texturing a procedurally created mesh

for a top down 2d project i made a script that creates a cone shaped mesh to represent a weapon's cone of fire.

the cone is procedurally made using some values of the weapon such as it's range and spread angle.

now i'm attemting to throw a texture on this cone, but im having trouble understanding how to actually do this.

all i need is for a square texture to appear on the cone surface while the parts of the texture that are outside the shape bounds not show.

can i avoid UV editing somehow? if not how would i go about it considering the procedural nature of this?

here is the mesh script

 using UnityEngine;
 using System.Collections.Generic;
 
 [RequireComponent(typeof(MeshFilter))]
 public class WeaponConeMesh : MonoBehaviour {
 
     Mesh mesh;
     MeshRenderer meshRenderer;
     Transform weapon;
 
     float range;
     float weaponAngle;
     float coneAngle;
     Vector3 projOrigin;
 
     int numOfTriangles = 10;
     List<Vector3> verticesList = new List<Vector3> ();
     List<int> trianglesList = new List<int> ();
 
     void Awake() {
         mesh = GetComponent<MeshFilter>().mesh;
         meshRenderer = GetComponent<MeshRenderer>();
         weapon = transform.parent;
     }
 
     void Start() {
         WeaponHandler weaponHandler = weapon.GetComponent<WeaponHandler>();
         range = weaponHandler.range;
         projOrigin = weaponHandler.projOrigin;
         if (weaponHandler.angle <= 1) {
             coneAngle = 1;
         } else {
             coneAngle = weaponHandler.angle;
         }
 
         transform.position = projOrigin;
 
         MakeMeshData();
         CreateMesh();
     }
 
     void MakeMeshData() {
         float angleMultiplier = 1f;
         float pointIncrement = 1f / (numOfTriangles / 2);
         for (int i = 0; i < numOfTriangles + 2; i++) {
             if (i == 0) {
                 verticesList.Add(new Vector3(0, 0, 0));
             }
             else {
                 verticesList.Add(Quaternion.Euler(new Vector3(transform.eulerAngles.x, transform.eulerAngles.y, coneAngle * angleMultiplier)) * Vector3.up * range);
                 angleMultiplier -= pointIncrement;
             }
         }
 
         for (int i = 0; i < numOfTriangles; i++) {
             List<int> singleTriangle = new List<int> ();
             singleTriangle.Add(0);
             singleTriangle.Add(1 + i);
             singleTriangle.Add(2 + i);
             trianglesList.AddRange(singleTriangle);
         }
 
     }
 
 
     void CreateMesh() {
         mesh.Clear();
         mesh.vertices = verticesList.ToArray();
         mesh.triangles = trianglesList.ToArray();
     }
 
 }