A 3D solid can be described completely by three flat drawings, each showing what you see when you look at it from a particular direction.
Front elevation — the view from the front
Side elevation — the view from the side
- Decide which face is the "front" — the question will usually mark it with an arrow.
- For the plan, imagine hovering above and looking straight down. Draw the outline you see.
- For the front elevation, look horizontally at the front face. Draw its outline at true height and width.
- For the side elevation, look horizontally from the side. Its width is the depth of the solid.
- Add lines wherever a step or a change of level would be visible.
- Use squared paper and keep every view to the same scale.
A cuboid is $6$ cm wide, $4$ cm deep and $3$ cm tall. Describe its three views.
Check: plan and front share the width $6$; front and side share the height $3$; plan and side share the depth $4$ ✓
A cylinder of radius $3$ cm and height $8$ cm stands on its circular base. Describe its three views.
A pyramid has a square base of side $10$ cm and vertical height $12$ cm. Describe its three views.
The most common exam question uses a shape built from unit cubes. The trick is to work out, for each column of the view, the maximum height or depth you can see.
• Plan — for each square of the base, mark it if there is any cube above it.
• Front elevation — for each column across, draw as tall as the tallest stack in that column.
• Side elevation — for each column of depth, draw as tall as the tallest stack at that depth.
A shape is built from unit cubes: a bottom row of $3$ cubes, with $2$ cubes stacked on the leftmost, and $1$ extra on the middle one. Describe the three elevations.
Harder questions give you the three views and ask you to identify or sketch the solid.
- Read the plan first — it tells you the "footprint" of the solid.
- Read the front elevation for the height and any steps across the width.
- Read the side elevation for the height and any steps through the depth.
- Combine: the solid must fit inside a box with those three dimensions.
- Sketch on isometric paper if you have it.
| Plan | Front | Side | Solid |
|---|---|---|---|
| Circle | Rectangle | Rectangle | Cylinder (standing up) |
| Circle | Circle | Circle | Sphere |
| Circle | Triangle | Triangle | Cone |
| Square | Square | Square | Cube |
| Square with diagonals | Triangle | Triangle | Square-based pyramid |
| Rectangle | Triangle | Rectangle | Triangular prism (lying down) |
The plan of a solid is a circle of diameter $8$ cm. The front elevation is a triangle of base $8$ cm and height $10$ cm. The side elevation is the same triangle. Name the solid and find its volume.
Plan
The view from directly above.
Front elevation
The view from the front, at true height and width.
Side elevation
The view from the side; its width is the depth.
Matching check
Plan and front share width; front and side share height; plan and side share depth.
Cube stacks
Each column of an elevation is as tall as the tallest stack behind it.
Cylinder
Circle plan; rectangle for both elevations.
Cone
Circle plan; triangle for both elevations.
Pyramid
Square plan with diagonals; triangular elevations.
No perspective
Draw flat outlines only — no shading, no vanishing points.
Describe the plan, front elevation and side elevation of a cube of side $5$ cm.
▶ Show solution
All three views are identical: a $5$ cm by $5$ cm square.
A cube looks the same from every direction perpendicular to a face.
A cuboid measures $10$ cm wide, $3$ cm deep and $7$ cm tall. Give the dimensions of each of the three views.
▶ Show solution
Plan: $10$ cm by $3$ cm rectangle.
Front elevation: $10$ cm by $7$ cm rectangle.
Side elevation: $3$ cm by $7$ cm rectangle.
The plan, front elevation and side elevation of a solid are all circles of the same size. Name the solid.
▶ Show solution
A sphere — the only solid that appears as an identical circle from every direction.
A cylinder of radius $5$ cm and length $12$ cm is lying on its side, with the circular ends facing left and right. Describe the three views.
▶ Show solution
Plan (from above): a rectangle $12$ cm by $10$ cm.
Front elevation: a rectangle $12$ cm wide by $10$ cm tall.
Side elevation: a circle of diameter $10$ cm — you are looking straight at a circular end.
A solid is built from $4$ unit cubes in a straight row. Describe the three views.
▶ Show solution
Plan: a $4 \times 1$ rectangle.
Front elevation: a $4 \times 1$ rectangle (four squares across, one high).
Side elevation: a single $1 \times 1$ square.
A solid has a square plan with both diagonals drawn, and both elevations are isosceles triangles. Name the solid.
▶ Show solution
A square-based pyramid.
The diagonals on the plan are the four sloping edges running up to the apex, which sits directly above the centre of the square.
A triangular prism has an equilateral triangular cross-section of side $6$ cm and length $15$ cm, lying with a rectangular face on the table and the triangles facing front and back. Describe the plan and both elevations.
▶ Show solution
Front elevation: an equilateral triangle of side $6$ cm (you are looking straight at a triangular end).
Side elevation: a rectangle $15$ cm long by the triangle's height, $\sqrt{6^2 - 3^2} = \sqrt{27} = 5.20$ cm.
Plan: a rectangle $6$ cm by $15$ cm, with a line down the middle where the top ridge of the prism is.
The three views of a solid are: plan a $3 \times 2$ rectangle, front elevation a $3 \times 4$ rectangle, side elevation a $2 \times 4$ rectangle. Name the solid and find its volume.
▶ Show solution
All three views are rectangles with consistent dimensions, so the solid is a cuboid.
Width $3$, depth $2$, height $4$.
$V = 3 \times 2 \times 4 = 24$ cubic units.
A shape is made from unit cubes. Looking from the front, the column heights (left to right) are $2$, $3$, $1$. The shape is $1$ cube deep throughout. Sketch-describe all three views and state how many cubes are used.
▶ Show solution
Front elevation: three columns of $2$, $3$ and $1$ squares — a stepped shape.
Plan: the footprint is $3$ squares in a row, so a $3 \times 1$ rectangle.
Side elevation: only $1$ deep, and the tallest column is $3$, so a $1 \times 3$ rectangle.
Number of cubes $= 2 + 3 + 1 = 6$.
A solid consists of a cuboid $8$ cm wide, $6$ cm deep and $4$ cm tall, with a cube of side $2$ cm sitting on top at the centre of the cuboid's top face.
(a) Describe the plan. (b) Describe the front elevation. (c) Find the total volume. (d) Find the total surface area.
▶ Show solution
(a) An $8$ cm by $6$ cm rectangle, with a $2$ cm by $2$ cm square drawn in the middle of it (the outline of the cube seen from above).
(b) An $8$ cm by $4$ cm rectangle with a $2$ cm by $2$ cm square sitting centrally on top — a stepped shape $6$ cm tall overall.
(c) Cuboid: $8 \times 6 \times 4 = 192\text{ cm}^3$. Cube: $2^3 = 8\text{ cm}^3$.
Total $= 192 + 8 = \mathbf{200\text{ cm}^3}$.
(d) Cuboid surface area $= 2(8\times6) + 2(8\times4) + 2(6\times4) = 96 + 64 + 48 = 208\text{ cm}^2$.
The cube adds its $4$ side faces ($4 \times 4 = 16\text{ cm}^2$) and its top ($4\text{ cm}^2$), but hides $4\text{ cm}^2$ of the cuboid's top face.
Net change $= 16 + 4 - 4 = 16\text{ cm}^2$.
Total $= 208 + 16 = \mathbf{224\text{ cm}^2}$.