A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area of the circle is equal to 3√32π ≈ 0.827. Can you see why the answer is exactly twice the answer for the inscribed equilateral triangle?
Answer: The (regular) hexagon splits into 6 equilateral triangles of side r: area = 6 × √34r2 = 3√32r2; ratio = 3√32π ≈ 0.827. It is twice 3√34π because the inscribed equilateral triangle (joining alternate corners of the hexagon) is exactly half the hexagon.
Step-by-step solution
Idea: Each central angle of a regular hexagon is 60°, so each of the 6 triangles from the centre is equilateral with side r.
- Join O to the 6 corners. Each angle at O is 360° ÷ 6 = 60°, and two sides of each triangle are radii, so each triangle is equilateral with side r.1 mark
- Area of hexagon = 6 × √34r2 = 3√32r2. Ratio = 3√32r2 ÷ πr2 = 3√32π ≈ 0.827.1 mark
- Why twice: name the corners A, B, C, D, E, F. Joining alternate corners A, C, E gives the equilateral triangle inscribed in the same circle. OABC is a rhombus (all sides r), and its diagonal AC splits it into two congruent triangles: △ABC ≅ △AOC.1 mark
- Similarly △CDE ≅ △COE and △EFA ≅ △EOA. So the three triangles outside ACE have the same total area as triangle ACE itself: hexagon = 2 × △ACE. Hence 3√32π = 2 × 3√34π.1 mark
Area of hexagon : area of circle = (3√3/2)r² : πr² = 3√3/(2π) ≈ 0.827, which is twice 3√3/(4π) because the inscribed equilateral triangle ACE is exactly half of the hexagon.
Check: 3√32π = 5.1966.283 ≈ 0.827 and 2 × 0.4135 = 0.827 ✓.
Answer to write in the exam
Central angle = 360°6 = 60° ⇒ 6 equilateral triangles of side r
ar(hexagon) = 6 × √34r2 = 3√32r2
Ratio = 3√32r2 ÷ πr2 = 3√32π ≈ 0.827
△ABC ≅ △AOC, △CDE ≅ △COE, △EFA ≅ △EOA (rhombuses OABC, OCDE, OEFA)
⇒ ar(hexagon) = 2 × ar(△ACE)
∴ 3√32π = 2 × 3√34π
Common mistakes that cost marks
- Treating the hexagon as made of 6 right triangles. They are equilateral.
- Thinking the inscribed triangle is a third of the hexagon; it is half.
- Using side 2r for the hexagon; its side equals the radius.
How this can come in the exam
MCQ (1 mark)
A regular hexagon is inscribed in a circle of radius 2 cm. Its area is
- 3√3 cm2
- 6√3 cm2
- 12 cm2
- 24√3 cm2
Show answer
(B) 6√3 cm2
3√32 × 4 = 6√3 cm2.
Try one yourself
A regular hexagon has side 10 cm. Find its area and the area of the triangle formed by joining its alternate corners. (Take √3 = 1.732.)
Show answer
Hexagon = 3√32 × 100 ≈ 259.8 cm2; triangle = half ≈ 129.9 cm2.
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