Math Problem Statement

Calculate the surface area of each prism. Order the prisms from greatest to least surface area. Show your work.

Solution

The image shows a triangular prism with the following dimensions:

  • Height (vertical side of the triangle): 6.1 cm
  • Base length of the triangle: 7.0 cm
  • Side length of the triangle: 5.0 cm

To calculate the surface area of the prism, we need to find the area of its two triangular bases and the area of its rectangular lateral faces. The surface area AA can be calculated using the following formula:

A=2×Area of Triangle+Perimeter of Triangle×Length of PrismA = 2 \times \text{Area of Triangle} + \text{Perimeter of Triangle} \times \text{Length of Prism}

1. Area of the Triangle:

We can use Heron's formula to calculate the area of the triangle, where the semi-perimeter ss is:

s=7.0+5.0+6.12=9.05cms = \frac{7.0 + 5.0 + 6.1}{2} = 9.05 \, \text{cm}

Now, apply Heron's formula for the area of the triangle:

Area of Triangle=s(s7.0)(s5.0)(s6.1)\text{Area of Triangle} = \sqrt{s(s - 7.0)(s - 5.0)(s - 6.1)}

2. Perimeter of the Triangle:

The perimeter PP of the triangle is:

P=7.0+5.0+6.1=18.1cmP = 7.0 + 5.0 + 6.1 = 18.1 \, \text{cm}

3. Surface Area of Prism:

Now, let's plug everything into the surface area formula:

A=2×Area of Triangle+18.1×6.1A = 2 \times \text{Area of Triangle} + 18.1 \times 6.1

Let me calculate the exact surface area for you. math

Given values

a = 7.0 # base of triangle (cm) b = 5.0 # side of triangle (cm) c = 6.1 # height of triangle (cm)

Semi-perimeter of the triangle

s = (a + b + c) / 2

Area of the triangle using Heron's formula

area_triangle = math.sqrt(s * (s - a) * (s - b) * (s - c))

Perimeter of the triangle

perimeter_triangle = a + b + c

Length of the prism (height of the prism)

length_prism = 6.1

Surface area of the prism

surface_area = 2 * area_triangle + perimeter_triangle * length_prism surface_areaThe surface area of Prism A is approximately 140.19 cm².

Would you like me to assist further with other prisms or additional details? Here are some related questions:

  1. How do you calculate the surface area for other types of prisms?
  2. What is the difference between surface area and volume of a prism?
  3. How do you apply Heron’s formula in different scenarios?
  4. How can you calculate the volume of this triangular prism?
  5. What is the effect of changing the side lengths on the surface area?

Tip: When using Heron's formula, always ensure that the semi-perimeter is calculated correctly before applying it to find the area.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Surface Area
Geometry
Heron's Formula

Formulas

Surface Area = 2 × Area of Triangle + Perimeter of Triangle × Length of Prism
Heron's Formula: Area of Triangle = √(s(s-a)(s-b)(s-c)), where s = (a + b + c) / 2

Theorems

Heron's Formula for the area of a triangle

Suitable Grade Level

Grades 7-9