Prism Calculator
Find the volume, total surface area, or lateral surface area of any prism — triangular, rectangular, or polygonal — from its base dimensions and height. Handy for construction, packaging, and geometry coursework.
About this calculator
A prism is a 3-D solid with two parallel, congruent polygonal bases connected by rectangular faces. The volume equals the base area multiplied by the prism's height: V = A_base × h. The lateral surface area — the area of the side faces only — equals the base perimeter times the height: A_lateral = P_base × h. The total surface area adds both bases: A_total = 2 × A_base + P_base × h. These formulas work universally regardless of base shape, which is why entering just the base area and perimeter is sufficient. Applications range from calculating concrete volumes in construction to cardboard needed for prismatic packaging.
How to use
Consider a triangular prism with a right-triangle base having legs 3 m and 4 m (hypotenuse 5 m) and a prism height of 10 m. Base area: A_base = (1/2) × 3 × 4 = 6 m². Base perimeter: P_base = 3 + 4 + 5 = 12 m. Volume: V = 6 × 10 = 60 m³. Lateral surface area: A_lateral = 12 × 10 = 120 m². Total surface area: A_total = 2 × 6 + 120 = 132 m². Enter base area = 6, base perimeter = 12, height = 10, and select your desired calculation.
Frequently asked questions
How do you calculate the volume of a triangular prism step by step?
First calculate the area of the triangular base using A = (1/2) × base × height of the triangle. Then multiply that base area by the length (height) of the prism: V = A_base × h. For example, a triangle with base 6 cm and height 4 cm has A_base = 12 cm²; if the prism is 8 cm long, V = 12 × 8 = 96 cm³. Make sure you distinguish between the triangle's internal height and the prism's length, as these are two separate dimensions.
What is the difference between lateral surface area and total surface area of a prism?
The lateral surface area includes only the rectangular side faces of the prism, calculated as the base perimeter times the height (P × h). The total surface area also includes the two polygonal bases, so A_total = 2 × A_base + P × h. For example, if you are painting only the sides of a prism-shaped column but not the top and bottom, you would use the lateral surface area. For wrapping an entire prismatic box in paper, use the total surface area.
Why does the prism volume formula work for any polygon base shape?
Cavalieri's principle states that if two solids have the same cross-sectional area at every height, they have the same volume. Since every horizontal cross-section of a prism is identical to the base, the total volume is simply the base area summed over the entire height, giving V = A_base × h. This applies regardless of whether the base is a triangle, hexagon, or any other polygon, making the formula universally applicable to all prism types.