Heptagon Area Calculator
Area, perimeter, circumradius, apothem, diagonals, and angles of a regular heptagon — or the area of an irregular 7-sided polygon from vertex coordinates.
Enter a positive length. Unit applies to all length outputs.
Area is computed with the shoelace formula; perimeter is the sum of the 7 edge lengths.
| Measurement | Value |
|---|---|
| Area A | |
| Perimeter P | |
| Side length s | |
| Circumradius R | |
| Apothem a | |
| Short diagonal (skips 1 vertex) | |
| Long diagonal (skips 2 vertices) | |
| Width across corners (2R) | |
| Width across flats (2a) | |
| Interior angle | 128.571428571° |
| Central / exterior angle | 51.428571428571° |
Formulas
Area: A = (7/4) × s² × cot(π/7) ≈ 3.633912444001589 × s²
From circumradius: s = 2R × sin(π/7) ≈ 0.8677674782351162 × R
From apothem: s = 2a × tan(π/7) ≈ 0.9631492376150572 × a
Diagonals: dk = 2R × sin(kπ/7); short d₂ ≈ 1.8019377358048383 × s, long d₃ ≈ 2.246979603717467 × s
Angles: interior = 900°/7 ≈ 128.57142857142858°, exterior = central = 360°/7 ≈ 51.42857142857143°; total diagonals = 14
Worked examples
Example 1 — Heptagon garden bed, side 5 m
A = 3.633912444001589 × 5² = 90.84781110003972 m². Perimeter 35 m, circumradius ≈ 5.7619121774062165 m, apothem ≈ 5.191303491430842 m.
Example 2 — Heptagon gazebo base, circumradius 10 ft
s = 2 × 10 × sin(π/7) ≈ 8.677674782351161 ft; A ≈ 3.633912444001589 × 8.677674782351161² ≈ 273.6410188638104 ft².
Example 3 — Bolt plate, apothem 2.5 cm
s = 2 × 2.5 × tan(π/7) ≈ 2.407873094037643 cm; A ≈ 3.633912444001589 × 2.407873094037643² ≈ 21.068889572829374 cm².
Example 4 — Irregular 7-point lot survey
Vertices (0,0), (5,0), (7,2), (7,6), (4,8), (1,7), (−1,3): shoelace area = 50.0 square units. (Try it in the “Irregular 7 points” mode.)
Heptagon FAQs
What is the formula for the area of a regular heptagon?
For a regular heptagon with side length s, the exact area is A = (7/4) × s² × cot(π/7), approximately 3.633912444001589 × s². For example, a heptagon with side 10 has area 363.3912444001589 square units.
How do you find the side length of a heptagon from its circumradius?
Use s = 2R × sin(π/7) ≈ 0.8677674782351162 × R. A heptagon with circumradius 10 therefore has side s ≈ 8.677674782351161 and area ≈ 273.6410188638104.
What is the apothem of a regular heptagon with side s?
The apothem (inradius) is a = s / (2 × tan(π/7)) ≈ 1.0382606982861684 × s. An apothem of 2.5 gives side s ≈ 2.407873094037643 and area ≈ 21.068889572829374.
What are the interior and exterior angles of a heptagon?
Each interior angle of a regular heptagon is 900°/7 ≈ 128.57142857142858°, each exterior angle is 360°/7 ≈ 51.42857142857143°, and the central angle subtended by one side at the center is also ≈ 51.42857142857143°.
How many diagonals does a heptagon have, and how long are they?
A heptagon has 14 diagonals (7 × 4 / 2). In a regular heptagon of side 1, the short diagonal (skipping one vertex) is ≈ 1.8019377358048383 and the long diagonal (skipping two vertices) is ≈ 2.246979603717467.
How do I calculate the area of an irregular heptagon?
Enter the (x, y) coordinates of its 7 vertices in order and use the shoelace formula: area = |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| / 2. The irregular mode of this calculator does that automatically, and also sums the 7 edge lengths for the perimeter.
What is the perimeter of a regular heptagon?
P = 7 × s. A heptagon with side 12.75 has perimeter 89.25, circumradius ≈ 14.692876052385852, apothem ≈ 13.237823903148646, and area ≈ 590.7378916780083.
What is the area of a heptagon with side 5?
A = 3.633912444001589 × 5² = 90.84781110003972 square units. Its perimeter is 35, circumradius ≈ 5.7619121774062165, and apothem ≈ 5.191303491430842.
Can I compute the area if I only know the apothem?
Yes: s = 2a × tan(π/7) ≈ 0.9631492376150572 × a, then A ≈ 3.633912444001589 × s². An apothem of 2.5 gives side ≈ 2.407873094037643 and area ≈ 21.068889572829374.
Why is the heptagon special in geometry?
The regular heptagon is the smallest regular polygon that cannot be constructed with compass and straightedge alone — Gauss proved this follows from 7 not being a Fermat prime. Its area constant 3.633912444001589 (per unit side) comes from (7/4) × cot(π/7) ≈ 1.75 × 2.076521396572337.
Free tool by NRTechStudio