先看看效果

分析
所谓五角星，其实就是均分圆上的5个点，相互连线而成的。

那怎么求圆上点的坐标呢？
圆心坐标：(x0,y0) 半径：r 角度：a 求：x1 = ? y1 = ?

答案就是三角函数

也就是说，只要是个直角三角形(三角函数只在直角三角形中有效),知道斜边c的长度，∠A的度数，就可以知道直角边a、b的长度。
边a = sin a * 斜边
边b = cos a * 斜边

也就是说：
x1 = x0 + r * cos∠A
y1 = y0 + r * sin∠A

在android中需要用到Math函数

public final class Math {

    public static final double PI = 3.14159265358979323846;

    /**
     * Returns the trigonometric sine of an angle.  Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the
     * result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the sine of the argument.
     */
    @CriticalNative
    public static native double sin(double a);

    /**
     * Returns the trigonometric cosine of an angle. Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the
     * result is NaN.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the cosine of the argument.
     */
    @CriticalNative
    public static native double cos(double a);
}

需要注意的是：在android中Math.sin(double a)、Math.cos(double a)参数一个弧度，而不是角度，需要用Math.toRadians(double angdeg)转化为角度。
x1 = x0 + r * Math.sin( Math.toRadians(270) )
y1 = y0 + r * Math.cos( Math.toRadians(270) )

编写代码

package com.noahedu.my;

import android.content.Context;
import android.graphics.Canvas;
import android.graphics.Color;
import android.graphics.Paint;
import android.graphics.Rect;
import android.support.annotation.Nullable;
import android.util.AttributeSet;
import android.view.View;

import com.noahedu.my.bean.PointBean;

import java.util.ArrayList;
import java.util.List;


public class PentacleView extends View {

    private Context context;
    private Paint mPaint;
    private Paint mBgPaint;
    private Rect mRect;

    public PentacleView(Context context) {
        this(context, null);
    }

    public PentacleView(Context context, @Nullable AttributeSet attrs) {
        this(context, attrs, -1);
    }

    public PentacleView(Context context, @Nullable AttributeSet attrs, int defStyleAttr) {
        super(context, attrs, defStyleAttr);

        init(context);
    }

    private void init(Context context) {

        this.context = context;

        mPaint = new Paint(Paint.ANTI_ALIAS_FLAG);
        mPaint.setStyle(Paint.Style.STROKE);
        mBgPaint = new Paint(Paint.ANTI_ALIAS_FLAG);
        mBgPaint.setColor(Color.WHITE);
        mPaint.setColor(Color.RED);
        mRect = new Rect();
    }

    @Override
    protected void onDraw(Canvas canvas) {
        super.onDraw(canvas);

        int width = canvas.getWidth();
        int height = canvas.getHeight();
        canvas.drawRect(0, 0, width, height, mBgPaint);

        int zhijing = width > height ? height : width;
        int radius = zhijing >> 1;

        float circleX = width >> 1;
        float circleY = height >> 1;

        drawPentacleView(canvas, mPaint, circleX, circleY, radius);
    }

    private List<PointBean> mPointBeanList;

    /**
     * 根据圆心和半径求出五角星5个点
     */
    private void drawPentacleView(Canvas canvas, Paint paint, float circleX, float circleY, float radius) {

        mPointBeanList = new ArrayList<>();
        for (int i = 270; i >= -22; i -= 72) {

            float x = (float) (circleX + radius * Math.cos(Math.toRadians(i)));
            float y = (float) (circleY + radius * Math.sin(Math.toRadians(i)));

            PointBean pointBean = new PointBean(x, y);
            mPointBeanList.add(pointBean);
        }

        drawPentacleView(canvas, paint, mPointBeanList);
    }

    /**
     * 画出五角星的5条线
     */
    private void drawPentacleView(Canvas canvas, Paint paint, List<PointBean> pointBeanList) {

        drawPentacleLine(canvas, paint, pointBeanList, 1, 4);
        drawPentacleLine(canvas, paint, pointBeanList, 4, 2);
        drawPentacleLine(canvas, paint, pointBeanList, 2, 0);
        drawPentacleLine(canvas, paint, pointBeanList, 0, 3);
        drawPentacleLine(canvas, paint, pointBeanList, 3, 1);
    }

    private void drawPentacleLine(Canvas canvas, Paint paint, List<PointBean> pointBeanList, int startIndex, int endIndex) {

        float startX = pointBeanList.get(startIndex).getPointX();
        float startY = pointBeanList.get(startIndex).getPointY();
        float endX = pointBeanList.get(endIndex).getPointX();
        float endY = pointBeanList.get(endIndex).getPointY();
        canvas.drawLine(startX, startY, endX, endY, paint);
    }
}

参考：
已知圆心，半径，角度，求圆上的点坐标