Polygons · Ages 13–14
Equilateral triangles, squares and hexagons
Triangle: two arcs of one radius; square: two diameters at right angles (a set square's corner); hexagon: the radius marked off 6 times round.
How to do it
Method
Triangle: two arcs of one radius; square: two diameters at right angles (a set square's corner); hexagon: the radius marked off 6 times round
AC = BC = AB: three equal sides
Another way
Six equilateral triangles
6 × 60° = 360°, each angle 60° + 60° = 120°
In full
The equilateral triangle and the hexagon need compasses and an unmarked straight edge only. An equilateral triangle on AB: with centre A and radius AB, draw an arc; with centre B and the same radius, draw another, crossing the first at C; join AC and BC. Why does it work? AC and BC are both the same radius, equal to AB, so all three sides are equal. A square: draw a circle, centre O, and a diameter AC; with a set square or a protractor, draw a right angle at O to get a second diameter BD; join A, B, C and D. Why? OA = OB = OC = OD are radii and the four angles at O are 90°; turn the figure 90° about O and A goes to B, B to C, C to D and D to A, landing exactly on itself, so the four sides are equal and so are the four angles, which make 360°, 90° each: ABCD is a square. A regular hexagon: draw a circle, centre O, radius r; take a point A on it, and with the compasses still open at r, mark off B on the circle from A, then C from B, and so on: the sixth mark lands back on A; join the six points in turn. Why? Each △OAB has three sides of r, so it is equilateral, its angle at the centre is 60°, and 6 × 60° = 360°, exactly round. So a regular hexagon's side equals the radius of its circle. Another way to see it: a regular hexagon is six equilateral triangles round the centre, and each of its angles is two 60° angles, 120°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A square ABCD is drawn on a circle of radius 5 cm, as shown; its diagonals AC and BD are both diameters. The diagonal AC =
Show the answer and working
Answer: 10 cm
AC is a diameter, two radii: 2 × 5 = 10 cm.
Example 2
On a circle of radius 8 cm, marking off 8 cm with the compasses comes back to the start after exactly 6 marks. Why?
- A. A regular hexagon has 6 lines of symmetry
- B. The circumference is exactly 6 times the radius
- C. OA = OB = AB = 8 cm, so △OAB is equilateral, the angle at the centre is 60°, and 6 × 60° = 360°
- D. A hexagon's angles add to 720° = 2 × 360°
Choose the right answer
Show the answer and working
Answer: C. OA = OB = AB = 8 cm, so △OAB is equilateral, the angle at the centre is 60°, and 6 × 60° = 360°
Each mark makes a triangle with O whose sides are all 8 cm: equilateral, 60° at the centre, and 6 of those make 360°.
Example 3
An arc with centre A and radius AB, and another with centre B and the same radius, cross at C. AB = 7 cm. The perimeter of △ABC =
Show the answer and working
Answer: 21 cm
AC and BC are both the radius, equal to AB: three sides of 7 cm, perimeter 3 × 7 = 21 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.On a circle of radius 7 cm, marking off 7 cm with the compasses comes back to the start after exactly 6 marks. Why?
- A. A hexagon's angles add to 720° = 2 × 360°
- B. The circumference is exactly 6 times the radius
- C. OA = OB = AB = 7 cm, so △OAB is equilateral, the angle at the centre is 60°, and 6 × 60° = 360°
- D. A regular hexagon has 6 lines of symmetry
Choose the right answer
Answer:
2.A square ABCD is drawn on a circle of radius 10 cm, as shown; its diagonals AC and BD are both diameters. The diagonal AC =
Answer:
3.A square ABCD is drawn on a circle of radius 6 cm, as shown; its diagonals are diameters at right angles. The area of ABCD =
Answer:
4.A regular hexagon ABCDEF has sides of 9 cm. The diagonal AD =
Answer:
5.On a circle of radius 4 cm, marking off 4 cm with the compasses comes back to the start after 6 marks, making a regular hexagon.
True or false?
Answer:
All the answers
- 1. C. OA = OB = AB = 7 cm, so △OAB is equilateral, the angle at the centre is 60°, and 6 × 60° = 360°
- 2. 20 cm
- 3. 72 cm²
- 4. 18 cm
- 5. True
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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