How do you find the lateral and surface area of a right triangular right prism. Base 8in by 6in by 10in and height 9 in.
The lateral area is equal to the perimeter of the base times height.
add the area of the base, and you have surface area.
To find the lateral and surface area of a right triangular prism, you need to know the dimensions of its base and its height.
The base of the right triangular prism you mentioned has dimensions of 8 inches by 6 inches by 10 inches.
Let's start by finding the lateral area of the prism:
1. Determine the perimeter of the triangular base:
- The base is a right triangle, so you can find the perimeter by adding the lengths of all three sides. In this case, the two shorter sides are 8 inches and 6 inches, and the longest side is the hypotenuse, which we can find using the Pythagorean theorem:
- Hypotenuse^2 = (8^2) + (6^2)
- Hypotenuse^2 = 64 + 36
- Hypotenuse^2 = 100
- Hypotenuse = √100
- Hypotenuse = 10 inches
- The perimeter is the sum of the three sides:
- Perimeter = 8 + 6 + 10
- Perimeter = 24 inches
2. Calculate the lateral area:
- The lateral area of the prism is equal to the perimeter of the base multiplied by the height of the prism:
- Lateral Area = Perimeter × Height
- Lateral Area = 24 × 9
- Lateral Area = 216 square inches
Now, let's calculate the surface area of the prism:
1. Determine the area of each face:
- The triangular base has an area of (1/2) × base × height:
- Base Area = (1/2) × 8 × 6
- Base Area = 24 square inches
- The two rectangular faces have areas of length × width:
- Side 1 Area = 8 × 9
- Side 1 Area = 72 square inches
- Side 2 Area = 6 × 9
- Side 2 Area = 54 square inches
2. Calculate the surface area:
- The surface area of the prism is the sum of the areas of all its faces:
- Surface Area = 2 × Base Area + 2 × Side 1 Area + 2 × Side 2 Area
- Surface Area = 2 × 24 + 2 × 72 + 2 × 54
- Surface Area = 48 + 144 + 108
- Surface Area = 300 square inches
Therefore, the lateral area of the right triangular right prism is 216 square inches, and the surface area is 300 square inches.
To find the lateral and surface area of a right triangular prism, you can follow these steps:
1. Calculate the lateral area:
Since the lateral area only includes the area of the sides, you can find it by finding the perimeter of the triangular base and multiplying it by the prism's height.
a) Find the perimeter of the triangular base:
The triangular base has sides measuring 8 inches, 6 inches, and 10 inches. The perimeter is the sum of these sides.
Perimeter = 8 + 6 + 10 = 24 inches
b) Calculate the lateral area:
Lateral area of a prism = Perimeter of the base × Height
Lateral area = 24 inches × 9 inches = 216 square inches
2. Calculate the surface area:
The surface area includes the lateral area as well as the area of the two triangular bases.
a) The area of each triangle is calculated as:
Area of a triangle = 0.5 × base × height
Area of Triangle 1 = 0.5 × 8 inches × 6 inches = 24 square inches
Area of Triangle 2 = 0.5 × 10 inches × 6 inches = 30 square inches
b) Calculate the surface area:
Surface area = 2 × Area of Triangle + Lateral Area
Surface area = 2 × (24 square inches + 30 square inches) + 216 square inches
Surface area = 2 × 54 square inches + 216 square inches
Surface area = 108 square inches + 216 square inches
Surface area = 324 square inches
Therefore, the lateral area of the right triangular right prism is 216 square inches, and the surface area is 324 square inches.