The 5-Sided Shape Area Calculator calculates the area of any five-sided shape (pentagon) — such as a lot, a garden bed or a room with one angled corner.
For an irregular pentagon, it splits the shape into three triangles using two diagonals from the same corner and adds their areas with Heron’s formula. You can also enter the five corner coordinates (shoelace formula), or use the regular pentagon mode, A ≈ 1.7205 × side², when all sides are equal.
5-Sided Shape Area Calculator (Irregular Pentagon)
Area of a five-sided shape from its sides and two diagonals, from corner coordinates, or of a regular pentagon from its side.
Label the corners A–E in order. Both diagonals start at corner A. Works for convex shapes (no corner pointing inward).
Formula
Five sides and two diagonals
A = T(AB, BC, AC) + T(AC, CD, AD) + T(AD, DE, EA)
Each triangle uses Heron’s formula: T = √(p(p − x)(p − y)(p − z)), where p = (x + y + z) ÷ 2.
Corner coordinates
A = ½ |Σ (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|
Regular pentagon
A = ¼√(5(5 + 2√5)) × s² ≈ 1.7205s²
Where: AB…EA = the five sides with corners labeled A–E in order; AC, AD = diagonals from corner A; s = side of a regular pentagon.
How the Calculation Works
Step 1: Label the corners A, B, C, D, E in order around the shape.
Step 2: Measure the two diagonals from corner A (to C and to D). They split the pentagon into three triangles.
Step 3: Find each triangle’s area from its three sides and add them.
Variables Explained
| Input | Meaning | Unit |
|---|---|---|
| AB, BC, CD, DE, EA | The five sides, in order | ft, m, yd… |
| AC, AD | Diagonals from corner A | ft, m, yd… |
| x, y | Corner coordinates | ft, m… |
| A | Area | ft², m², acres… |
Example Calculation
Irregular pentagon
Given: AB = 10, BC = 8, CD = 9, DE = 7, EA = 8 ft; AC = 14, AD = 12 ft
Triangle ABC (10, 8, 14) ≈ 39.19 ft²
Triangle ACD (14, 9, 12) ≈ 53.51 ft²
Triangle ADE (12, 7, 8) ≈ 26.91 ft²
Result: A ≈ 119.61 ft² (perimeter 42 ft)
Regular pentagon
Given: side = 6 ft
Result: A ≈ 1.7205 × 36 ≈ 61.94 ft²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10.7639 ft² |
| 1 yd² | 9 ft² |
| 1 acre | 43,560 ft² |
Shape Properties
A pentagon’s interior angles add up to 540°. In a regular pentagon each angle is 108°. Irregular pentagons can have any combination of sides and angles, which is why five side lengths alone do not fix the area — the two diagonals are needed.
Convex vs concave
The diagonals method works when no corner points inward (a convex shape). For a pentagon with an inward corner, use the coordinates mode.
Dimensions Explained
Measuring the diagonals
Both diagonals must start at the same corner (A). Measure straight across the shape from A to C and from A to D.
Frequently Asked Questions
How do you find the area of a 5-sided shape?
Split it into three triangles with two diagonals from one corner, find each triangle’s area and add them.
How do you find the area of an irregular pentagon?
Use the sides and two diagonals, or the coordinates of the five corners with the shoelace formula.
What is the area of a regular pentagon?
About 1.7205 × side². A pentagon with 6 ft sides covers about 61.94 ft².
Can you find the area of a pentagon with only the sides?
Not for an irregular pentagon — you also need two diagonals or the corner positions.
What do the angles of a pentagon add up to?
540°.
