Question
Jack bought a jewelry box. He wants to paint all 5 sides of the jewelry box. How much paint does he need?
Do you need to find the surface area or volume of the prism in order to solve this problem?
Right triangular prism. Base of the triangular base measures 6 in, its height measures 8 in and the hypotenuse measures 10 in. The height of the prism measures 12 in.
Volume
Surface Area
Nice!
This jewelry box is in the shape of a prism. One way you can find the surface area is by adding the area of the 2 bases, and the area of the long rectangle along the edges of the bases. Start by finding the area of the base on paper, then record your answer on Zearn. You can use the calculator if you need to.
Right triangular prism. Base of the triangular base measures 6 in, its height measures 8 in and the hypotenuse measures 10 in. The height of the prism measures 12 in.
The area of one base is
in2.
Do you need to find the surface area or volume of the prism in order to solve this problem?
Right triangular prism. Base of the triangular base measures 6 in, its height measures 8 in and the hypotenuse measures 10 in. The height of the prism measures 12 in.
Volume
Surface Area
Nice!
This jewelry box is in the shape of a prism. One way you can find the surface area is by adding the area of the 2 bases, and the area of the long rectangle along the edges of the bases. Start by finding the area of the base on paper, then record your answer on Zearn. You can use the calculator if you need to.
Right triangular prism. Base of the triangular base measures 6 in, its height measures 8 in and the hypotenuse measures 10 in. The height of the prism measures 12 in.
The area of one base is
in2.
Answers
GPT-4o mini
To find out how much paint Jack needs for the jewelry box in the shape of a right triangular prism, we will need to calculate the surface area rather than the volume since we're interested in covering the surface with paint.
### Step 1: Calculate the area of the triangular base.
The base of the triangular prism is a right triangle with:
- Base = 6 in
- Height = 8 in
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Plugging in the values:
\[
A = \frac{1}{2} \times 6 \, \text{in} \times 8 \, \text{in} = \frac{1}{2} \times 48 \, \text{in}^2 = 24 \, \text{in}^2
\]
So, the area of one triangular base is \( 24 \, \text{in}^2 \).
### Step 2: Find the surface area of the prism.
A right triangular prism has two triangular bases and three rectangular lateral faces.
- **Area of the two triangular bases:**
\[
\text{Area of two bases} = 2 \times 24 \, \text{in}^2 = 48 \, \text{in}^2
\]
- **Area of the three rectangular lateral faces:**
- The dimensions of the rectangle along the base (6 in) is the height of the prism (12 in):
\[
\text{Area}_{\text{base}} = 6 \, \text{in} \times 12 \, \text{in} = 72 \, \text{in}^2
\]
- The dimensions of the rectangle along the height (8 in) is the height of the prism (12 in):
\[
\text{Area}_{\text{height}} = 8 \, \text{in} \times 12 \, \text{in} = 96 \, \text{in}^2
\]
- The dimensions of the rectangle along the hypotenuse (10 in) is the height of the prism (12 in):
\[
\text{Area}_{\text{hypotenuse}} = 10 \, \text{in} \times 12 \, \text{in} = 120 \, \text{in}^2
\]
- **Total area of the rectangular faces:**
\[
\text{Total rectangular area} = 72 \, \text{in}^2 + 96 \, \text{in}^2 + 120 \, \text{in}^2 = 288 \, \text{in}^2
\]
### Step 3: Calculate the total surface area of the prism.
\[
\text{Total Surface Area} = \text{Area of the bases} + \text{Area of the rectangular faces}
\]
\[
\text{Total Surface Area} = 48 \, \text{in}^2 + 288 \, \text{in}^2 = 336 \, \text{in}^2
\]
Jack will need to paint **336 square inches** of the jewelry box.
### Step 1: Calculate the area of the triangular base.
The base of the triangular prism is a right triangle with:
- Base = 6 in
- Height = 8 in
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Plugging in the values:
\[
A = \frac{1}{2} \times 6 \, \text{in} \times 8 \, \text{in} = \frac{1}{2} \times 48 \, \text{in}^2 = 24 \, \text{in}^2
\]
So, the area of one triangular base is \( 24 \, \text{in}^2 \).
### Step 2: Find the surface area of the prism.
A right triangular prism has two triangular bases and three rectangular lateral faces.
- **Area of the two triangular bases:**
\[
\text{Area of two bases} = 2 \times 24 \, \text{in}^2 = 48 \, \text{in}^2
\]
- **Area of the three rectangular lateral faces:**
- The dimensions of the rectangle along the base (6 in) is the height of the prism (12 in):
\[
\text{Area}_{\text{base}} = 6 \, \text{in} \times 12 \, \text{in} = 72 \, \text{in}^2
\]
- The dimensions of the rectangle along the height (8 in) is the height of the prism (12 in):
\[
\text{Area}_{\text{height}} = 8 \, \text{in} \times 12 \, \text{in} = 96 \, \text{in}^2
\]
- The dimensions of the rectangle along the hypotenuse (10 in) is the height of the prism (12 in):
\[
\text{Area}_{\text{hypotenuse}} = 10 \, \text{in} \times 12 \, \text{in} = 120 \, \text{in}^2
\]
- **Total area of the rectangular faces:**
\[
\text{Total rectangular area} = 72 \, \text{in}^2 + 96 \, \text{in}^2 + 120 \, \text{in}^2 = 288 \, \text{in}^2
\]
### Step 3: Calculate the total surface area of the prism.
\[
\text{Total Surface Area} = \text{Area of the bases} + \text{Area of the rectangular faces}
\]
\[
\text{Total Surface Area} = 48 \, \text{in}^2 + 288 \, \text{in}^2 = 336 \, \text{in}^2
\]
Jack will need to paint **336 square inches** of the jewelry box.