Library
Mathematics

Euler's Formula and Surface Area of Prisms

This lesson explains Euler's formula for polyhedra, which relates the number of faces, vertices, and edges, and walks through finding the number of faces given edges and vertices. Then, it explains how to calculate the total surface area of a prism.

A Simple Polyhedron Vertex (Corner) Face (Flat surface) Edge (Straight line)
A polyhedron is simply a 3D shape with flat polygonal faces, straight edges, and sharp corners called vertices. Think of a box or a pyramid.
Step-by-step solver
1

Understand Euler's Formula

Euler's formula states that for any convex polyhedron, the number of faces (F), vertices (V), and edges (E) are related by the equation F + V - E = 2.

F+V−E=2F + V - E = 2
2

Substitute known values

We are given that the polyhedron has 12 edges (E = 12) and 6 vertices (V = 6). Plug these into the formula.

F+6−12=2F + 6 - 12 = 2
3

Solve for Faces (F)

Simplify the equation: 6 - 12 is -6. Then add 6 to both sides to find F.

F−6=2  ⟹  F=8F - 6 = 2 \implies F = 8
4

Understand Surface Area of a Prism

The total surface area of a prism is the sum of the areas of all its faces. This consists of two identical bases and a number of rectangular lateral faces.

SA=2B+LSA = 2B + L
5

Express Lateral Area

The lateral area (L) can be found by multiplying the perimeter of the base (P) by the height of the prism (h).

SA=2B+PhSA = 2B + Ph

Original question

1)A polyhedron ha s12 edges and 6 vertices. How many faces does it have? 2)total surface area of each prism

Follow-up chat

Ask me anything about this lesson. I'll answer using what we just covered.