Hexagon worksheet

1. Area: 41.57 square units

1. The long diagonal is two sides long, so the side is a = D ÷ 2

   $a$$=\frac{8}{2}$$=4$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 4$$=24$

3. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times 4^2$$=41.569219$

4. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 4$$=6.928203$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 4$$=3.464102$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=4$

2. Area: 48.79 square units

1. The perimeter is six equal sides, so each side is a = P ÷ 6

   $a$$=\frac{26}{6}$$=4.333333$

2. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times {4.333333}^2$$=48.786098$

3. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 4.333333$$=7.505553$

4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 4.333333$$=8.666667$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 4.333333$$=3.752777$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=4.333333$

3. Area: 29.13 square units

1. The short diagonal is √3 (about 1.732) times the side, so the side is a = d ÷ √3

   $a$$=\frac{5.8}{\sqrt{3}}$$=3.348632$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 3.348632$$=20.091789$

3. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times {3.348632}^2$$=29.133095$

4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3.348632$$=6.697263$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 3.348632$$=2.9$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3.348632$

4. Area: 28.87 square units

1. The perimeter is six equal sides, so each side is a = P ÷ 6

   $a$$=\frac{20}{6}$$=3.333333$

2. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times {3.333333}^2$$=28.867513$

3. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 3.333333$$=5.773503$

4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3.333333$$=6.666667$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 3.333333$$=2.886751$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3.333333$

5. Side length (a): 3.4

1. Work the area formula A = (3√3 ÷ 2)a² backwards to get the side, a = √(2A ÷ 3√3)

   $a$$=\sqrt{\frac{2\times 30}{3\times \sqrt{3}}}$$=3.398088$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 3.398088$$=20.388531$

3. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 3.398088$$=5.885662$

4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3.398088$$=6.796177$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 3.398088$$=2.942831$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3.398088$

6. Area: 31.18 square units

1. The inradius is the height of one of the six triangles, (√3 ÷ 2)a, so the side is a = 2r ÷ √3

   $a$$=\frac{2\times 3}{\sqrt{3}}$$=3.464102$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 3.464102$$=20.78461$

3. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times {3.464102}^2$$=31.176915$

4. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 3.464102$$=6$

5. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3.464102$$=6.928203$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3.464102$

7. Area: 36.59 square units

1. The inradius is the height of one of the six triangles, (√3 ÷ 2)a, so the side is a = 2r ÷ √3

   $a$$=\frac{2\times 3.25}{\sqrt{3}}$$=3.752777$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 3.752777$$=22.51666$

3. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times {3.752777}^2$$=36.589573$

4. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 3.752777$$=6.5$

5. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3.752777$$=7.505553$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3.752777$

8. Area: 41.57 square units

1. The circumradius equals the side, because the six triangles have all sides equal: a = R

   $a$$=4$$=4$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 4$$=24$

3. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times 4^2$$=41.569219$

4. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 4$$=6.928203$

5. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 4$$=8$

6. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 4$$=3.464102$

9. Side length (a): 3.8

1. Work the area formula A = (3√3 ÷ 2)a² backwards to get the side, a = √(2A ÷ 3√3)

   $a$$=\sqrt{\frac{2\times 37.5}{3\times \sqrt{3}}}$$=3.799178$

2. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 3.799178$$=22.795071$

3. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 3.799178$$=6.58037$

4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3.799178$$=7.598357$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 3.799178$$=3.290185$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3.799178$

10. Area: 23.38 square units

1. Perimeter: add up the six equal sides, so P = 6a

   $P$$=6\times 3$$=18$

2. Area: the hexagon is six equal-sided triangles, which add up to A = (3√3 ÷ 2) × a²

   $A$$=\frac{3\times \sqrt{3}}{2}\times 3^2$$=23.382686$

3. Short diagonal (skips one corner, same as the width across the flats): d = √3 × a

   $d$$=\sqrt{3}\times 3$$=5.196152$

4. Long diagonal (corner to opposite corner, through the center): two sides long, so D = 2a

   $D$$=2\times 3$$=6$

5. Inradius (center to the middle of a side): r = (√3 ÷ 2) × a

   $r$$=\frac{\sqrt{3}}{2}\times 3$$=2.598076$

6. Circumradius (center to a corner) equals the side: R = a

   $R$$=a$$=3$