Geometrical constructions Revision Notes

Accurate measurement and drawing are fundamental in geometry. A ruler is used to measure lengths and draw straight lines. When measuring, align the zero mark of the ruler exactly at the start point and read the measurement at the end point. For drawing, place the ruler firmly and draw a line along its edge to ensure straightness.

Angles can be measured using a protractor, but this objective focuses on using a ruler for straight lines and angles. Constructing perpendicular or angle bisectors is not required here, so the emphasis is on precise measurement and drawing of given lines and angles.

Example: To draw a line segment of length 7 cm, place the ruler on the paper, mark points at 0 cm and 7 cm, then connect these points with a straight line using the ruler.

• Always start measuring from the zero mark on the ruler, not the edge.
• Hold the ruler steady to avoid errors in length or angle.
• Use a sharp pencil for clear, precise lines.

Constructing a triangle when all three side lengths are known requires a ruler and compass. This is a classic ruler-and-compass construction.

Method: 1. Draw one side of the triangle using the ruler. 2. With the compass set to the length of the second side, draw an arc from one endpoint of the base. 3. With the compass set to the length of the third side, draw an arc from the other endpoint of the base. 4. The intersection of these arcs determines the third vertex of the triangle. 5. Connect this vertex to the endpoints of the base with straight lines using the ruler.

Including construction arcs is essential to show the method clearly and earn full marks in exams. The arcs demonstrate how the compass was used to locate the third vertex accurately, ensuring the triangle's sides match the given lengths exactly. This method guarantees the triangle is unique if the side lengths satisfy the triangle inequality (the sum of any two sides is greater than the third). If not, no triangle can be constructed.

• Always label points clearly (e.g., A, B, C).
• Set compass widths carefully to the exact side lengths.
• Draw arcs lightly so construction lines are distinguishable from final lines.
• Check the triangle inequality before construction to confirm feasibility.

A net is a two-dimensional layout that can be folded to form a three-dimensional solid. Drawing nets helps visualise the shape's faces and is useful for calculating surface area and volume.

Common solids include cubes, cuboids, prisms, and pyramids. Each has characteristic nets:

Cube: Six equal squares arranged so they can fold into a cube. Cuboid: Six rectangles (some may be squares) arranged similarly. Prism: Two identical polygonal faces connected by rectangular faces. Pyramid: A polygonal base with triangular faces meeting at a vertex. Drawing nets requires careful measurement of each face's dimensions to ensure the net folds correctly into the solid.

• Use a ruler to draw each face to scale.
• Label edges to show which faces connect.
• Ensure all faces are drawn flat and connected along edges.
• Practice folding paper nets to understand the 3D shape.

Once a net is drawn or given, it can be used to find the surface area by summing the areas of all faces. For volume, use the relevant formula based on the solid's shape.

Surface area calculation steps: 1. Calculate the area of each face using length × width for rectangles or appropriate formulas for triangles and polygons. 2. Add all face areas together.

Volume formulas: Cube: side³ Cuboid: length × width × height Prism: area of cross-sectional face × length Pyramid: (1/3) × area of base × height Use measurements from the net to find areas and lengths needed in these formulas. Always include units (e.g., cm² for surface area, cm³ for volume). Check calculations carefully to avoid errors.

• Calculate each face area separately before summing.
• Use correct units and convert if necessary.
• Double-check measurements taken from the net.
• Apply the correct volume formula for the solid type.