Regular Polygon worksheet

1. Area: 130.78 square units

1. The interior angles of any 5-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 5 corners

   $\mathrm{interior}$$=\frac{\left(5-2\right)\times 180}{5}$$=108\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{5}$$=72\text{°}$

3. The apothem a and half a side form a right triangle with a 180°/n angle at the center. So the side is s = 2a·tan(180°/n)

   $s$$=2\times 6\times \text{tan}36\text{°}$$=8.71851$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{8.71851}{2\times \text{tan}36\text{°}}$$=6$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{8.71851}{2\times \text{sin}36\text{°}}$$=7.416408$

6. Perimeter P: add up the 5 equal sides

   $P$$=5\times 8.71851$$=43.592552$

7. Area: the shape is 5 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{43.592552\times 6}{2}$$=130.777655$

2. Area: 50 square units

1. The interior angles of any 4-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 4 corners

   $\mathrm{interior}$$=\frac{\left(4-2\right)\times 180}{4}$$=90\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{4}$$=90\text{°}$

3. The circumradius R is the long slanted side of that right triangle. So the side is s = 2R·sin(180°/n)

   $s$$=2\times 5\times \text{sin}45\text{°}$$=7.071068$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{7.071068}{2\times \text{tan}45\text{°}}$$=3.535534$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{7.071068}{2\times \text{sin}45\text{°}}$$=5$

6. Perimeter P: add up the 4 equal sides

   $P$$=4\times 7.071068$$=28.284271$

7. Area: the shape is 4 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{28.284271\times 3.535534}{2}$$=50$

3. Area: 3.37 square units

1. The interior angles of any 5-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 5 corners

   $\mathrm{interior}$$=\frac{\left(5-2\right)\times 180}{5}$$=108\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{5}$$=72\text{°}$

3. All 5 sides are equal, so the side is the perimeter divided by 5

   $s$$=\frac{7}{5}$$=1.4$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{1.4}{2\times \text{tan}36\text{°}}$$=0.963467$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{1.4}{2\times \text{sin}36\text{°}}$$=1.190911$

6. Area: the shape is 5 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{7\times 0.963467}{2}$$=3.372136$

4. Area: 3.54 square units

1. The interior angles of any 6-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 6 corners

   $\mathrm{interior}$$=\frac{\left(6-2\right)\times 180}{6}$$=120\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{6}$$=60\text{°}$

3. All 6 sides are equal, so the side is the perimeter divided by 6

   $s$$=\frac{7}{6}$$=1.166667$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{1.166667}{2\times \text{tan}30\text{°}}$$=1.010363$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{1.166667}{2\times \text{sin}30\text{°}}$$=1.166667$

6. Area: the shape is 6 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{7\times 1.010363}{2}$$=3.53627$

5. Area: 1.72 square units

1. The interior angles of any 5-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 5 corners

   $\mathrm{interior}$$=\frac{\left(5-2\right)\times 180}{5}$$=108\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{5}$$=72\text{°}$

3. All 5 sides are equal, so the side is the perimeter divided by 5

   $s$$=\frac{5}{5}$$=1$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{1}{2\times \text{tan}36\text{°}}$$=0.688191$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{1}{2\times \text{sin}36\text{°}}$$=0.850651$

6. Area: the shape is 5 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{5\times 0.688191}{2}$$=1.720477$

6. Area: 49 square units

1. The interior angles of any 4-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 4 corners

   $\mathrm{interior}$$=\frac{\left(4-2\right)\times 180}{4}$$=90\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{4}$$=90\text{°}$

3. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{7}{2\times \text{tan}45\text{°}}$$=3.5$

4. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{7}{2\times \text{sin}45\text{°}}$$=4.949747$

5. Perimeter P: add up the 4 equal sides

   $P$$=4\times 7$$=28$

6. Area: the shape is 4 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{28\times 3.5}{2}$$=49$

7. Area: 6.01 square units

1. The interior angles of any 7-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 7 corners

   $\mathrm{interior}$$=\frac{\left(7-2\right)\times 180}{7}$$=128.571429\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{7}$$=51.428571\text{°}$

3. All 7 sides are equal, so the side is the perimeter divided by 7

   $s$$=\frac{9}{7}$$=1.285714$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{1.285714}{2\times \text{tan}25.714286\text{°}}$$=1.334907$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{1.285714}{2\times \text{sin}25.714286\text{°}}$$=1.481635$

6. Area: the shape is 7 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{9\times 1.334907}{2}$$=6.00708$

8. Area: 127.31 square units

1. The interior angles of any 6-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 6 corners

   $\mathrm{interior}$$=\frac{\left(6-2\right)\times 180}{6}$$=120\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{6}$$=60\text{°}$

3. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{7}{2\times \text{tan}30\text{°}}$$=6.062178$

4. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{7}{2\times \text{sin}30\text{°}}$$=7$

5. Perimeter P: add up the 6 equal sides

   $P$$=6\times 7$$=42$

6. Area: the shape is 6 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{42\times 6.062178}{2}$$=127.305734$

9. Area: 59.44 square units

1. The interior angles of any 5-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 5 corners

   $\mathrm{interior}$$=\frac{\left(5-2\right)\times 180}{5}$$=108\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{5}$$=72\text{°}$

3. The circumradius R is the long slanted side of that right triangle. So the side is s = 2R·sin(180°/n)

   $s$$=2\times 5\times \text{sin}36\text{°}$$=5.877853$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{5.877853}{2\times \text{tan}36\text{°}}$$=4.045085$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{5.877853}{2\times \text{sin}36\text{°}}$$=5$

6. Perimeter P: add up the 5 equal sides

   $P$$=5\times 5.877853$$=29.389263$

7. Area: the shape is 5 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{29.389263\times 4.045085}{2}$$=59.441032$

10. Side length: 1.52

1. The interior angles of any 5-sided polygon add up to (n − 2) × 180°; a regular one shares that equally among its 5 corners

   $\mathrm{interior}$$=\frac{\left(5-2\right)\times 180}{5}$$=108\text{°}$

2. Each exterior angle is what remains of a straight line, and the exterior angles add up to 360°

   $\mathrm{exterior}$$=\frac{360}{5}$$=72\text{°}$

3. Work backwards from the area formula, area = n·s² ÷ (4 tan(180°/n)), to find the side s

   $s$$=\sqrt{\frac{4\times 4\times \text{tan}36\text{°}}{5}}$$=1.524774$

4. Apothem a (center to the middle of a side): the side of the right triangle next to the 180°/n angle at the center

   $a$$=\frac{1.524774}{2\times \text{tan}36\text{°}}$$=1.049336$

5. Circumradius R (center to a corner): the long slanted side of that right triangle

   $R$$=\frac{1.524774}{2\times \text{sin}36\text{°}}$$=1.29705$

6. Perimeter P: add up the 5 equal sides

   $P$$=5\times 1.524774$$=7.623871$

7. Area: the shape is 5 triangles, each with base s and height a. So area = perimeter × apothem ÷ 2

   $\mathrm{area}$$=\frac{7.623871\times 1.049336}{2}$$=4$

8. Check: substituting the side back gives the area you entered

   $\frac{5\times {1.524774}^2}{4\times \text{tan}36\text{°}}$$=4$

9. So the side length is

   $s$$=1.524774$