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# How to Find the Surface Area of Right-Angled and Isosceles Triangular Prisms

Author:

Mark, a math enthusiast, loves writing tutorials for stumped students and those who need to brush up on their math skills.

## What Is a Prism?

A prism is a three-dimensional object whose two end faces are identical and whose sides are parallelograms (a four-sided shape with two pairs of parallel sides). The type of prism is determined by the shape of its ends. Hence, a prism with a triangle at each end is called a triangular prism. It doesn't matter if that prism is right-angled or isosceles, the way we find the surface area is the same for both types.

## How Do We Find the Surface Area?

The surface area of any prism is the total area of all its sides and faces. A triangular prism has three rectangular sides and two triangular faces. To find the area of the rectangular sides, use the formula A = lw, where A = area, l = length, and h = height. To find the area of the triangular faces, use the formula A = 1/2bh, where A = area, b = base, and h = height. Once you have the areas of all sides and faces, you simply add them together to get the surface area.

## Formulas You'll Need to Complete This Lesson

 Shape Formula Area of a triangle A = 1/2bh Area of a rectangle A = lw Surface area of triangular prism SA = bh + (s1 + s2 + s3)H

## Example 1: Find the Surface Area of the Right-Angled Triangular Prism Above

Let’s begin with the triangular faces. Both faces have the same area because they are congruent! Just multiply the base and height and divide the answer by 2:

Area of triangular faces

= 1/2(base × height)
= 1/2(3 × 4) = 6 cm²

Next work out the area of the rectangular sides. Each side is a different size, and can be calculated by multiplying the length by the width:

Area of sloping rectangular side

= length x width
= 11 x 5
= 55 cm²

Area of back side

= 11 x 3
=33 cm²

Area of bottom side

= 11 x 4
= 44 cm²

All you need to do is total all these areas:

6 + 6 + 55 + 33 + 44 = 144 cm²

So the total surface area of this triangular prism is 144 cm²

What Is the Perimeter of a Shape?

The perimeter is the total distance around a two-dimensional shape. For example, a triangle whose sides are all 3 inches long has a perimeter of 9 inches (3 + 3 + 3, or 3 x 3).

## Using a Formula to Find the Surface Area

Now that we've covered the basics, it's time to introduce a less tedious method. There is a single formula you can use to calculate the surface area of a triangular prism:

SA = bh + (s1 + s2 + s3)H

In the above formula, b = the base and h = the height of the triangle, s1, s2, and s3 = the length of each side of the triangle, and H = the prism's height (which is the same as the rectangles' length).

You might be wondering how we came up with this formula. Well, it's pretty simple. If you'll recall, the surface area is found by adding together the area of each side and face. Let's start with the two triangles on the ends. The area of each triangle is 1/2bh. Since they are both identical, we can double this formula to find both of their areas at the same time.

The area of both triangles

= 2(1/2bh)
= 2/2bh
= bh

Typically to work out the area of the three rectangular sides, you would multiply each one's length by its respective width. However, this isn't necessary because the sides of the triangles are equal to the widths of the three rectangles. Similarly, the prism's height, H, is equal to the length of each rectangle. Therefore, multiplying the height, H, of the prism (length of the rectangles) by the perimeter (the three rectangular widths) of its base will give us the area of each rectangle.

The area of the rectangular sides

= (s1 + s2 + s3)H

Therefore, the area of a triangular prism

= the area of the triangular faces + the area of the rectangular sides
= bh + (s1 +s2 +s3)H

## Example 1.1

Let's use our new formula to redo the example above!

The surface area

= bh + (s1 + s2 + s3)H
= 4(3) + (3 + 5 + 4)(11)
= 12 + 12(11)
= 12 + 132
= 144 cm2

As you can see, our answer matches the one above. Now that we know our formula works, let's put it to use in the next example.

## Example 2: Find the Surface Area of the Isosceles Triangular Prism Above

First, plug the known values into the equation.

SA = bh + (s1 + s2 + s3)H
SA = 4(6) + (4 + 7 + 7)(12)

Next, calculate the perimeter of the triangles (add together the three sides), followed by their area (base times height).

