Quick answer
A pentagon has five sides and a 540° interior-angle sum. A hexagon has six sides and a 720° interior-angle sum.
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Regular pentagons and regular hexagons are polygons with equal side lengths and equal interior angles. They appear often in art, architecture, crystals, games, and nature because their symmetry is visually strong and mathematically useful. Studying them helps students connect angle formulas, construction methods, and area formulas to real shapes.
These two polygons also show an important contrast: the regular hexagon tiles the plane perfectly, while the regular pentagon does not.
Understanding Pentagon vs Hexagon
Diagonals reveal a major difference between these shapes. A diagonal joins two vertices that are not next to each other. The number of diagonals in any polygon can be found by multiplying the number of sides by the number that is three less, then dividing by two.
This gives five diagonals in a pentagon and nine in a hexagon. In a regular pentagon, the diagonals form a star pattern. Their crossings create smaller similar shapes, and their lengths are linked to the golden ratio.
This is one reason pentagon patterns appear in decorative art and logos. In a hexagon, diagonals can split the shape into triangles, rhombuses, or smaller hexagons. These subdivisions make calculations and designs easier to organize.
The tiling difference comes from what happens around one point. Angles that meet at a point must add to 360 degrees with no gaps or overlaps. Three hexagon corners fit exactly because each corner has an angle of 120 degrees.
Pentagon corners leave a gap when three are placed together. A regular pentagon can still be part of a repeating pattern, but it needs other shapes to fill the remaining space. This matters in floor tiles, paving, computer game maps, and engineering diagrams.
Honeycomb cells are often close to hexagons because hexagonal packing covers an area efficiently while using little wall material. Real honeycomb is not drawn with a ruler, so individual cells may be slightly uneven.
Area methods show why the center of a regular polygon is useful. Joining the center of a regular hexagon to every vertex makes six equal equilateral triangles. Each triangle has the same base and height, so the total area comes from adding six triangle areas.
A regular pentagon can be divided into five equal isosceles triangles from its center. The apothem is the short perpendicular distance from the center to a side. It acts as the height of each triangle.
Adding the five triangle areas leads to the rule that area equals one half of the apothem times the perimeter. Students should draw these center lines first instead of trying to measure an irregular-looking diagram directly.
Symmetry helps check whether a drawing is truly regular. A regular pentagon has five mirror lines and matches itself after turns of one fifth of a full rotation. A regular hexagon has six mirror lines and matches after turns of one sixth of a full rotation.
These turns connect directly to exterior angles. A common error is mixing up an interior angle, which lies inside the shape, with an exterior turning angle, which is made while walking around its boundary.
Another common error is counting a side as a diagonal. When solving problems, label vertices, mark known equal lengths, and decide whether the task needs perimeter, area, angle, symmetry, or a tiling argument before choosing a formula.
Key Facts
- For any n-sided polygon, sum of interior angles = (n - 2)180°.
- Each interior angle of a regular n-gon is ((n - 2)180°)/n.
- Regular pentagon: interior angle = 108° and exterior angle = 72°.
- Regular hexagon: interior angle = 120° and exterior angle = 60°.
- Regular hexagon with side length s can be divided into 6 equilateral triangles, so A = (3√3/2)s^2.
- Regular pentagon area can be found with A = (1/2)ap, where a is apothem and p is perimeter.
Vocabulary
- Regular polygon
- A polygon with all sides equal in length and all interior angles equal in measure.
- Interior angle
- An angle formed inside a polygon by two adjacent sides.
- Exterior angle
- An angle formed by one side of a polygon and the extension of an adjacent side.
- Apothem
- The perpendicular distance from the center of a regular polygon to one of its sides.
- Tessellation
- A repeating pattern of shapes that covers a plane with no gaps and no overlaps.
Common Mistakes to Avoid
- Using 360° as the sum of interior angles for every polygon is wrong because 360° is the sum of one exterior angle at each vertex, not the interior angle sum.
- Assuming a regular pentagon tiles the plane is wrong because three 108° angles total 324° and four total 432°, so they cannot meet perfectly around a point.
- Confusing side length with apothem is wrong because the apothem is measured from the center perpendicular to a side, not along the boundary.
- Treating any hexagon like a regular hexagon is wrong because formulas such as A = (3√3/2)s^2 only apply when all sides and angles are equal.
Practice Questions
- 1 Find the sum of the interior angles of a pentagon, then find each interior angle if the pentagon is regular.
- 2 A regular hexagon has side length 8 cm. Find its perimeter and area using A = (3√3/2)s^2.
- 3 Explain why regular hexagons can tessellate a flat surface but regular pentagons cannot, using the measures of their interior angles.