Geometrical constructions

1. Overview

Geometrical constructions involve drawing accurate shapes using only a ruler, compasses, and sometimes a protractor. This topic is crucial for developing precise drawing skills, essential not only for IGCSE Mathematics but also for fields like engineering and design. You'll learn to construct lines, angles, triangles, and nets of 3D shapes, all while adhering to strict accuracy standards.

Key Definitions

• Construct: To draw a shape or line using only a ruler and compasses (and sometimes a protractor, unless specified "ruler and compasses only").
• Net: A 2D pattern that can be folded to create a 3D solid.
• Arc: A portion of the circumference of a circle, drawn with a compass.
• Vertex: A corner point where two or more lines meet (plural: vertices).
• Face: A flat surface of a 3D solid.
• Edge: The line segment where two faces of a 3D solid meet.

---

Core Content

1. Measuring and Drawing Lines and Angles

Accuracy is vital. In IGCSE exams, the tolerance is usually within $\pm 2\text{mm}$ for lengths and $\pm 2^\circ$ for angles.

• To draw a line: Use a sharp pencil and a ruler. Ensure you start measuring from the $0$ mark, not the edge of the ruler.
• To measure/draw an angle:
  1. Place the center of the protractor on the vertex.
  2. Align the $0^\circ$ line with one of the sides of the angle.
  3. Read the scale that starts at $0$ (inner or outer).

2. Constructing a Triangle (SSS - Side, Side, Side)

When you are given three side lengths, you must use a pair of compasses. If you simply use a ruler to "guess" where the lines meet, you will lose all construction marks.

Worked example 1 — Constructing a triangle given three sides

Construct triangle $PQR$ with sides $PQ = 8\text{cm}$, $QR = 5\text{cm}$, and $RP = 6\text{cm}$.

1. Step 1: Draw the longest side ($PQ = 8\text{cm}$) using a ruler. Label the points $P$ and $Q$.
2. Step 2: Set your compass to a radius of $6\text{cm}$ (the length of $RP$).
3. Step 3: Place the point of the compass on $P$ and draw an arc above the line.
4. Step 4: Set your compass to a radius of $5\text{cm}$ (the length of $QR$).
5. Step 5: Place the point of the compass on $Q$ and draw an arc that intersects the first arc.
6. Step 6: Label the intersection point $R$ and join it to $P$ and $Q$ using a ruler.

Mark Alert: Never erase your construction arcs! markers look for these to award marks for the correct method.

Worked example 2 — Constructing a triangle given three sides

Construct triangle $XYZ$ where $XY = 4.5\text{cm}$, $YZ = 7\text{cm}$ and $ZX = 5.5\text{cm}$.

1. Step 1: Draw the longest side $YZ = 7\text{cm}$ using a ruler. Label the points $Y$ and $Z$.
2. Step 2: Set your compass to $4.5\text{cm}$ (the length of $XY$).
3. Step 3: Place the compass point on $Y$ and draw an arc. This arc represents all possible locations of point $X$ that are $4.5\text{cm}$ from $Y$.
4. Step 4: Set your compass to $5.5\text{cm}$ (the length of $ZX$).
5. Step 5: Place the compass point on $Z$ and draw another arc. The intersection of this arc with the previous arc will be point $X$.
6. Step 6: Label the intersection point $X$. Use a ruler to draw a straight line from $X$ to $Y$ and another from $X$ to $Z$.
7. Step 7: The triangle $XYZ$ is now constructed. Ensure all construction arcs are visible.

3. Drawing and Interpreting Nets

A net must have the correct number of faces and they must be arranged so that they "fold" without overlapping.

Worked Example 3: Net of a Triangular Prism

Draw the net of a prism with a length of $10\text{cm}$ and an equilateral triangle base with sides of $3\text{cm}$.

• Step 1: Draw three rectangles side-by-side for the "sides" of the prism. Each will be $10\text{cm} \times 3\text{cm}$.
• Step 2: Attach two equilateral triangles (sides $3\text{cm}$) to the top and bottom of the middle rectangle.

---

Extended Content (Extended Curriculum Only)

While the learning objectives are similar to Core, Extended questions often combine constructions with other topics like Loci (Topic 4.3) or require higher precision with complex 3D shapes. Extended questions may also require you to apply geometrical properties to calculate lengths before constructing.

Extended Example 1: Net of a Square-Based Pyramid

Draw the net of a pyramid where the square base is $4\text{cm} \times 4\text{cm}$ and the slant height of each triangular face is $5\text{cm}$.

