How Do You Calculate the Total Surface Area of a Triangular Prism?
The concept of Surface Area of a Triangular Prism Formula is a vital geometry topic for students in middle and high school. Understanding how to calculate the surface area helps in solving practical problems in school exams, competitive tests, and even real-life situations like designing, packaging, and construction.
Understanding the Surface Area of a Triangular Prism
A triangular prism is a three-dimensional (3D) shape with two identical triangular bases joined by three rectangular faces. You can imagine it as a tent or a Toblerone chocolate bar. The surface area of a triangular prism is the total area of all its external faces including the triangles and rectangles. This concept plays a key role in measurement topics covered in classes 7, 8, 9, and 10, and helps students visualize connections between two-dimensional and three-dimensional figures.
Formula for the Surface Area of a Triangular Prism
To find the total surface area of a triangular prism, we need to add the areas of all three rectangular faces and both triangular bases. Here is the main formula:
- Surface Area = (Perimeter of the triangular base × Length of the prism) + (2 × Area of the triangular base)
Written with variables:
- Surface Area = (S1 + S2 + S3) × L + 2 × (Area of Triangle)
Where:
- S1, S2, S3 = Sides of the triangular base
- L = Length (height) of the prism
- Area of Triangle = ½ × base × height (or use Heron's formula if all sides are known)
This formula can be applied to any triangular prism, including prisms with equilateral, isosceles, or scalene triangles as bases.
Step-by-Step Example: Calculating Surface Area
Let's work through a real example:
- Identify the sides of the triangle: Suppose the triangular base has sides of 5 cm, 12 cm, and 13 cm.
- Find the perimeter of the base:
Perimeter = 5 + 12 + 13 = 30 cm - Find the area of the triangle (using ½ × base × height):
Base = 5 cm, Height = 12 cm
Area = ½ × 5 × 12 = 30 cm² - Find the length of the prism: Suppose the prism has a length of 11 cm.
- Plug into the formula:
Surface Area = (30 × 11) + 2 × 30 = 330 + 60 = 390 cm²
| Triangle Side 1 | Triangle Side 2 | Triangle Side 3 | Triangle Height | Length of Prism | Perimeter | Base Area | Surface Area |
|---|---|---|---|---|---|---|---|
| 5 cm | 12 cm | 13 cm | 12 cm | 11 cm | 30 cm | 30 cm² | 390 cm² |
Lateral Surface Area of a Triangular Prism
The lateral surface area (LSA) means the area of only the rectangle faces, not including the triangular bases. The formula is simple:
- Lateral Surface Area = (Perimeter of base) × (Length of the prism)
- LSA = (S1 + S2 + S3) × L
For the example above: LSA = 30 × 11 = 330 cm²
Surface Area of a Right Triangular Prism
A right triangular prism has bases that are right triangles. The calculation is the same, but it's often easier because you know one angle is 90°. For right triangles, area = ½ × base × height.
- Surface Area = (Perimeter of base × Length) + (2 × ½ × base × height)
- Surface Area = (S1 + S2 + S3) × L + (base × height)
This works for any right triangular prism. If you have three side lengths and it's a right triangle, use the sides directly for perimeter.
Worked Examples
Example 1
Find the surface area of a right triangular prism with base = 3 cm, height = 4 cm (triangle), hypotenuse = 5 cm, and prism length = 10 cm.
- Perimeter = 3 + 4 + 5 = 12 cm
- Area of triangle = ½ × 3 × 4 = 6 cm²
- Surface Area = (12 × 10) + (2 × 6) = 120 + 12 = 132 cm²
Example 2
If the area of each triangular base is 30 cm², perimeter is 11 cm, and prism length is 25 cm, what is the total surface area?
- Combined area of triangles = 2 × 30 = 60 cm²
- Area of rectangles = 11 × 25 = 275 cm²
- Total Surface Area = 60 + 275 = 335 cm²
Practice Problems
- A triangular prism has base sides 6 cm, 8 cm, and 10 cm. Its length is 7 cm. Find the surface area. (Use Heron's formula if needed)
- If all sides of the triangular base are 5 cm and prism length is 12 cm, what is the surface area?
- Base area = 15 cm², perimeter = 12 cm, prism length = 8 cm. Find surface area.
- The LSA of a prism is 90 cm² and the length is 5 cm. What is the perimeter of the base?
- If the height of the triangle is not given, but all sides are 7 cm, prism length is 9 cm, find the surface area.
Common Mistakes to Avoid
- Mixing up the base of the triangle and the length of the prism. Always identify which measurement is which!
- Forgetting to include both triangular bases in total surface area calculations.
- Incorrectly finding the area of the triangle – use the correct formula (½ × base × height or Heron's formula).
- Using wrong units or forgetting to label the answer in cm², m², etc.
- Confusing lateral surface area with total surface area – remember LSA excludes the bases.
Real-World Applications
The surface area of a triangular prism is used when calculating the amount of material needed for tents, aquariums, roof structures, and packaging. For example, to paint a Toblerone-shaped chocolate box or design a triangular wedge in construction, engineers and designers use this formula. At Vedantu, we teach surface area concepts with practical examples to help students connect mathematics to the world around them.
