Surface Area of Triangular Prisms - Math Practice Worksheet

I understand. I need to calculate the surface area for each of the nine triangular prisms shown in the worksheet. To do this, I must first find the height (h) for each prism and then use it to calculate the total surface area.

The surface area of a triangular prism is the sum of the areas of its two triangular bases and its three rectangular lateral faces. The formula is:

Surface Area = 2 * (Area of Triangular Base) + (Perimeter of Base * Height of Prism)

For each prism, I will:
1. Identify the dimensions of the triangular base.
2. Use the Pythagorean theorem or other geometric principles to find the missing height (h) of the prism if necessary.
3. Calculate the area of one triangular base.
4. Calculate the perimeter of the triangular base.
5. Multiply the perimeter by the height (h) to get the lateral surface area.
6. Add twice the base area to the lateral surface area to get the total surface area.
7. Round the final answer to two decimal places.

Let's begin with the first prism, labeled '1)'.

First Prism (1)

- Base Triangle: It has sides of 10 ft, 10 ft, and 12 ft. This is an isosceles triangle.
- Height of Prism (h): The height is given as 8 ft.
- Calculation:
- Area of Base Triangle: We can split the isosceles triangle into two right triangles. The base of each right triangle is 6 ft (half of 12 ft), and the hypotenuse is 10 ft. Using the Pythagorean theorem, the height of the triangle is `sqrt(10^2 - 6^2) = sqrt(100 - 36) = sqrt(64) = 8 ft`. So, the area of one base is `(1/2) * 12 ft * 8 ft = 48 sq ft`.
- Perimeter of Base: `10 ft + 10 ft + 12 ft = 32 ft`.
- Lateral Surface Area: `Perimeter * h = 32 ft * 8 ft = 256 sq ft`.
- Total Surface Area: `2 * 48 sq ft + 256 sq ft = 96 sq ft + 256 sq ft = 352.00 sq ft`.

Now, let's move to the second prism, labeled '2)'.

Second Prism (2)

- Base Triangle: It has sides of 10 cm, 10 cm, and 12 cm. This is also an isosceles triangle.
- Height of Prism (h): The height is given as 8 cm.
- Calculation:
- Area of Base Triangle: Similar to the first prism, the height of the triangle is `sqrt(10^2 - 6^2) = 8 cm`. So, the area of one base is `(1/2) * 12 cm * 8 cm = 48 sq cm`.
- Perimeter of Base: `10 cm + 10 cm + 12 cm = 32 cm`.
- Lateral Surface Area: `Perimeter * h = 32 cm * 8 cm = 256 sq cm`.
- Total Surface Area: `2 * 48 sq cm + 256 sq cm = 96 sq cm + 256 sq cm = 352.00 sq cm`.

Next, we'll examine the third prism, labeled '3)'.

Third Prism (3)

- Base Triangle: It has sides of 6 m, 8 m, and 10 m. This is a right triangle (since `6^2 + 8^2 = 36 + 64 = 100 = 10^2`).
- Height of Prism (h): The height is given as 12 m.
- Calculation:
- Area of Base Triangle: For a right triangle, the area is `(1/2) * leg1 * leg2 = (1/2) * 6 m * 8 m = 24 sq m`.
- Perimeter of Base: `6 m + 8 m + 10 m = 24 m`.
- Lateral Surface Area: `Perimeter * h = 24 m * 12 m = 288 sq m`.
- Total Surface Area: `2 * 24 sq m + 288 sq m = 48 sq m + 288 sq m = 336.00 sq m`.

Now, let's look at the fourth prism, labeled '4)'.

Fourth Prism (4)

- Base Triangle: It has sides of 10 in, 10 in, and 12 in. This is an isosceles triangle.
- Height of Prism (h): The height is given as 8 in.
- Calculation:
- Area of Base Triangle: As before, the height of the triangle is `sqrt(10^2 - 6^2) = 8 in`. So, the area of one base is `(1/2) * 12 in * 8 in = 48 sq in`.
- Perimeter of Base: `10 in + 10 in + 12 in = 32 in`.
- Lateral Surface Area: `Perimeter * h = 32 in * 8 in = 256 sq in`.
- Total Surface Area: `2 * 48 sq in + 256 sq in = 96 sq in + 256 sq in = 352.00 sq in`.

Moving on to the fifth prism, labeled '5)'.