1. Step 1: Draw a central square $4\text{cm} \times 4\text{cm}$.
2. Step 2: Each triangle has a base of $4\text{cm}$ and a height (the slant height) of $5\text{cm}$, so each sloping edge is $\sqrt{5^2 + 2^2} = \sqrt{29} \approx 5.4\text{cm}$. Set your compass to $5.4\text{cm}$ and draw arcs from both ends of each side of the square; where the arcs cross is the apex of that triangle.
3. Step 3: Ensure all four triangles are identical.

Extended Example 2: Combining Construction and Calculation

A tent is in the shape of a triangular prism. The triangular faces are right-angled, with sides $3\text{m}$ and $4\text{m}$ forming the right angle. The length of the tent is $6\text{m}$. Draw an accurate net of the tent, using a scale of $1\text{cm} = 0.5\text{m}$.

1. Step 1: Calculate the hypotenuse of the triangular face.
   Using Pythagoras' Theorem:
   $a^2 + b^2 = c^2$
   $3^2 + 4^2 = c^2$
   $9 + 16 = c^2$
   $25 = c^2$
   $c = \sqrt{25}$
   $c = 5\text{m}$

2. Step 2: Calculate the scaled dimensions for the net.

   • $3\text{m}$ becomes $3 / 0.5 = 6\text{cm}$
   • $4\text{m}$ becomes $4 / 0.5 = 8\text{cm}$
   • $5\text{m}$ becomes $5 / 0.5 = 10\text{cm}$
   • $6\text{m}$ becomes $6 / 0.5 = 12\text{cm}$

3. Step 3: Draw the net. The net consists of two right-angled triangles (6cm and 8cm sides) and three rectangles. One rectangle is 12cm x 6cm, another is 12cm x 8cm, and the third is 12cm x 10cm. Arrange the net so that it can be folded to form the triangular prism.

---

Key Equations

In this topic, "equations" are rare, but geometric properties are used to check accuracy:

Sum of angles in a triangle: $a + b + c = 180^\circ$ (Not on formula sheet - MEMORIZE)

Pythagoras’ Theorem (for right-angled triangles): $a^2 + b^2 = c^2$ (Not on formula sheet - MEMORIZE)

• Example: In a $3-4-5$ triangle, $3^2 + 4^2 = 5^2 \implies 9 + 16 = 25$.

Euler's Formula (for interpreting nets): $F + V - E = 2$ (Not on formula sheet - MEMORIZE)

• $F$ = Number of Faces
• $V$ = Number of Vertices
• $E$ = Number of Edges

---

Common Mistakes to Avoid

• ❌ Wrong: Using a blunt pencil, resulting in thick, imprecise lines.
  • ✓ Right: Use a 2H or H pencil, sharpened to a fine point, for thin, accurate lines.
• ❌ Wrong: Erasing construction arcs once the shape is finished, thus losing marks.
  • ✓ Right: Leave all arcs visible; they are evidence of your method and demonstrate understanding.
• ❌ Wrong: Measuring from the edge of the ruler instead of the '0' mark, leading to inaccurate lengths.
  • ✓ Right: Always align the '0' mark with the start of your line for precise measurements.
• ❌ Wrong: Drawing a net without considering how the faces will fold, resulting in overlaps or gaps.
  • ✓ Right: Mentally "fold" the net in your head (or sketch a rough 3D shape) to ensure no two faces occupy the same space and that all edges connect correctly.
• ❌ Wrong: Forgetting to use compasses for triangle construction when only side lengths are given.
  • ✓ Right: Always use compasses to construct triangles given SSS (side, side, side); freehand drawing will not earn marks.

---

Exam Tips

• Command Words:
  • "Draw" usually allows the use of a protractor and ruler.
  • "Construct" means draw accurately, using a ruler and compasses. A protractor may also be used unless the question says "ruler and compasses only", as when constructing a triangle from three sides.
• Calculator vs Non-Calculator: This topic is usually found in the non-calculator paper or the early parts of the calculator paper. You won't need a calculator for the drawing itself, but you might need it to calculate a missing side using Pythagoras before drawing.
• Real-World Contexts: Expect questions about "Scale Drawings" (e.g., "Construct a map of a garden where $1\text{cm} = 2\text{m}$").
• The "Check" Step: After constructing a triangle, measure the angles with a protractor. If they don't add up to $180^\circ$ (within a degree or two), your side lengths are likely inaccurate.
• Read Carefully: Pay close attention to the units given in the question and the units required in the answer, especially in scale drawing problems. Convert units before you start drawing.