For more geometry concepts, check out our pages on Surface Area of Prism and Heron's Formula for irregular triangles. You can also practice more using this surface area worksheet (PDF).
In this topic, we learned how to use the Surface Area of a Triangular Prism Formula to solve geometry problems. Mastering this formula helps in exams and practical life. For more in-depth lessons and personalized learning, explore Vedantu’s courses and resources to excel in maths and beyond.
FAQs on Surface Area of a Triangular Prism: Formula, Steps & Examples
1. What is a triangular prism and what are its fundamental properties?
A triangular prism is a three-dimensional polyhedron with two parallel and congruent triangular bases and three rectangular faces connecting the corresponding sides of the bases. Its fundamental properties include:
- Faces: It has a total of 5 faces (2 triangular bases and 3 rectangular lateral faces).
- Vertices: It has 6 vertices (corners).
- Edges: It has 9 edges (where the faces meet).
2. What is the main formula to calculate the total surface area (TSA) of a triangular prism?
The total surface area (TSA) of a triangular prism is the sum of the areas of all its five faces. The standard formula is: Total Surface Area (TSA) = (Perimeter of the base × Length of the prism) + (2 × Area of the base). This formula neatly combines the area of the three rectangular sides and the two triangular ends.
3. What is the difference between the total surface area (TSA) and lateral surface area (LSA) of a triangular prism?
The key difference lies in which faces are being measured:
- The Total Surface Area (TSA) is the area of all five faces combined—the three rectangular sides plus the two triangular bases.
- The Lateral Surface Area (LSA) is the area of only the three rectangular faces, excluding the top and bottom triangular bases. Its formula is LSA = Perimeter of the base × Length of the prism.
4. What are the key steps to calculate the surface area of a triangular prism?
To find the surface area of a triangular prism step-by-step, follow this process:
- Step 1: Identify the dimensions of the triangular base (the lengths of its three sides a, b, c) and the length (or height) of the prism (L).
- Step 2: Calculate the perimeter of the triangular base by adding its side lengths: P = a + b + c.
- Step 3: Calculate the area of one triangular base (A). If it's a right-angled triangle, use (1/2) × base × height. If not, you may need to use Heron's formula.
- Step 4: Calculate the lateral surface area using the formula: LSA = P × L.
- Step 5: Calculate the total surface area by adding the area of the two bases to the lateral surface area: TSA = LSA + (2 × A).
5. How does the calculation change if the triangular base is a right-angled or equilateral triangle?
The main surface area formula remains the same, but calculating the base area becomes simpler.
- For a right-angled triangle, the area is easily found with the formula A = (1/2) × base × height, using the two sides that form the right angle.
- For an equilateral triangle with side length 's', the area can be calculated directly using the formula A = (√3/4) × s².
6. What are some common real-world examples of a triangular prism?
Triangular prisms are found in many real-world objects and structures. Some common examples include a camping tent, a slice of a triangular cake, the shape of a Toblerone chocolate bar, a roof gable on a house, and the glass prisms used in science to disperse light into a spectrum.
7. Why is the surface area formula expressed as (Perimeter × Length) + (2 × Base Area)?
This formula is a conceptual shortcut. Imagine you 'unroll' the three rectangular faces of the prism. They would form one large rectangle whose width is the length of the prism and whose length is the sum of the sides of the triangular base, which is the perimeter. So, (Perimeter × Length) is the lateral surface area. The term `(2 × Base Area)` is then added to account for the two triangular bases at each end.
8. How is the surface area of a triangular prism different from that of a triangular pyramid?
The primary difference comes from their structure. A triangular prism has two parallel triangular bases and three rectangular side faces. A triangular pyramid has one triangular base and three triangular side faces that meet at a single point (apex). Consequently, their surface area calculations are different:
- Prism TSA: Area of 3 rectangles + Area of 2 triangles.
- Pyramid TSA: Area of 1 base triangle + Area of 3 side triangles.
9. What is a common mistake when calculating the surface area of a prism with a scalene triangular base?
A common mistake is incorrectly calculating the area of the scalene triangular base. When only the three side lengths (a, b, c) are known and it's not a right-angled triangle, students often forget that they cannot simply multiply two sides. They must use Heron's formula, which involves first calculating the semi-perimeter (s = (a+b+c)/2) and then finding the area using the formula A = √[s(s-a)(s-b)(s-c)].
10. If all dimensions of a triangular prism are doubled, how does its total surface area change?
If every linear dimension (the sides of the base and the length of the prism) is doubled, the new total surface area will be four times the original. This is because surface area is a two-dimensional quantity (measured in square units). When linear dimensions are scaled by a factor of 'k' (in this case, k=2), the area is scaled by a factor of k², so the new area becomes 2² = 4 times larger.
11. In what practical scenarios is knowing the lateral surface area more important than the total surface area?
Knowing the lateral surface area (LSA) is more important in situations where only the sides of the prism are being considered, and the bases are excluded. For example:
- When painting the walls of a room shaped like a triangular prism, you only need to cover the walls (LSA), not the floor or ceiling.
- When manufacturing a cardboard sleeve or label to wrap around a triangular prism-shaped product, you only need material for the lateral faces.
