Surface Area of a Triangular Prism Calculator

Published on: July 28, 2024
Final Answer: Free • Full Steps: Plus

This Surface Area of a Triangular Prism Calculator finds the total area of the outside of a right triangular prism from the base and height of its triangular end and the length of the prism, and shows the working line by line. The sloping side of the triangular end is found first, then the three sides are added and multiplied by the length, and the two triangular ends are added on. Following the same pattern every time is what makes a prism easy to check and repeat.

Triangular prism labeled b, h, ℓ

In the diagram, the triangular end has legs \(b\) and \(h\), and the prism extends by \(\ell\).

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the base and the height under the square root.
  4. Work out the sloping side of the triangular end.
  5. Substitute every length into the formula.
  6. Add the three side lengths inside the brackets.
  7. Work out the two products.
  8. Add the two products to get the surface area.

Formula:

\(SA = (b + h + \sqrt{b^{2} + h^{2}})\ell + bh\)
See Example 1 Hide Example 1

Problem

\(b = 9, h = 12, \ell = 7\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the base and the height under the square root.
  4. Work out the sloping side of the triangular end.
  5. Substitute every length into the formula.
  6. Add the three side lengths inside the brackets.
  7. Work out the two products.
  8. Add the two products to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a right triangular prism whose triangular end has base \(b = 9\) and height \(h = 12\), and whose length is \(\ell = 7\).

\(b = 9,\; h = 12,\; \ell = 7\)

Step 2

Step 2 - Write the formula.

In this problem: The prism is covered by two triangular ends and three rectangular side faces, so the three side lengths of the triangle are added, multiplied by the length, and the two triangular ends are added on.

\(SA = (b + h + \sqrt{b^{2} + h^{2}})\ell + bh\)

Step 3

Step 3 - Substitute the base and the height under the square root.

In this problem: The third side of the triangular end is the sloping one, and it is the square root of the two known sides squared and added, so replacing \(b\) with \(9\) and \(h\) with \(12\) gives \(c = \sqrt{9^{2} + 12^{2}}\).

\(c = \sqrt{9^{2} + 12^{2}}\)

Step 4

Step 4 - Work out the sloping side of the triangular end.

In this problem: \(9^{2} = 81\) and \(12^{2} = 144\), then \(81 + 144 = 225\), and \(\sqrt{225} = 15\).

\(c = \sqrt{9^{2} + 12^{2}} = 15\)

Step 5

Step 5 - Substitute every length into the formula.

In this problem: Replacing \(b\), \(h\), the sloping side and \(\ell\) in \(SA = (b + h + \sqrt{b^{2} + h^{2}})\ell + bh\) gives \(SA = \left(9 + 12 + 15\right) \times 7 + 9 \times 12\).

\(SA = \left(9 + 12 + 15\right) \times 7 + 9 \times 12\)

Step 6

Step 6 - Add the three side lengths inside the brackets.

In this problem: Adding the three side lengths of the triangular end gives \(9 + 12 + 15 = 36\).

\(SA = 36 \times 7 + 9 \times 12\)

Step 7

Step 7 - Work out the two products.

In this problem: Two products are still to be worked out, so each one is taken in turn.

\(SA = 36 \times 7 + 9 \times 12\)

Step 7a

Step 7a - Work out the two products.

In this problem: Multiplying out the three side faces gives \(36 \times 7 = 252\).

\(SA = 252 + 9 \times 12\)

Step 7b

Step 7b - Work out the two products.

In this problem: Multiplying out the two triangular ends gives \(9 \times 12 = 108\).

\(SA = 252 + 108\)

Step 8

Step 8 - Add the two products to get the surface area.

In this problem: Adding the side faces to the triangular ends gives \(252 + 108 = 360\).

\(SA = 252 + 108 = 360\)

Final Answer

\(SA = 360\)
See Example 2 Hide Example 2

Problem

\(b = 5, h = 12, \ell = 3\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the base and the height under the square root.
  4. Work out the sloping side of the triangular end.
  5. Substitute every length into the formula.
  6. Add the three side lengths inside the brackets.
  7. Work out the two products.
  8. Add the two products to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a right triangular prism whose triangular end has base \(b = 5\) and height \(h = 12\), and whose length is \(\ell = 3\).

\(b = 5,\; h = 12,\; \ell = 3\)

Step 2

Step 2 - Write the formula.

In this problem: The prism is covered by two triangular ends and three rectangular side faces, so the three side lengths of the triangle are added, multiplied by the length, and the two triangular ends are added on.

\(SA = (b + h + \sqrt{b^{2} + h^{2}})\ell + bh\)

Step 3

Step 3 - Substitute the base and the height under the square root.

In this problem: The third side of the triangular end is the sloping one, and it is the square root of the two known sides squared and added, so replacing \(b\) with \(5\) and \(h\) with \(12\) gives \(c = \sqrt{5^{2} + 12^{2}}\).

\(c = \sqrt{5^{2} + 12^{2}}\)

Step 4

Step 4 - Work out the sloping side of the triangular end.

In this problem: \(5^{2} = 25\) and \(12^{2} = 144\), then \(25 + 144 = 169\), and \(\sqrt{169} = 13\).

\(c = \sqrt{5^{2} + 12^{2}} = 13\)

Step 5

Step 5 - Substitute every length into the formula.

In this problem: Replacing \(b\), \(h\), the sloping side and \(\ell\) in \(SA = (b + h + \sqrt{b^{2} + h^{2}})\ell + bh\) gives \(SA = \left(5 + 12 + 13\right) \times 3 + 5 \times 12\).

\(SA = \left(5 + 12 + 13\right) \times 3 + 5 \times 12\)

Step 6

Step 6 - Add the three side lengths inside the brackets.

In this problem: Adding the three side lengths of the triangular end gives \(5 + 12 + 13 = 30\).

\(SA = 30 \times 3 + 5 \times 12\)

Step 7

Step 7 - Work out the two products.

In this problem: Two products are still to be worked out, so each one is taken in turn.

\(SA = 30 \times 3 + 5 \times 12\)

Step 7a

Step 7a - Work out the two products.

In this problem: Multiplying out the three side faces gives \(30 \times 3 = 90\).

\(SA = 90 + 5 \times 12\)

Step 7b

Step 7b - Work out the two products.

In this problem: Multiplying out the two triangular ends gives \(5 \times 12 = 60\).

\(SA = 90 + 60\)

Step 8

Step 8 - Add the two products to get the surface area.

In this problem: Adding the side faces to the triangular ends gives \(90 + 60 = 150\).

\(SA = 90 + 60 = 150\)

Final Answer

\(SA = 150\)

Example Problems

Click one to load it into the box and solve it.