己知正四面体ABCD的各边长为a，请问外接球和内接球半经公式是？ 正四面体棱长为a，求其内切球与外接球的表面积
\u5df1\u77e5\u6b63\u56db\u9762\u4f53ABCD\u7684\u5404\u8fb9\u957f\u4e3aa\uff0c\u7403o\u4e0e\u5404\u68f1\u76f8\u5207\uff0c\u90a3\u4e48\u8be5\u7403\u7684\u534a\u7ecfr\u516c\u5f0f\u662f\uff1f\u7b54\uff1a\u8fc7\u9876\u70b9\u5411\u5e95\u9762\u505a\u5782\u7ebfL\uff0c\u4ee5\u4fa7\u68f1\u4e2d\u70b9\u4e3a\u5782\u8db3\u505a\u5782\u7ebf\u4ea4L\u4e8e\u70b9O\uff0c\u4e3a\u7403\u5fc3\uff0c\u540e\u6765\u505a\u7684\u5782\u7ebf\u5373\u534a\u5f84r\u3002\u5229\u7528\u4e09\u89d2\u5f62\u76f8\u4f3c\uff0c\u53ef\u6c42\u7684r=(2^0.5)*a/4 \u3002\u56db\u5206\u4e4b\u6839\u4e0b\u4e8c\u500da\u3002
\u89e3\u7b54\uff1a\u89e3\uff1a\u8bbe\u6b63\u56db\u9762\u4f53\u7684\u9762BCD\u548c\u9762ACD\u7684\u4e2d\u5fc3\u5206\u522b\u4e3aO1\uff0cO2\uff0c\u8fde\u7ed3AO2\u4e0eBO1\u5e76\u5ef6\u957f\uff0c\u5fc5\u4ea4\u4e8eCD\u7684\u4e2d\u70b9E\uff0c\u53c8BE=32a\uff0cO2E=36a\uff0c\u8fde\u63a5BO2\uff0c\u5728Rt\u25b3BO2E\u4e2d\uff0cBO2=63\uff0c\u8fde\u7ed3AO1\u4e0eBO2\u4ea4\u4e8eO3\uff0c\u7531Rt\u25b3AO2O3\u224cRt\u25b3BO1O2\uff0c\u2234O3O2=O3O1\uff0cO3A=O3B\uff0c\u540c\u7406\u53ef\u8bc1O3C=O3D=O3A\uff0cO3\u5230\u53e6\u4e8c\u9762\u7684\u8ddd\u79bb\u4e5f\u7b49O3O1\uff0c\u2234O3\u4e3a\u56db\u9762\u4f53\u5916\u63a5\u7403\u4e0e\u5185\u63a5\u7403\u7684\u7403\u5fc3\uff0c\u7531\u25b3BO1O3\u223d\u25b3BO2E\uff0c\u2234O1O3=612a\uff0c\u2234R\u5916=64a\uff0cS\u5916=32\u03c0a2\uff0cr\u5185=612a\uff0cS\u5185=16\u03c0a2\uff0e
己知正四面体ABCD的各边长为 a体对角线 √3a
外接球半径为体对角线一半 √3/2a
内接球半经为连长一半 1/2a
底面高h1=√3a/2,侧棱射影=h1*2/3=√3a/2*(2/3)=√3a/3,
高h=√[a^2-(√3a/3)^2]=√6a/3,
从侧棱作高的垂直平分线交高于O，O点就是外接球球心，a*a/2=R*h,R=√6a/4,
内切球半径r=h-R=√6a/3-√6a/4=√6a/12,
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