Pentagon Calculator
Solve a regular pentagon from any one measurement: side, perimeter, area, or diagonal. Enter it in any unit and get the full set back — area, apothem, height, and the diagonal, which is always exactly φ (1.618) times the side.
Example: with What do you know? Side length (a) · Value 10 → Area: 172.048 square units.
- Side (a)10 units
- Perimeter (5a)50 units
- Diagonal (φ·a)16.18 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
Regular pentagon: A = ¼√(5(5+2√5))·a² ≈ 1.7205a². The diagonal-to-side ratio is exactly the golden ratio φ = 1.618.
The most golden of polygons
A regular pentagon has five equal sides meeting at 108° corners, and its area is A = ¼√(5(5+2√5))·a² ≈ 1.7205a². The shape is soaked in the golden ratio: every diagonal is exactly φ ≈ 1.618 times a side, any two crossing diagonals cut each other in golden proportion, and the five diagonals together trace a pentagram whose inner core is another, smaller regular pentagon.
Measured dimensions come in three flavors and this page reports them all. The apothem (center to mid-side, ≈ 0.688a) sets the inscribed circle; the height from a flat base to the opposite apex is ≈ 1.539a; and the diagonal (≈ 1.618a) is what you get taping across the shape corner to corner. The most famous example is literal: the Pentagon building's outer walls are 921 ft each, enclosing about 1.46 million square feet.
How it’s calculated
A = ¼√(5(5+2√5))·a² ≈ 1.720477a²; perimeter = 5a; diagonal = φ·a with φ = (1+√5)/2 ≈ 1.618034; apothem = a/(2·tan 36°) ≈ 0.688191a; height = apothem + circumradius = a/(2·tan 36°) + a/(2·sin 36°) ≈ 1.538842a. Reverse solving: a = P/5, a = √(A/1.720477), a = d/1.618034.
Regular pentagons only — five equal sides and 108° angles. Home-plate shapes and irregular five-sided lots need to be split into triangles.
Regular pentagon dimensions
| Side a | Area | Diagonal (φa) | Apothem |
|---|---|---|---|
| 1 | 1.72 | 1.62 | 0.69 |
| 5 | 43.01 | 8.09 | 3.44 |
| 10 | 172.05 | 16.18 | 6.88 |
| 12 | 247.75 | 19.42 | 8.26 |
| 921 ft (Pentagon building) | ≈ 1.46 million ft² (33.5 acres) | 1,490 ft | 634 ft |
Computed with A ≈ 1.720477a² and diagonal = 1.618034a; the Pentagon's published outer wall length is 921 ft.
Common mistakes
- Entering a diagonal as the side — the diagonal is 1.618× longer, so area comes out 2.6× too large.
- Using the pentagon-shaped home plate as a regular pentagon; its sides are not equal, so none of these formulas apply.
- Mixing up apothem (0.688a, center to side) with circumradius (0.851a, center to corner) when laying out from a center stake.
- Estimating area as "about a² times 2" — the true coefficient is 1.7205, a tempting shortcut that overshoots by 16%.
Frequently asked questions
What is the area formula for a pentagon?
For a regular pentagon with side a: A = ¼√(5(5+2√5))·a² ≈ 1.7205a². Equivalently ½ × perimeter × apothem, since the apothem is a/(2·tan 36°).
How is the golden ratio connected to a pentagon?
Each diagonal is exactly φ = (1+√5)/2 ≈ 1.618 times the side. A pentagon with 10-unit sides has 16.18-unit diagonals, and crossing diagonals divide each other in the same 1.618 ratio.
What is the interior angle of a regular pentagon?
108° at every corner — from (n−2)×180°/n with n = 5. The exterior angles are 72° each, which is also the wedge angle when you slice the pentagon into five triangles from the center.
How tall is a pentagon compared to its side?
From a flat base to the top vertex it stands about 1.539 × the side — the apothem (0.688a) plus the circumradius (0.851a). A pentagon with 10-in sides is about 15.4 in tall.
How big is the Pentagon building?
Each outer wall runs 921 ft, giving a 4,605-ft perimeter and a regular-pentagon footprint of roughly 1.46 million square feet — about 33.5 acres — consistent with its published size.