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Daily Math Minute

Geometry

Composing Shapes

Combining shapes to form a new shape.

Foundational15 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Defining & Non-Defining Attributes

Building New Shapes from Old Ones

Now that you can recognize a shape by its defining attributes, you can also build with shapes — putting two or more together to make a brand-new shape, the same way blocks combine to build something bigger.

Definition — Composing Shapes

Putting two or more shapes together, edge to edge with no gaps or overlaps, to form a new, larger shape.

When shapes are combined this way, the new shape has its own defining attributes, separate from the smaller shapes that built it. Two triangles joined along one matching side, for example, can form a square or a larger triangle, depending on how they're placed.

Worked Example — Combining Two Triangles into a Square

Two identical right triangles are placed together, matching their longest sides exactly. The combined shape has 4 equal sides and 4 corners — it's a square, even though it was built from two triangles.

Worked Example — Combining Squares into a Rectangle

Two identical squares are placed side by side, touching along one full edge with no gaps. The combined shape has 4 sides, but two of them are twice as long as the other two — it's a rectangle, not a square.

Geometry Canvas

Construct
Objects
  1. 1.

    P1: a free point, draggable on the plane

  2. 2.

    P2: a free point, draggable on the plane

  3. 3.

    P3: a free point, draggable on the plane

  4. 4.

    poly1: the polygon through P1, P2, P3

Measurements
  • poly1area = 15perimeter = 17.66

Tip

When combining shapes, match up edges that are exactly the same length — mismatched edges leave a gap or an overlap, and the result isn't a clean new shape.

Common Mistakes

  • Overlapping two shapes when combining them instead of joining them edge to edge.

    Line shapes up so their edges touch exactly, with no part of one shape covering part of the other.

  • Assuming the new combined shape keeps the name of the smaller shapes used to build it, such as calling a combined shape 'two triangles' instead of naming what it actually looks like now.

    Name the combined shape by counting its own sides and corners — the new shape has its own defining attributes, separate from its pieces.

Key Takeaways

  • Composing shapes means joining two or more shapes edge to edge to build a new shape.
  • The new shape has its own defining attributes, based on its own sides and corners.
  • How shapes are joined — which edges match up — determines what new shape is formed.

Summary

Composing shapes builds something new out of simpler pieces joined together. The final lesson does the reverse — dividing a whole shape into smaller, equal pieces.

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