Triangular Prism Calculator
Calculate volume, surface area, and height of any triangular prism instantly
Enter Prism Dimensions
Key Formulas
V = (h/4) × √[(a+b+c)(b+c-a)(c+a-b)(a+b-c)]A = (1/4) × √[(a+b+c)(b+c-a)(c+a-b)(a+b-c)]Alat = h × (a + b + c)Atot = 2 × Abase + AlatCommon Triangle Types as Prism Bases
| Triangle Type | Condition | Base Area Formula | Example (a=b=c=5, h=10) |
|---|---|---|---|
| Equilateral | a = b = c | (√3 / 4) × a² | V ≈ 108.25 cm³ |
| Isosceles | a = c ≠ b | Heron’s formula | Varies by sides |
| Right Triangle | a² + b² = c² | (1/2) × base × height | V = (1/2) × b × H × h |
| Scalene | a ≠ b ≠ c | Heron’s formula | Use calculator above |
What Is a Triangular Prism?
A triangular prism is a three-dimensional geometric solid with two parallel and congruent triangular faces (bases) connected by three rectangular lateral faces. It belongs to the family of prisms, where the cross-section remains constant along the length of the shape.
Triangular prisms appear in architecture, engineering, optics, and everyday objects like wedges, rooftops, and Toblerone packaging. The shape is defined by three edge lengths of the triangular base (a, b, c) and the perpendicular height h of the prism.
Parts of a Triangular Prism
- Base triangle: the two congruent triangles at top and bottom
- Lateral faces: three rectangles connecting the two triangular bases
- Edges: nine total (three on each triangle plus three lateral edges)
- Vertices: six total (three on each triangular base)
How to Calculate Triangular Prism Volume
Triangular prism volume is calculated by multiplying the area of the triangular base by the prism height. When only the three side lengths are known, Heron’s formula is used to find the base area without needing the triangle’s altitude.
Step-by-Step Example
Given a = 5 cm, b = 6 cm, c = 7 cm, h = 10 cm:
- Semi-perimeter: s = (5 + 6 + 7) / 2 = 9
- Base area: A = √[9 × 4 × 3 × 2] = √216 ≈ 14.697 cm²
- Volume: V = 14.697 × 10 ≈ 146.97 cm³
- Lateral surface: 10 × (5 + 6 + 7) = 180 cm²
- Total surface: 2 × 14.697 + 180 ≈ 209.39 cm²
When Sides Form an Invalid Triangle
A valid triangle requires that the sum of any two sides must exceed the third side. For example, sides 1, 2, and 5 cannot form a triangle because 1 + 2 = 3, which is less than 5. The calculator will flag this automatically.
Surface Area of a Triangular Prism
Surface area of a triangular prism includes all five faces: two triangular bases and three rectangular sides. The total depends on the perimeter of the triangle and the prism height.
Three Surface Area Components
- Top and bottom (Atop + Abot): each equals the base triangle area. Total = 2 × Abase
- Lateral area (Alat): sum of three rectangles = h × (a + b + c)
- Total surface area (Atot): Atop + Abot + Alat = 2Abase + h(a+b+c)
The lateral surface area alone is useful when the top and bottom are not exposed, for example when calculating material needed to wrap the side of a roof or a ramp.
Frequently Asked Questions
A triangular prism is a three-dimensional solid with two parallel triangular bases and three rectangular lateral faces. The triangular bases are congruent and the height of the prism is the perpendicular distance between the two bases.
The volume of a triangular prism equals the area of the triangular base multiplied by the prism height. Using Heron’s formula, V = (h / 4) × √[(a+b+c)(b+c-a)(c+a-b)(a+b-c)], where a, b, c are the triangle side lengths and h is the prism height.
The total surface area of a triangular prism equals two times the base triangle area plus the lateral surface area. Written as a formula: Atot = 2 × Abase + h × (a + b + c). The lateral area h(a+b+c) covers the three rectangular side faces.
The lateral surface area of a triangular prism is the combined area of the three rectangular side faces only, not including the triangular top and bottom. It equals h × (a + b + c), where h is the prism height and a, b, c are the triangle side lengths.
The calculator works with any consistent unit of measurement. If side lengths are entered in centimeters, volume will be in cubic centimeters and surface area in square centimeters. The unit selector is for display labeling only and does not convert between units.
No. For three lengths to form a valid triangle, each side must be less than the sum of the other two (the triangle inequality). If this condition is not met, no triangle can exist and the prism volume and surface area cannot be calculated.