Constructing a square with a compass and straightedge is a classic geometry skill because it shows how exact shapes can be built from simple rules. Instead of measuring with a ruler or protractor, you create equal lengths and right angles using arcs and lines. This matters because it connects visual drawing to logical proof.
A correct construction gives a square whose sides are equal and whose angles are all 90 degrees.
Key Facts
- A square has four equal sides and four right angles.
- If AB is the chosen side, then each side of the square has length AB.
- A perpendicular line forms a 90 degree angle with the original line.
- Compass arcs can copy a length exactly without using a ruler scale.
- For square ABCD, AB = BC = CD = DA and angle A = angle B = angle C = angle D = 90 degrees.
- The diagonal of a square with side length s is d = s√2.
Vocabulary
- Compass
- A drawing tool used to make circles and arcs and to copy distances exactly.
- Straightedge
- A tool used to draw straight lines without measuring length.
- Perpendicular
- Two lines are perpendicular when they meet at a right angle of 90 degrees.
- Arc
- An arc is part of a circle drawn by a compass from a fixed center.
- Diagonal
- A diagonal is a segment connecting two nonadjacent vertices of a polygon.
Common Mistakes to Avoid
- Using the straightedge as a ruler, which is wrong because compass and straightedge construction depends on copying distances with arcs, not measuring with marks.
- Changing the compass width while copying a side length, which is wrong because the copied side will no longer match the original side exactly.
- Drawing a line that only looks perpendicular, which is wrong because a square needs exact 90 degree angles constructed from equal arcs or a valid perpendicular method.
- Connecting the final vertices before checking equal distances, which is wrong because a small construction error can create a rectangle or skew quadrilateral instead of a square.
Practice Questions
- 1 A square is constructed with starting side AB = 6 cm. What is the length of each of the other three sides?
- 2 A square has side length 8 cm. Use d = s√2 to find the exact diagonal length and then approximate it using √2 ≈ 1.414.
- 3 Explain why constructing perpendicular lines at the endpoints of the starting side and copying the same compass width onto those lines produces a square rather than just a rectangle.