A solid angle cannot be constructed from only two planes. Therefore at least three faces need to come together to form a vertex. Let $P$ be a platonic solid. Let the polygon which forms each face of $P$ be a equilateral triangles. We have that: :each vertex of a regular tetrahedron is composed of $3$ equilateral triangles :each vertex of a regular octahedron is composed of $4$ equilateral triangles :each vertex of a regular icosahedron is composed of $5$ equilateral triangles. $6$ equilateral triangles, placed together at a vertex, form $4$ right angles. From {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}: :a solid angle is contained by plane angles which total less than $4$ right angles. Thus it is not possible to form $P$ such that its vertices are formed by $6$ equilateral triangles. For the same reason, it is not possible to form $P$ such that its vertices are formed by more than $6$ equilateral triangles. Hence there are only $3$ possible platonic solids whose faces are equilateral triangles. We have that each vertex of a cube is composed of $3$ squares. $4$ squares, placed together at a vertex, form $4$ right angles. Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}: :it is not possible to form $P$ such that its vertices are formed by $4$ squares. For the same reason, it is not possible to form $P$ such that its vertices are formed by more than $4$ squares. Hence there is only $1$ possible platonic solid whose faces are squares. We have that each vertex of a regular dodecahedron is composed of $3$ regular pentagons. From {{EuclidPropLink|book = XIII|prop = 18|qual=Lemma|title = Comparison of Sides of Five Platonic Figures}}: :the vertices of a regular pentagon equal $1 \dfrac 1 5$ right angles. $4$ regular pentagons, placed together at a vertex, form $4 \dfrac 4 5$ right angles. Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}: :it is not possible to form $P$ such that its vertices are formed by $4$ regular pentagons. For the same reason, it is not possible to form $P$ such that its vertices are formed by more than $4$ regular pentagons. Hence there is only $1$ possible platonic solid whose faces are regular pentagons. $3$ regular hexagons, placed together at a vertex, form $4$ right angles. Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}: :it is not possible to form $P$ such that its vertices are formed by $3$ or more regular hexagons. Regular polygons with more than $6$ sides have vertices which are greater than those of a regular hexagon. Therefore $3$ such regular polygons, placed together at a vertex, form more than $4$ right angles. Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}: :it is not possible to form $P$ such that its vertices are formed by $3$ or more regular polygons with more than $6$ sides. Hence the $5$ possible platonic solids have been enumerated and described. {{qed}}