The Pentagonal Prism Surface Area Calculator calculates the total surface area of a regular pentagonal prism by adding its two pentagon bases to its five rectangular sides: A = 2B + 5sh.
The pentagon base area is B = ¼√(5(5 + 2√5)) × s² ≈ 1.7205s². The same calculator also handles regular hexagonal and octagonal prisms. Enter the side length and prism height below.
Pentagonal Prism Surface Area Calculator
Surface area of regular pentagonal, hexagonal and octagonal prisms from side length and height.
Formula
A = 2B + n × s × h
| Prism | Sides n | Base area B |
|---|---|---|
| Pentagonal | 5 | ¼√(5(5 + 2√5)) × s² ≈ 1.7205s² |
| Hexagonal | 6 | (3√3 ÷ 2) × s² ≈ 2.5981s² |
| Octagonal | 8 | 2(1 + √2) × s² ≈ 4.8284s² |
Where:
- A = total surface area
- B = area of one regular polygon base
- s = side length of the base; n = number of sides
- h = prism height; n × s × h = lateral area
How the Calculation Works
Step 1: Find the area of one regular pentagon (or hexagon, octagon) from its side length.
Step 2: Find the lateral area: the base perimeter (n × s) times the height.
Step 3: Add two bases and the lateral area.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| s | Side length of the base | cm, m, in, ft |
| h | Height of the prism | cm, m, in, ft |
| B | Base area | cm², m², in², ft² |
| A | Total surface area | cm², m², in², ft² |
Example Calculation
Pentagonal prism
Given: side = 4 cm, height = 10 cm
Base area: B = 1.72048 × 16 ≈ 27.53 cm²
Lateral area: 5 × 4 × 10 = 200 cm²
Result: A = 2 × 27.53 + 200 ≈ 255.06 cm²
Hexagonal prism
Given: side = 4 cm, height = 10 cm
Result: A = 2 × 41.57 + 240 ≈ 323.14 cm²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 m² | 10.7639 ft² |
Shape Properties
A pentagonal prism has 7 faces (2 pentagons and 5 rectangles), 15 edges and 10 vertices. “Regular” means all five base sides and angles are equal; each interior angle of a regular pentagon is 108°.
Irregular pentagonal prisms
If the pentagon is irregular, find its area with the 5-Sided Shape Area Calculator, then use A = 2B + (perimeter × height).
Dimensions Explained
The side length is one edge of the pentagon. The height is the distance between the two pentagon faces. The apothem (center to the middle of a side) is not needed — the calculator works it out from the side length.
Frequently Asked Questions
How do you find the surface area of a pentagonal prism?
Add two pentagon bases to the five side rectangles: A = 2B + 5sh.
What is the area of a regular pentagon?
B = ¼√(5(5 + 2√5)) × s², about 1.7205 × side². Or B = ½ × perimeter × apothem.
How many faces does a pentagonal prism have?
Seven: two pentagons and five rectangles.
What is the lateral area of a pentagonal prism?
5 × side × height.
How do you find the surface area of a hexagonal prism?
A = 3√3 × s² + 6sh. Select the Hexagonal mode.
