作図、展开図と立体 ⟦number⟧

Building with ruler and compasses

• Ancient Greek geometers built every shape using just a straight edge 棱 and compasses — no rulers with markings, no protractors.
• Today, construction questions test whether you can use these tools precisely and leave your working visible.

Constructing triangles

• To construct a triangle from three sides:
• 1. Draw the base with a ruler.
• 2. Set compasses to the second side length and swing an arc from one end.
• 3. Set compasses to the third side and swing from the other end.
• 4. The intersection is the third vertex. Leave the arcs showing.

Scale drawings 比例图

• A scale drawing represents a real object at a fixed ratio (e.g. $1\text{ cm}:5\text{ m}$).
• Measure on the drawing, then multiply by the scale factor to get the real measurement.

Nets 展开图 and solids

• A net is a flat shape that folds up into a 3D solid — handy for surface area.
• Face 面: a flat side. Edge: where two faces meet. Vertex: a corner.
• A cube has $6$ faces, $12$ edges, and $8$ vertices.

Naming solids

Coordinate grids help with accurate constructions — shapes can be placed, measured, and transformed precisely.

Use a net to calculate

• A $3\times2\times1$ cm cuboid needs two faces of each size $3\times2$, $3\times1$, $2\times1$. Surface area $S=2(6+3+2)=22\text{ cm}^2$; volume $V=3(2)(1)=6\text{ cm}^3$.
• A triangular prism net has two identical triangles and three rectangles. Their widths match the three triangle sides, and each rectangle length matches the prism length.

A rhombus from two triangles

• Draw a 6 cm diagonal AB. Draw 5 cm radius arcs from A and B above and below it, meeting at C and D. Join A-C-B-D-A: all four edges are 5 cm, so the shape is a rhombus. Keep the arcs.
• For a square-based pyramid net, draw a square with one triangle attached to each edge. A base of side 4 cm and face height 3 cm has $S=b^2+4(bh/2)=16+24=40\text{ cm}^2$. Current Core and Extended construction outcomes do not require bisector constructions.

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