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Year 4 Maths 2D Shapes & Symmetry

Combine & split shapes

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Theory

Combining and splitting shapes is a Year 4 maths geometry skill: joining smaller 2D shapes edge to edge to build a larger shape, and cutting a larger shape with straight lines to split it into smaller shapes. This page shows how triangles join to make rectangles, squares and larger triangles, how shapes split into triangles and rectangles, and how a hexagon is built from pattern blocks. Curriculum: VC2M4SP02 / AC9M4SP01.

When you combine shapes, you join them edge to edge with no gaps and no overlaps. The pieces together make one new, larger shape. For example, three equal squares in a row make a long rectangle.

When you split a shape, you draw a straight line to cut it into smaller shapes. One diagonal (a line from corner to opposite corner) splits a square or a rectangle into two triangles. A line straight across a hexagon makes two trapeziums.

A shape made of smaller shapes is called a composite shape. To name the whole shape, count its outside sides, not the pieces or cut lines inside it. A shape with 3 sides is a triangle, 4 sides a quadrilateral, 5 sides a pentagon and 6 sides a hexagon.

A rectangle split into two triangles by a diagonal A rectangle with a red dashed diagonal from the bottom-left corner to the top-right corner. The diagonal cuts the rectangle into two equal triangles, one shaded navy and one shaded grey. triangle triangle 1 rectangle = 2 triangles
One diagonal splits a rectangle into 2 equal triangles.
A hexagon split into six equal triangles A regular hexagon with red dashed lines from the centre to each corner, dividing it into six equal triangles that are shaded in alternating navy and grey. 1 hexagon = 6 triangles
Six equal triangles meet at the centre to make a hexagon.

These shape facts come from pattern blocks and tangram pieces. They help you build and split shapes quickly.

Whole shapeCan be made from
Rectangle2 identical right-angled triangles
Square2 identical right-angled triangles with equal short sides
Hexagon6 triangles
Hexagon2 trapeziums
Hexagon3 rhombuses
Trapezium (half hexagon)3 triangles
Rhombus (pattern block)2 triangles
Count sides to name a shape. 3 sides is a triangle, 4 sides a quadrilateral, 5 sides a pentagon and 6 sides a hexagon. Count only the outside edges of the whole shape.

How to combine or split shapes

  1. Decide the job. Are you joining pieces to build a bigger shape (combine), or cutting one shape into pieces (split)?
  2. Match the edges. To combine, line up sides that are the same length so there are no gaps and no overlaps. To split, draw a straight line from one point to another.
  3. Name the pieces. Count the sides of each small shape to name it: 3 sides a triangle, 4 sides a quadrilateral.
  4. Name the whole shape. Count the outside sides of the finished shape, ignoring any cut or join lines inside it.

How to build a hexagon from pattern blocks

  1. Count in triangles. A hexagon is worth 6 triangles, a trapezium 3 and a rhombus 2.
  2. Add the blocks you have. Add up the triangles they are worth.
  3. Fill the gap. Subtract from 6 to see how many triangles are still needed.
Example 1 — Combine two triangles
Two identical right-angled triangles are joined back to back along their upright sides. What shape do they make?
Two right-angled triangles joined along a short side Two identical right-angled triangles placed back to back along their upright sides. A red dashed line shows the join. Together they make one large triangle.
Solution

Count the outside sides of the new shape: the long bottom side and the two slanted sides.

\(3\) outside sides, so the two triangles make one large triangle. The dashed join line is inside, so it is not a side.

Example 2 — Split a square
A square is cut along both of its diagonals. How many pieces are made, and what are they?
A square cut along both diagonals A square with two red dashed diagonals that cross in the middle, cutting it into four triangles.
Solution

The two cuts cross in the middle and make \(4\) pieces.

Each piece has \(3\) straight sides, so the square is split into \(4\) triangles of the same size.

Example 3 — Build a hexagon
A trapezium block is worth \(3\) triangle blocks and a rhombus block is worth \(2\). A hexagon is made from \(1\) trapezium, \(1\) rhombus and some triangles. How many triangles are needed?
A hexagon made from three pattern blocks A hexagon made from a trapezium block across the bottom half, a rhombus block and a triangle block in the top half. trapezium rhombus triangle
Solution

A hexagon is the same as \(6\) triangle blocks.

The trapezium and rhombus cover \(3 + 2 = 5\) triangles.

\(6 - 5 = 1\), so \(1\) triangle block is needed.

Example 4 — Name a joined shape
A triangle is joined to each end of a rectangle. How many outside sides does the new shape have, and what is its name?
A rectangle with a triangle on each end A long rectangle with a triangle joined to its left end and another to its right end. Red dashed lines show the joins. The outline has six sides.
Solution

Count only the outline: the top, the bottom and \(2\) slanted sides at each end.

\(1 + 1 + 2 + 2 = 6\) outside sides, so the shape is a hexagon.

Example 5 — Two cuts
This trapezium is cut along the two dashed lines. Name the pieces.
A trapezium cut by two dashed lines A trapezium with a long bottom side. Two red upright dashed lines cut it into a rectangle in the middle and a triangle at each end.
Solution

The two cuts make \(3\) pieces.

The middle piece has \(4\) square corners, so it is a rectangle.

The two end pieces each have \(3\) sides, so they are triangles: \(1\) rectangle and \(2\) triangles.

Common pitfalls

No gaps, no overlaps. When you combine shapes, the edges must meet exactly. Leftover gaps, or pieces sitting on top of each other, do not make a proper composite shape.
Count the outside, not the pieces. A rectangle cut into 2 triangles is still 1 shape with 4 sides. The cut line inside is not an outside side.
Splitting keeps the same space. Cutting a shape into pieces does not make it bigger or smaller. All the pieces together fill exactly the whole shape.

Frequently asked questions

What does it mean to combine shapes?

To combine shapes you join them edge to edge with no gaps and no overlaps. The pieces together make one new, larger shape. For example, three equal squares in a row make a long rectangle.

What does it mean to split a shape?

To split a shape you draw a straight line to cut it into smaller shapes. For example, a line down the middle of a square makes two rectangles.

How many rhombus blocks make a hexagon?

A rhombus block is worth 2 triangles and a hexagon is worth 6 triangles, so \(3\) rhombus blocks make a hexagon.

How many sides does a shape have after it is cut?

Count only the outside edges of the whole shape. A rectangle cut into two triangles is still one shape with four sides, because the cut line is inside the shape, not an outside side.

Does splitting a shape change its size?

No. Cutting a shape into pieces does not make it bigger or smaller. All the pieces together fill exactly the same space as the whole shape.

What shape do two triangles make?

It depends on the triangles and how they are joined. Two identical right-angled triangles can make a rectangle (a square if their short sides are equal), a larger triangle or a parallelogram.

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