Calculate the total surface area of any prism. Enter the perimeter of the base and the height of the prism.
Prism Dimensions
Result
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Square Units
Understanding the Surface Area of a Prism
A prism is a three-dimensional geometric shape with two identical and parallel bases, connected by rectangular or parallelogram-shaped faces. Common examples of prisms include triangular prisms, rectangular prisms (cuboids), pentagonal prisms, and hexagonal prisms. The surface area of a prism is the sum of the areas of all its faces, including its two bases.
The Formula
The formula for calculating the surface area of any prism is derived from its structure. It consists of the area of the two bases plus the area of all the lateral (side) faces.
Surface Area (SA) = (2 × Area of Base) + (Perimeter of Base × Height of Prism)
SA = 2B + Ph
B represents the area of one of the prism's bases. Since a prism has two identical bases, we multiply this area by 2.
P represents the perimeter of one of the prism's bases. This is the total length of the boundary of the base.
h represents the height of the prism, which is the perpendicular distance between the two bases.
How it Works
The lateral surface area (the area of the sides, excluding the bases) can be visualized as unfolding the prism's sides into a single large rectangle. The height of this rectangle is the height of the prism (h), and its width is the perimeter of the base (P). Thus, the lateral surface area is P × h.
To get the total surface area, we add the area of the two identical bases (2 × B) to this lateral surface area.
Use Cases
Calculating the surface area of a prism is useful in various practical applications:
Packaging Design: Determining the amount of material needed to wrap a box or container shaped like a prism.
Construction: Estimating the amount of paint, wallpaper, or cladding required for walls or structures shaped like prisms.
Architecture: Calculating the surface area of rooms or buildings with prismatic shapes for insulation or material cost estimations.
Manufacturing: In design and engineering, understanding the surface area is crucial for calculating heat transfer, fluid dynamics, or material stress.
Example Calculation
Let's consider a hexagonal prism where:
The perimeter of the hexagonal base (P) is 36 cm.
The area of one hexagonal base (B) is approximately 93.53 square cm.
The height of the prism (h) is 15 cm.
Using the formula:
SA = 2B + Ph
SA = (2 × 93.53 cm²) + (36 cm × 15 cm)
SA = 187.06 cm² + 540 cm²
SA = 727.06 cm²
Therefore, the total surface area of this hexagonal prism is approximately 727.06 square centimeters.
function calculateSurfaceArea() {
var basePerimeter = parseFloat(document.getElementById("basePerimeter").value);
var baseArea = parseFloat(document.getElementById("baseArea").value);
var prismHeight = parseFloat(document.getElementById("prismHeight").value);
var resultElement = document.getElementById("result");
// Clear previous result and errors
resultElement.innerText = "–";
// Input validation
if (isNaN(basePerimeter) || basePerimeter <= 0) {
alert("Please enter a valid positive number for the Base Perimeter.");
return;
}
if (isNaN(baseArea) || baseArea <= 0) {
alert("Please enter a valid positive number for the Area of One Base.");
return;
}
if (isNaN(prismHeight) || prismHeight <= 0) {
alert("Please enter a valid positive number for the Prism Height.");
return;
}
// Calculate surface area
var lateralSurfaceArea = basePerimeter * prismHeight;
var totalSurfaceArea = (2 * baseArea) + lateralSurfaceArea;
// Display the result
resultElement.innerText = totalSurfaceArea.toFixed(2);
}