

The front elevation is what you see from the Front arrow. The plan is what you see looking straight down.

Part (a)

1. From the front you see the triangular end of the prism.
2. The front elevation is a triangle.

Part (b)

1. From above you see the two sloping faces, side by side.
2. Together they make a rectangle, with a line down the middle where the faces meet. The plan is a rectangle.

Look down on the solid from above. How many cubes touch the table, and where is the cube that is not in the front row?

1. From above you see four squares: the three cubes in the front row, and one cube behind the right-hand end.
2. The cube on top of the middle cube is directly above another cube, so it adds no new square to the plan.
3. That is diagram B. Diagram C is the front elevation, and diagram A has the back cube on the wrong side.

From the side, a cube hides any cube directly behind it in the same row. Count how tall the solid looks at the front and at the back.

1. From the side you see two columns of squares: one for the front row of cubes and one for the back row.
2. The front row is 1 cube tall. The back row is 3 cubes tall at its tallest.
3. The side elevation has 1+3=4 squares.

From above you see the whole base, and the four sloping edges meeting at the point. From the front you see one sloping face.

Part (a)

1. From above, the outline is the square base.
2. The four sloping edges run from the corners to the point in the middle, so you see the two diagonals of the square. That is diagram C.

Part (b)

1. From the front you see the square base edge along the bottom and the point at the top.
2. The front elevation is a triangle.

For the front elevation, each column of the plan shows only its tallest stack, because the taller stack hides the shorter one.

Part (a)

1. Add the numbers: 1+3+2+2+1+1=10 cubes.

Part (b)

1. From the front, each column shows its tallest stack.
2. Left column: the taller of 1 and 2 is 2. Middle column: the taller of 3 and 1 is 3. Right column: the taller of 2 and 1 is 2.
3. The front elevation has 2+3+2=7 squares.

From above you see both sloping faces. Where do they meet, and which way does that edge run?

1. Looking down, you see the two sloping faces, one at the front and one at the back.
2. They meet along the top edge, which runs from the left-hand end to the right-hand end, halfway back.
3. So the plan is a rectangle with a line across the middle from left to right. That is diagram A.

The front elevation gives the length and the height. Volume = length \times depth \times height.

Part (a)

1. From the grid, the front elevation is 5 cm long and 3 cm high.
2. Area of the front =5\times3=15\text{ cm}^2.
3. Depth =60\div15=4 cm.

Part (b)

1. The side elevation shows the depth and the height: a rectangle 4 cm by 3 cm.
2. Area =4\times3=12\text{ cm}^2.

From the front, how high does the doorstop reach? A sloping length looks shorter when you view it from straight in front.

Part (a)

1. The sloping face rises to the top of the 3 cm vertical edge.
2. Seen from the front, the doorstop is only 3 cm high. Tom used the sloping length, 5 cm, as the height.

Part (b)

1. The correct front elevation is a rectangle 6 cm wide and 3 cm high.
2. Area =6\times3=18\text{ cm}^2.

The plan tells you the shape of the base. The elevations tell you what is on top.

1. The plan is a circle with a point in the middle, so the solid is round and comes to a point at the centre.
2. Both elevations are a rectangle with a triangle on top.
3. So the solid is a cone on top of a cylinder. (A pyramid on a cube would have a square plan with diagonals.)

Volume of a prism = area of the cross-section \times length.

Part (a)

1. From the grid, the triangle has a base of 6 cm and a height of 4 cm.
2. Area of the cross-section =\frac{1}{2}\times6\times4=12\text{ cm}^2.
3. Length =96\div12=8 cm.

Part (b)

1. From above you see the full 6 cm width of the triangle and the length of the prism, 8 cm.
2. Area of the plan =6\times8=48\text{ cm}^2.

Every 1 m on the shed is 2 cm on the drawing. From the side, the shed looks as tall as the top of its roof.

Part (a)

1. The shed is 3 m tall to the top of the roof.
2. On the drawing: 3\times2=6 cm.

Part (b)

1. From the side you see the 4 m length of the shed, and the roof rising to 3 m above the ground.
2. On the drawing the rectangle is 4\times2=8 cm wide and 3\times2=6 cm high.
3. Area =8\times6=48\text{ cm}^2.

Look down on the solid for the plan. For the front elevation, the cube gives a square and the pyramid a triangle on top of it.

Part (a)

1. The sloping edges meet at the point of the pyramid, which is directly above the centre of the square.
2. The centre of the square is halfway between (1, 1) and (5, 5): the point (3, 3).

Part (b)

1. The front elevation is a 4 cm square with a triangle on top. The triangle has base 4 cm and height 3 cm.
2. Area =4\times4+\frac{1}{2}\times4\times3=16+6=22\text{ cm}^2.

The surface is made of the two L-shaped ends and the rectangles around the sides. Each rectangle is as long as the prism.

1. From the front elevation, the L shape is a 4 cm by 1 cm rectangle with a 1 cm by 2 cm rectangle on top. Its area is 4+2=6\text{ cm}^2.
2. From the side elevation, the prism is 5 cm long.
3. Going round the L shape, its perimeter is 4+1+3+2+1+3=14 cm, so the rectangles around the sides have a total area of 14\times5=70\text{ cm}^2.
4. Total surface area =2\times6+70=82\text{ cm}^2.

Use the plan to see where the stacks are. Then use the side elevation to decide how tall the front row and the back row can be.

1. The plan shows 6 stacks: a front row of 3 and a back row of 3.
2. The side elevation shows the front row is only 1 cube high, so each stack in the front row has 1 cube.
3. The front elevation shows the tallest stacks from left to right are 3, 2 and 1. These must be the back row, since the front row is only 1 high.
4. Number of cubes =1+1+1+3+2+1=9.