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Math & Conversions Prism Calculator Prism Calculator (Volume and Surface Area) Calculate the volume ($V$), lateral surface area ($LSA$), and total surface area ($TSA$) for the two most common types of prisms: **Rectangular** and **Triangular**. Select the prism type below to input your dimensions. Rectangular Prism (Cuboid) Triangular Prism Base is a Rectangle: $V = l \cdot w \cdot h$ Length (l) Width (w) Height of Prism (h) Base is a Triangle: $V = \frac{1}{2} b \cdot h_{base} \cdot L$ Triangular Base Dimensions: Base Side ($b$) Base Height ($h_{base}$) Prism Length/Height ($L$) Base Side Lengths (for Area Calculation): Side 1 ($s_1$) Side 2 ($s_2$) Side 3 is the Base Side ($b$). For TSA, enter the three sides of the triangular base. Calculate Prism Results Lateral Surface Area ($LSA$) Total Surface Area ($TSA$) Volume ($V$) Step-by-Step Solution Universal Prism Formulas A prism is a polyhedron whose two bases are parallel, congruent polygons, and whose faces are parallelograms. All calculations stem from the base properties. 1. Volume ($V$) The volume formula is consistent for all prisms, requiring only the area of the base ($B$) and the height/length of the prism ($L$): $$V = B \cdot L$$ Where $B$ is the area of the polygonal base and $L$ is the distance between the two bases. 2. Surface Area ($TSA$ and $LSA$) The lateral surface area ($LSA$) is the area of all the sides, excluding the top and bottom bases. The total surface area ($TSA$) includes the bases. Lateral Surface Area ($LSA$): $$LSA = P \cdot L \quad (\text{where } P \text{ is the perimeter of the base})$$ Total Surface Area ($TSA$): $$TSA = LSA + 2B = P \cdot L + 2B$$ Specific Prism Formulas Rectangular Prism (Cuboid) For a rectangular base with length $l$ and width $w$, and prism height $h$: **Base Area ($B$):** $B = l \cdot w$ **Base Perimeter ($P$):** $P = 2l + 2w$ **Volume ($V$):** $V = l \cdot w \cdot h$ **Total Area ($TSA$):** $TSA = 2(lw + lh + wh)$ Triangular Prism For a triangular base with base side $b_{tri}$ and height $h_{tri}$, and prism length $L$: **Base Area ($B$):** $B = \frac{1}{2} b_{tri} \cdot h_{tri}$ **Volume ($V$):** $V = \frac{1}{2} b_{tri} \cdot h_{tri} \cdot L$ **Total Area ($TSA$):** You must sum the area of the three rectangular sides (Lateral Area) and the two triangular bases. Frequently Asked Questions (FAQ) What is the universal formula for the volume of any prism? The volume of any prism, regardless of the shape of its base, is found by multiplying the area of the base ($B$) by the height ($L$) of the prism: $V = B \cdot L$. What is the difference between total surface area and lateral surface area? The **lateral surface area** ($LSA$) is the area of all the sides, excluding the top and bottom bases. The **total surface area** ($TSA$) includes the area of the two bases plus the lateral surface area: $TSA = LSA + 2B$. How do I find the volume of a rectangular prism? A rectangular prism (or cuboid) has a rectangular base. The base area ($B$) is length times width ($l \cdot w$). Therefore, the volume is $V = l \cdot w \cdot h$, where $h$ is the prism height. What is a right prism versus an oblique prism? A **right prism** has lateral faces that are rectangles, meaning the lateral edges are perpendicular (at $90^\circ$) to the base. An **oblique prism** has lateral faces that are parallelograms and are not perpendicular to the base. The volume formula $V=B \cdot L$ only works for the oblique prism if $L$ is the *perpendicular* distance between the bases. Universal Prism Formulas Volume $$V = B \cdot L$$ Total Surface Area $$TSA = P \cdot L + 2B$$ Base Area (Rectangle) $$B = l \cdot w$$ Related Solid Geometry Tools Rectangular Prism Volume Pyramid Calculator Cylinder Volume Calculator Sphere Volume Calculator Volume Calculator (General)
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