Regular polygons have equal sides and equal angles, from equilateral triangles to hexagons and beyond. Enter the number of sides and the side length to get every angle, the perimeter and the area.
Enter the number of sides.
Enter the side length.
Read the angles, perimeter and area.
Interior angle: (n − 2) × 180° ÷ n
Area: A = n s² ÷ (4 tan(π ÷ n))
A regular polygon has all sides equal and all interior angles equal. The equilateral triangle, square, regular pentagon, hexagon and octagon are the familiar ones. Given the number of sides and the length of one side, this calculator returns the interior angle, the exterior angle, the perimeter and the area.
Regular polygons appear in honeycombs, nuts and bolts, paving stones, the Ashoka Chakra's spokes, gazebo floors, coins with flat edges and the tiled floors of old havelis. Carpenters and masons need the angle to cut each piece, and anyone flooring or painting the shape needs its area. School questions about the sum of angles and the number of sides also start here.
1. Count the sides n and measure one side s.
2. Exterior angle: 360° ÷ n.
3. Interior angle: 180° minus the exterior angle, which is the same as (n − 2) × 180° ÷ n.
4. Perimeter: P = n × s.
5. Apothem, the distance from the centre to the middle of a side: a = s ÷ (2 tan(180° ÷ n)).
6. Area: A = ½ × P × a, which simplifies to A = n s² ÷ (4 tan(180° ÷ n)).
Walk along the edge of any convex polygon. At every corner you turn by the exterior angle, and by the time you are back at the start you have turned through one full revolution, 360°. For a regular polygon all these turns are equal, so each is 360° ÷ n. The interior and exterior angles at a corner lie on a straight line, so the interior angle is 180° minus the exterior. For an octagon this gives 45° and 135°. The sum of interior angles, (n − 2) × 180°, follows by multiplying.
Join the centre to every vertex. The polygon splits into n congruent isosceles triangles, each with base s and apex angle 360° ÷ n. Dropping a perpendicular from the centre to the middle of a side halves that angle, giving a right triangle with angle 180° ÷ n and opposite side s/2. The perpendicular, called the apothem, is therefore (s/2) ÷ tan(180° ÷ n). Each triangle has area ½ × s × apothem, and n of them give n s² ÷ (4 tan(180° ÷ n)).
Keep the distance from the centre to a vertex fixed and increase n. The polygon hugs its circumscribed circle more and more closely, and its area approaches πR². Archimedes used 96-sided polygons this way to show that π lies between 3 10/71 and 3 1/7. The circumradius itself is s ÷ (2 sin(180° ÷ n)). The calculator needs a whole number of sides of at least three; a fractional entry is rounded to the nearest whole number.
A landscaper in Mysuru is building a regular octagonal gazebo floor with each side 12 feet long, and needs the floor area for tiles and the angle at which to cut the corner joints.
Exterior angle = 360° ÷ n: = 360 ÷ 8 = 45°
Interior angle = 180° − exterior: = 135°
Perimeter = n × s: = 8 × 12 = 96
Area = n s² ÷ (4 tan(180° ÷ n)): = 8 × 12² ÷ (4 × tan 22.5°) = 695.2935
Answer: Area 695.2935; Interior angle 135°; Exterior angle 45°
Using the interior angle where the exterior angle is needed, or the other way round, when cutting mitre joints.
Applying the formulas to an irregular polygon whose sides or angles differ.
Evaluating tan(180° ÷ n) with the calculator set to radians.
Confusing the apothem, centre to mid-side, with the circumradius, centre to vertex.
Assuming the interior angle sum is always 360°, which is true only for quadrilaterals.
Planning hexagonal paver blocks and octagonal floor tiles.
Cutting frames, gazebos and table tops with the correct joint angles.
Designing nuts, bolts and machine parts with hexagonal heads.
Class 8 and Class 9 questions on angles of polygons.
Drawing mandalas, logos and rangoli patterns with n-fold symmetry.
What is the sum of interior angles?
(n − 2) × 180°. A hexagon's interior angles add up to 720°.
Does this work for irregular polygons?
No, only for polygons with all sides and angles equal.