Math Problem Statement

Find the surface area and volume of the triangular prism. The dimensions are 4.0 cm, 2.4 cm, 2.0 cm, and 3.2 cm.

Solution

The solid in the image appears to be a triangular prism. To find both the surface area and the volume of this prism, we need to follow specific formulas based on the dimensions provided.

1. Surface Area of the Triangular Prism:

The surface area AA of a triangular prism can be calculated using the following formula:

A=2×Base Area+Lateral AreaA = 2 \times \text{Base Area} + \text{Lateral Area}

  • The base of the prism is a triangle, and its area can be calculated as:

Base Area=12×base×height\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} where base = 3.2 cm and height = 2.4 cm.

Base Area=12×3.2×2.4=3.84cm2\text{Base Area} = \frac{1}{2} \times 3.2 \times 2.4 = 3.84 \, \text{cm}^2

  • The lateral area is the sum of the areas of the three rectangular faces. Each rectangular face has a width equal to the height (or slant) of the prism and a length equal to one of the triangle's sides:

    • First side: 3.2cm3.2 \, \text{cm}
    • Second side: 2.0cm2.0 \, \text{cm}
    • Third side: 4.0cm4.0 \, \text{cm}

    So, the lateral area is:

Lateral Area=(3.2×4)+(2.0×4)+(4.0×4)=12.8+8+16=36.8cm2\text{Lateral Area} = (3.2 \times 4) + (2.0 \times 4) + (4.0 \times 4) = 12.8 + 8 + 16 = 36.8 \, \text{cm}^2

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Math Problem Analysis

Mathematical Concepts

Geometry
Surface Area
Volume
Triangular Prism

Formulas

Surface area of a triangular prism: A = 2 * Base Area + Lateral Area
Base Area of a triangle: (1/2) * base * height
Lateral Area: Sum of areas of three rectangular faces
Volume of a triangular prism: Base Area * height of prism

Theorems

Area formula for triangles
Surface area and volume formulas for prisms

Suitable Grade Level

Grades 6-8