SA= 24 + 18(12)

Then, multiply the perimeter by the height of the prism.

SA = 24 + 216

SA = 240 cm2

## Example 2.1: Let's Check Our Work!

Triangular Face (TF1)TF2Rectangular Side 1 (RS1)RS2Rectangular BaseTotal

A = 1/2bh

A = 1/2bh

A = lw

A = lw

A = lw

A = 1/2(4 x 6)

A = 1/2(4 x 6)

A = 12(7)

A = 12(7)

A = 12(4)

A = 12

A = 12

A = 84

A = 84

A = 48

12 +

12 +

84 +

84 +

48 =

240 cm^2

## Review Questions

I. Use the diagram below to solve the following problems.

1. Alan wants to surprise his sister with a giant Toblerone for passing her math class (Fig. 1). Alan needs to know the surface area of the Toblerone to buy the right amount of wrapping paper. What is its surface area?
2. John just bought a brand new roof for his shed. Unfortunately, he hates that it's neon green. He would like to repaint his roof but doesn't know how much paint he should buy. He is on a pretty tight budget. Using the image above (Fig. 2), find the surface area of the roof (including the bottom).
3. Jackie wants to build a tent for her daughter. She has already constructed its frame but does not know how much fabric she needs to cover it. Find the surface area of the tent (Fig. 3) using the image above.
4. Katie's boss wants her to purchase concrete for the ramp that they are building. He gave her the blueprints, but she is still stumped. Find the surface area of the image above (Fig. 4) so Katie doesn't lose her job.

II. Find the surface area of the following:

1. A prism whose triangular ends have a height of 6 inches with a 4-inch base and each rectangular side is 5 inches long and 6 inches wide.
2. A prism whose triangular ends have a height of 10 meters with a 5-meter base and each rectangular side is 4 meters long and 10 meters wide.
3. A prism whose triangular ends have a height of 10 inches with a 15-inch base and each rectangular side is 12 inches long and 10 inches wide.
4. A prism whose triangular ends have a height of 6 meters with an 8-meter base and each rectangular side is 15 meters long and 6 meters wide.

Section I

1. 3,702 cm2
2. 62 ft2
3. 158 ft2
4. 60 m2

Section II

1. 114 in2
2. 170 m2
3. 510 in2
4. 318 m2

Question: What is the formula for finding the Total Surface Area of a Prism?

Answer: It depends on the type of prism, so there isn't one formula which works for all.

Question: How do you find the surface area of the right triangular prism with two numbers?

Answer: You may need to apply Pythagoras on the triangular face to work out a missing side length if you are only given two lengths to start off with.

Question: How would you work out the surface area of a scalene triangle, and what if that is a prism?

Answer: It's very similar to the right-angled triangular prism. Work out the area of the two triangles at either end and then add on the area of the three rectangles around the middle.

Question: The base length of the triangular face is 5cm, the perpendicular height is 2.4cm and the length of the prism is 7, how to calculate the surface area of that triangular prism?

Answer: The area of the triangular face is 5 times 2.4 divided by 2 which is 6cm^2.

The area of the triangular face at the back of the prism is also 6cm^2.

The area of the rectangular bottom face is 5 times 7 which is 35cm^2.

The area of the rectangular vertical face is 2.4 times 7 which is 16.8 cm^2.

Before you can work out the rectangular sloping face apply Pythagoras to give the other side length which will be 5.5cm

So the sloping rectangular face will be 5.5 times 7 which is 38.5 cm^2.

Adding up these areas will give a final answer of 102.3 cm^2.

Question: How do you work out the surface area for a right-angled triangular prism?

Answer: Work out the area of the triangles at the front and back of the prism, using 1/2 times base times height.

(These triangles will have the same area).

Next work out the area of the 3 rectangular faces of the prism using length times width for each rectangle.

Now add up the 5 areas to give the surface area of the triangular prism.

Question: How do I find the total surface area of a cube?

Answer: Work out the area of one of the square faces (length times width).

Then multiply this answer by 6, as there are 6 square faces which make the cube.