Problem Analysis:
The image shows a triangular prism. The task is likely to calculate either the
surface area or the
volume of the prism. Let's break it down step by step.
#### Key Dimensions from the Image:
1. The base of the triangular prism is a triangle with:
- One side = 20 cm
- Another side = 29 cm
- Height of the triangle (perpendicular to the base of 20 cm) = 21 cm
2. The slanted height of the triangular face is also given as 29 cm.
3. The length of the prism (height of the prism) = 31 cm.
#### Objective:
We will calculate both the
surface area and the
volume of the prism, as these are common tasks for such problems.
---
Step 1: Calculate the Area of the Triangular Base
The base of the prism is a triangle with:
- Base = 20 cm
- Height = 21 cm
The area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substitute the values:
\[
A = \frac{1}{2} \times 20 \times 21 = 210 \, \text{cm}^2
\]
So, the area of one triangular base is \( 210 \, \text{cm}^2 \).
Since the prism has two identical triangular bases, the total area of the two bases is:
\[
2 \times 210 = 420 \, \text{cm}^2
\]
---
Step 2: Calculate the Perimeter of the Triangular Base
To find the surface area of the rectangular faces, we need the perimeter of the triangular base. The sides of the triangle are:
- 20 cm
- 29 cm
- 29 cm (slanted height)
The perimeter \( P \) of the triangle is:
\[
P = 20 + 29 + 29 = 78 \, \text{cm}
\]
---
Step 3: Calculate the Surface Area of the Rectangular Faces
The prism has three rectangular faces, each with a height equal to the length of the prism (31 cm) and a width equal to one of the sides of the triangular base.
The areas of the three rectangular faces are:
1. Rectangle with width 20 cm:
\[
\text{Area} = 20 \times 31 = 620 \, \text{cm}^2
\]
2. Rectangle with width 29 cm (one side):
\[
\text{Area} = 29 \times 31 = 899 \, \text{cm}^2
\]
3. Rectangle with width 29 cm (other side):
\[
\text{Area} = 29 \times 31 = 899 \, \text{cm}^2
\]
The total area of the three rectangular faces is:
\[
620 + 899 + 899 = 2418 \, \text{cm}^2
\]
---
Step 4: Calculate the Total Surface Area
The total surface area \( S \) of the prism is the sum of the areas of the two triangular bases and the three rectangular faces:
\[
S = 420 + 2418 = 2838 \, \text{cm}^2
\]
---
Step 5: Calculate the Volume of the Prism
The volume \( V \) of a prism is given by:
\[
V = \text{Base Area} \times \text{Height of the Prism}
\]
The base area is \( 210 \, \text{cm}^2 \) and the height of the prism is \( 31 \, \text{cm} \):
\[
V = 210 \times 31 = 6510 \, \text{cm}^3
\]
---
Final Answers:
\[
\boxed{2838 \, \text{cm}^2 \text{ and } 6510 \, \text{cm}^3}
\]
Parent Tip: Review the logic above to help your child master the concept of surface area of a triangular prism worksheet with answers.
