GCSE 3D solids, plans and elevations Practice Questions

Picture the solid and decide what its faces are.

A face is a flat surface, an edge is where two faces meet and a vertex is a corner where edges meet. Counting them carefully beats guessing.

Count the faces.

A prism has one face for each side of its cross-section, so $4$ rectangles round the outside, plus the two $4$-sided ends: $6$ faces.

State the number of faces.

The solid has $6$ faces.

Write down the rules for the edges of a prism and of a pyramid.

A prism with an $n$-sided cross-section has $n$ edges at each end and $n$ joining them, so $3n$ in all. A pyramid on an $n$-sided base has $n$ base edges and $n$ sloping edges, so $2n$.

Apply the rule to each solid in the list.

A triangular prism has $9$ edges, which is what the question asks for.

Name the solid.

A triangular prism has exactly $9$ edges.

Write down Euler's formula.

For every convex polyhedron, faces minus edges plus vertices comes to $2$. Any triple that fails this cannot be a solid.

Test the triple that works.

With $F = 6$, $E = 12$ and $V = 8$, Euler's formula gives $2$, so this triple is possible.

Test F equals 6, E equals 12 and V equals 9.

This gives $3$, not $2$, so no convex polyhedron has these numbers.

Test F equals 5, E equals 9 and V equals 5.

This gives $1$, not $2$, so no convex polyhedron has these numbers.

Test F equals 7, E equals 12 and V equals 8.

This gives $3$, not $2$, so no convex polyhedron has these numbers.

State the possible triple.

Only $6$ faces, $12$ edges and $8$ vertices satisfies Euler's formula.

Say what you see when you look at the solid from the front.

Looking from the front you look along the rows, so a stack is hidden behind any taller stack in the same column of the base.

Work out the height of each column of the front elevation.

Looking from the front, a column is as tall as the TALLEST stack behind it, so take the largest number in each column of the grid: $2, 2, 3, 1$.

Rule out reading the front row only.

The front row is $2, 1, 3, 1$, but a taller stack behind it still shows in the elevation, so the front row is not the elevation.

Rule out adding the stacks in each column.

Adding the stacks in a column would count cubes that are hidden behind one another. An elevation is a view, not a total.

Rule out reading the columns from the wrong end.

The question asks for the columns from left to right, which is the order the stack heights are given in.

Write the stack heights out as a grid, row by row from the front.

The base is $4$ squares wide and $2$ squares deep, so there are $8$ stacks in all (a stack of $0$ means that square is empty).

Check the tallest column.

The tallest column of the elevation, $3$, must equal the tallest stack in the solid, and it does.

Check against the side elevation.

The side elevation reads $3, 2$ and also reaches $3$, so both views agree about the height of the solid.

Check the number of columns.

The base is $4$ squares wide, so the front elevation has exactly $4$ columns.

State the front elevation.

From left to right the columns are $2, 2, 3, 1$.

Say what you see when you look at the solid from the side.

Looking from the side you look along the width of the solid, so each ROW of the base gives one column of the elevation.

Work out the height of each column of the side elevation.

Looking from the side, a column is as tall as the tallest stack in that row of the base, so take the largest number in each row: $3, 3, 2$.

Write the stack heights out as a grid, row by row from the front.

The base is $3$ squares wide and $3$ squares deep, so there are $9$ stacks in all (a stack of $0$ means that square is empty).

Rule out using the columns of the base.

Taking the largest number in each column of the grid gives the FRONT elevation, $3, 2, 3$, not the side elevation.

Rule out adding the cubes in each row.

Adding a row counts cubes hidden behind one another. Only the tallest stack in the row is seen.

Rule out reading the rows from the back.

The question asks for the columns from the front of the solid to the back, so the front row must come first.

Check the number of columns.

The base is $3$ squares deep, so the side elevation has $3$ columns — not $3$, which is what the front elevation has.

Check the tallest column.

The tallest column of the side elevation must be the tallest stack of the solid, $3$ cubes.

Work out the area of the side elevation.

The elevation covers $8$ squares, so it has an area of $8\text{ cm}^2$.

Count the base squares that have at least one cube on them.

The plan is the view from directly above, so a base square appears in the plan exactly when a cube stands on it: $9$ of the $9$ squares are covered.

Check that the elevation cannot show every cube.

The solid has $17$ cubes but the side elevation shows only $8$ squares, because the cubes behind hide each other.

Check the front elevation reaches the same height.

Both elevations are views of the same solid, so their tallest columns must agree.

Add up the cubes in each row of the base.

Row by row from the front, the stacks hold $6, 6 and 5$ cubes.

Add the row totals to get the number of cubes.

Altogether the solid uses $17$ cubes.

State the side elevation.

From the front to the back the columns are $3, 3, 2$.