Area Pentagon Calculator: Formula & Examples

This calculator works out the area of a regular pentagon from the side length you enter, using the standard formula A = (1/4)√(5(5+2√5)) × s². Type in the side length in your chosen unit, and you get the area in square units, plus the apothem (inradius) and circumradius for reference. The result is exact for a regular pentagon, where all five sides and angles are equal.

Formula

The area of a regular pentagon with side length s is given by:

A = (1/4) × √(5(5 + 2√5)) × s²

Where:

• A = area (in square units, e.g., m², ft²)
• s = length of one side (in the same linear unit)
• √(5(5+2√5)) ≈ 6.8819, so the constant (1/4)×√(5(5+2√5)) ≈ 1.72048

For a regular pentagon, the apothem (the distance from the center to the midpoint of a side) is:

a = s / (2 × tan(π/5)) ≈ 0.68819 × s

The circumradius (distance from center to a vertex) is:

R = s / (2 × sin(π/5)) ≈ 0.85065 × s

These formulas come from dividing the pentagon into 5 congruent isosceles triangles with a common vertex at the center. Each triangle has a base equal to the side length and a height equal to the apothem. The area of one triangle is (1/2) × base × height, so the total area is 5 × (1/2) × s × a = (5/2) × s × a. Substituting the apothem expression gives the formula above.

Worked Example

Using the default side length of 5 meters:

1. Calculate the constant: (1/4) × √(5(5+2√5)) = (1/4) × √(5(5+4.4721)) = (1/4) × √(47.3607) ≈ (1/4) × 6.8819 = 1.72048
2. Square the side: 5² = 25
3. Multiply: 1.72048 × 25 = 43.012 m²

The calculator shows 43.01 m² (rounded to 2 decimals). The apothem is 5 / (2 × tan(π/5)) = 5 / (2 × 0.7265) ≈ 3.44 m, and the circumradius is 5 / (2 × sin(π/5)) = 5 / (2 × 0.5878) ≈ 4.25 m.

When to Use a Pentagon Area Calculator

You might need the area of a regular pentagon in several real situations:

• Construction and carpentry: Laying out a pentagonal deck, table, or garden bed. Knowing the area helps estimate materials like paving stones, wood, or paint.
• Architecture and design: Pentagon-shaped rooms, windows, or facades. The U.S. Department of Defense building (The Pentagon) is a famous example, though its central courtyard is actually a pentagon with sides of about 280 meters.
• Mathematics and geometry homework: Students often need to compute areas of regular polygons.
• Graphics and game design: Calculating the area of a pentagonal tile or texture region.
• Landscaping: A pentagonal flower bed or lawn area, to determine how much soil, seed, or fertilizer is needed.

This calculator is specifically for regular pentagons. If you have an irregular pentagon (sides not all equal), you cannot use this formula; you would need to break it into triangles or use a surveyor’s formula.

How to Read the Results

The main output is the area in square units. If you enter the side in meters, the area is in square meters; if you enter feet, the area is in square feet. The apothem and circumradius are in the same linear unit as the side.

For a quick check: a regular pentagon with side length 1 has an area of approximately 1.72048 square units. So for any side length, the area is about 1.72 times the square of the side. This gives a rough sanity check—if your result is wildly different, you may have entered the wrong number.

The apothem is useful if you need to draw the pentagon or calculate the area using the alternative formula A = (5/2) × s × a. The circumradius is useful if you are inscribing the pentagon in a circle or vice versa.

Common Mistakes and How to Avoid Them

• Using the diameter instead of the side length: Many people confuse the side length with the diagonal or the distance between non-adjacent vertices. Make sure you measure one side only.
• Unit mix-ups: If you measure the side in inches, the area is in square inches, not square feet. To get square feet, convert the side to feet first (divide by 12) or convert the area at the end (1 ft² = 144 in²).
• Using the formula for a regular hexagon: The hexagon area formula is different (A = (3√3/2)s²). Don’t mix them up.
• Rounding too early: If you round the constant 1.72048 to 1.72, the error is small for typical side lengths, but for very large sides it can add up. Use the full precision in calculations.
• Assuming the apothem is the same as the circumradius: They are different. The apothem is smaller (about 0.688s) than the circumradius (about 0.851s).

Limitations and Alternatives

This formula only works for regular pentagons. If your shape is irregular, you have other options:

• Triangulation: Divide the pentagon into three triangles by drawing diagonals from one vertex. Calculate each triangle’s area (using Heron’s formula or base×height/2) and sum them.
• Coordinate method (shoelace formula): If you have the coordinates of the vertices, use the shoelace formula: Area = 0.5 × |Σ(x_i·y_{i+1} – x_{i+1}·y_i)|. This works for any polygon.
• Approximate by a circle: For a rough estimate, you could use the area of a circle with a similar radius, but this is not accurate for pentagons.

Also note that the formula assumes a flat, two-dimensional pentagon. For a 3D pentagonal prism or pyramid, you would need additional formulas for surface area and volume.

Frequently Asked Questions

What is the formula for the area of a regular pentagon?

The area is A = (1/4)√(5(5+2√5)) × s², where s is the side length. This simplifies to approximately 1.72048 × s².

How do I find the area of an irregular pentagon?

For an irregular pentagon, you cannot use the regular pentagon formula. Instead, divide the shape into triangles and sum their areas, or use the shoelace formula if you know the vertex coordinates.

What is the apothem of a regular pentagon?

The apothem is the distance from the center to the midpoint of a side. For a regular pentagon with side s, the apothem is a = s / (2·tan(36°)) ≈ 0.68819 × s. It is the height of each of the five isosceles triangles that make up the pentagon.

Can I use this calculator for a pentagon with side length in inches?

Yes. Enter the side length in inches, and the area will be in square inches. To convert to square feet, divide the area by 144, or convert the side to feet before entering.

What is the area of a pentagon with side length 10 cm?

Using the formula: A = 1.72048 × 10² = 172.048 cm². So the area is approximately 172.05 square centimeters.

Why is the constant 1.72048 used?

The constant comes from the trigonometric derivation of the area formula. It equals (1/4)√(5(5+2√5)), which arises from the geometry of a regular pentagon. It is the area of a pentagon with side length 1.

Related

• Chronological Age Calculator: How It Works
• Aerial View Square Footage Calculator: Measure Area
• 4th Order Polynomial Calculator: Find Quartic Roots

in s | Side length | m | 5 | 0
out area | Area | m² | 0.25 * sqrt(5 * (5 + 2 * sqrt(5))) * s^2 | 2
out apothem | Apothem (inradius) | m | s / (2 * tan(pi / 5)) | 2
out circumradius | Circumradius | m | s / (2 * sin(pi / 5)) | 2