Draw solids and read their nets

Read a solid sketch

A drawing fits on a flat page, but it can represent a solid. Sloping lines show edges going away from you. A square face may look like a parallelogram in the sketch. Its appearance on the page does not change the actual face.

A cube has six equal square faces, twelve edges and eight vertices. In this sketch, solid lines show visible edges and broken lines show hidden edges. The hidden edges still belong to the cube.

A cuboid has three matching pairs of rectangular faces. A cube is a special cuboid with all its edge lengths equal. Here the cuboid has different lengths in its three directions.

To draw a cube, begin with a square for the front face. Draw an equal square offset behind it, then join corresponding corners. The four joining edges must go in the same direction. Use broken lines for the edges hidden behind the visible faces. For a cuboid, use a rectangle for the front face and choose a suitable depth. Keep corresponding edges parallel.

Draw one of each on paper. Check the whole sketch against the models: eight corners, twelve joined edges, a matching front and back face, and consistent depth. The page provides a model for comparison; checking your pencil drawing is a separate action.

Distinguish a prism from a pyramid

A prism has two equal, matching polygonal ends in parallel planes. Its other faces join corresponding sides of those ends. The triangular prism shown here has two triangular ends and three rectangular side faces. Turning it onto a different face does not change its name.

A pyramid has one polygonal base. Its triangular side faces meet at one common top vertex. A square-based pyramid has one square face and four triangular faces.

Both of these solids have five faces. The face count alone cannot distinguish them. Look for two matching ends in the prism, or one base and a common top vertex in the pyramid. A cuboid is also a prism; when choosing from cube, cuboid, prism and pyramid, use the most specific name that fits.

Draw a triangular prism by sketching two equal triangles offset in the same direction, then joining their corresponding corners. Check for two matching triangular ends and three joining edges. To draw a square-based pyramid, sketch its base as a sloping four-sided face, place one top vertex above it, and join that vertex to all four base corners. Mark hidden edges with broken lines. Compare both drawings with the models.

Read circular ends in a solid sketch

A cone has one circular base and one pointed vertex. Its other surface curves around the solid. The two sloping outlines in this sketch show the outside of that curved surface; they do not enclose a flat triangular face. The rear part of the circular rim is hidden and is drawn with a broken line.

A cylinder has two equal circular ends in parallel planes and a curved surface joining them. The ends look like ellipses because of the viewing angle. They are circular on the actual solid. A straight outline along each side shows the limit of what you can see; it is not a join between flat side faces.

A pyramid also has a point, but its base is a polygon and its side faces are flat triangles. A prism also has matching ends, but those ends are polygons and its joining faces are flat. These features distinguish the pairs even when their sketches are turned.

Check your understanding 1 Name a solid when the drawing is turned

Name solids A and B. For each, describe the feature that distinguishes it from a prism or pyramid. Do not choose by which way the drawing points.

Your solid names and distinguishing features

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A is a cylinder. It has two matching circular ends joined by a curved surface; a prism would have polygonal ends and flat joining faces. B is a cone. One circular end narrows to a point, even though the point is below the base in this drawing. A pyramid would have a polygonal base and separate flat triangular faces.

Turning the drawing changes its orientation, not the solid. An ellipse in either sketch represents a circle seen at an angle, rather than an oval end on the solid.

The flat-face nets below concern cubes, cuboids, prisms and pyramids.

A net must close without overlaps

A net is a connected flat arrangement of faces that folds into the solid without gaps or overlapping faces. Shared edges in these arrangements are dashed; the outside edges are solid. The numbers identify faces so you can follow where each one goes.

Check your understanding 2 Six squares are only the starting point

Which of arrangements A, B and C forms a cube? Explain where the faces would go as you fold. You can trace a copy or make a paper model.

Your arrangement letters and folding reasons

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A forms a cube. The row of four squares wraps round the four side faces. Faces 5 and 6 close the two remaining faces, each appearing once.

B does not form a cube. Four squares meet at each interior junction, but only three cube faces meet at a cube vertex. Those joins cannot all fold into the required corners. C wraps back round the same side faces and leaves the ends open. All three arrangements have six squares; you also need the right connections.

Match the edges of a cuboid net

A cuboid net needs three matching pairs of rectangles. Neighbouring edges must have equal lengths where they join, and the faces must land in different places when folded. Six arbitrary rectangles are insufficient.

Imagine keeping face 1 still and folding its neighbours up. Follow the strip round the cuboid, then check that faces 5 and 6 fit the remaining openings. A face with the wrong width could leave a gap even when the total face count was correct.

Check your understanding 3 Match the particular solid

A cube is also a cuboid. Would the equal-square arrangement A make a model with the same face proportions as the cuboid pictured below? Explain which feature decides, even if you make the model larger or smaller.

Your decision and reason

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No. Arrangement A has six equal square faces and makes a cube. This pictured cuboid has three different edge lengths, so its faces have different proportions. Enlarging all the squares together still makes a cube. The rectangular-face arrangement above fits this cuboid’s proportions.

Check the order of a prism’s faces

A triangular prism needs both matching triangular ends and three side faces that fit their sides. In the first arrangement below, the triangle’s sides have relative lengths 3, 4 and 5. Starting along its 3-unit side and travelling round the triangle, the next side is 5 units, then 4 units. The side-face strip follows that order.

Check your understanding 4 Matching counts can hide a wrong join

The next arrangement still has two matching triangles and three rectangles, but the 4-unit and 5-unit side faces have exchanged places. Would it form the same triangular prism? Explain what would fail at a corner. What would be missing if you removed one triangular face?

Your answer or drawing on paper

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It does not form the prism. Starting from the triangle’s 3-unit side, the next side requires a 5-unit rectangle, but this strip puts the 4-unit rectangle there. A pair of edges that should meet would have different lengths. Rearrange the side faces to follow the triangle’s corners, rather than just counting them.

Removing a triangle leaves one end open. Adding an arbitrary triangle anywhere would not necessarily repair it: both ends must match, and each must attach where it can close its own opening.

Follow a pyramid’s common top vertex

Keep the square still and fold the four triangles up. Their outside tips meet at a single top vertex. Their sloping side edges meet in pairs. A strip of rectangles could not replace those triangular faces.

On paper, draw a 6 cm square and attach one triangle to each side. Give each triangle a 6 cm base and a perpendicular height of 5 cm. Fold the triangles towards one another and check that their tips can meet. Glue tabs, if used, lie outside the faces and do not count as extra faces. Ask an adult to help if you cut out the model.

Read a different arrangement

Check your understanding 5 Name the solid from its net

Choose the most specific name from cube, cuboid, prism and pyramid for the solid made by this arrangement. Describe how its faces close. Its layout differs from arrangement A.

Your solid name and folding reason

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It makes a cube. The stem of four squares wraps round four sides; the other two close the remaining sides. Every face is an equal square. Cube is the most specific name in the list, although every cube is also a cuboid and a prism.