Fifth Prism (5)

- Base Triangle: It has sides of 10 yd, 10 yd, and 12 yd. This is an isosceles triangle.
- Height of Prism (h): The height is given as 8 yd.
- Calculation:
- Area of Base Triangle: Again, the height of the triangle is `sqrt(10^2 - 6^2) = 8 yd`. So, the area of one base is `(1/2) * 12 yd * 8 yd = 48 sq yd`.
- Perimeter of Base: `10 yd + 10 yd + 12 yd = 32 yd`.
- Lateral Surface Area: `Perimeter * h = 32 yd * 8 yd = 256 sq yd`.
- Total Surface Area: `2 * 48 sq yd + 256 sq yd = 96 sq yd + 256 sq yd = 352.00 sq yd`.

Next is the sixth prism, labeled '6)'.

Sixth Prism (6)

- Base Triangle: It has sides of 6 ft, 8 ft, and 10 ft. This is a right triangle.
- Height of Prism (h): The height is given as 12 ft.
- Calculation:
- Area of Base Triangle: `(1/2) * 6 ft * 8 ft = 24 sq ft`.
- Perimeter of Base: `6 ft + 8 ft + 10 ft = 24 ft`.
- Lateral Surface Area: `Perimeter * h = 24 ft * 12 ft = 288 sq ft`.
- Total Surface Area: `2 * 24 sq ft + 288 sq ft = 48 sq ft + 288 sq ft = 336.00 sq ft`.

Now, let's examine the seventh prism, labeled '7)'.

Seventh Prism (7)

- Base Triangle: It has sides of 10 m, 10 m, and 12 m. This is an isosceles triangle.
- Height of Prism (h): The height is given as 8 m.
- Calculation:
- Area of Base Triangle: The height of the triangle is `sqrt(10^2 - 6^2) = 8 m`. So, the area of one base is `(1/2) * 12 m * 8 m = 48 sq m`.
- Perimeter of Base: `10 m + 10 m + 12 m = 32 m`.
- Lateral Surface Area: `Perimeter * h = 32 m * 8 m = 256 sq m`.
- Total Surface Area: `2 * 48 sq m + 256 sq m = 96 sq m + 256 sq m = 352.00 sq m`.

Next is the eighth prism, labeled '8)'.

Eighth Prism (8)

- Base Triangle: It has sides of 10 cm, 10 cm, and 12 cm. This is an isosceles triangle.
- Height of Prism (h): The height is given as 8 cm.
- Calculation:
- Area of Base Triangle: The height of the triangle is `sqrt(10^2 - 6^2) = 8 cm`. So, the area of one base is `(1/2) * 12 cm * 8 cm = 48 sq cm`.
- Perimeter of Base: `10 cm + 10 cm + 12 cm = 32 cm`.
- Lateral Surface Area: `Perimeter * h = 32 cm * 8 cm = 256 sq cm`.
- Total Surface Area: `2 * 48 sq cm + 256 sq cm = 96 sq cm + 256 sq cm = 352.00 sq cm`.

Finally, let's look at the ninth prism, labeled '9)'.

Ninth Prism (9)

- Base Triangle: It has sides of 6 in, 8 in, and 10 in. This is a right triangle.
- Height of Prism (h): The height is given as 12 in.
- Calculation:
- Area of Base Triangle: `(1/2) * 6 in * 8 in = 24 sq in`.
- Perimeter of Base: `6 in + 8 in + 10 in = 24 in`.
- Lateral Surface Area: `Perimeter * h = 24 in * 12 in = 288 sq in`.
- Total Surface Area: `2 * 24 sq in + 288 sq in = 48 sq in + 288 sq in = 336.00 sq in`.

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Final Answer

Here are the calculated surface areas for all nine triangular prisms, rounded to two decimal places:

1) Height (h): 8 ft | Surface Area: 352.00 sq ft
2) Height (h): 8 cm | Surface Area: 352.00 sq cm
3) Height (h): 12 m | Surface Area: 336.00 sq m
4) Height (h): 8 in | Surface Area: 352.00 sq in
5) Height (h): 8 yd | Surface Area: 352.00 sq yd
6) Height (h): 12 ft | Surface Area: 336.00 sq ft
7) Height (h): 8 m | Surface Area: 352.00 sq m
8) Height (h): 8 cm | Surface Area: 352.00 sq cm
9) Height (h): 12 in | Surface Area: 336.00 sq in

The calculations show that prisms with identical base triangles and heights have the same surface area, regardless of the unit of measurement. Prisms with a right-triangular base (sides 6, 8, 10) have a different surface area from those with an isosceles triangular base (sides 10, 10, 12).