AMC12 2025 A

AMC12 2025 A · Q20

AMC12 2025 A · Q20. It mainly tests 3D geometry (volume), 3D geometry (surface area).

The base of the pentahedron shown below is a $13 \times 8$ rectangle, and its lateral faces are two isosceles triangles with base of length $8$ and congruent sides of length $13$, and two isosceles trapezoids with bases of length $7$ and $13$ and nonparallel sides of length $13$. What is the volume of the pentahedron?

如下图所示的五面体的底面为 $13 \times 8$ 矩形，其侧面为两个底边长 $8$、等腰边长 $13$ 的等腰三角形，以及两个底边长分别为 $7$ 和 $13$、非平行边长 $13$ 的等腰梯形。该五面体的体积是多少？

(A) 416 416

(B) 520 520

(D) 676 676

(E) 832 832

Answer

Correct choice: (C)

正确答案：（C）

Solution

Notice that the triangular faces have a slant height of $\sqrt{13^2-4^2}=\sqrt{153}$ and that the height is therefore $\sqrt{153-(\frac{13-7}{2})^2} = 12$. Then we can split the pentahedron into a triangular prism and two pyramids. The pyramids each have a volume of $\frac{1}{3}(3)(8)(12) = 96$ and the prism has a volume of $\frac{1}{2}(8)(12)(7) = 336$. Thus the answer is $336+96 \cdot 2 = \boxed{\textbf{(C) } 528}$

注意到三角形侧面的斜高为 $\sqrt{13^2-4^2}=\sqrt{153}$，高度因此为 $\sqrt{153-(\frac{13-7}{2})^2} = 12$。然后可以将五面体分解为一个三角柱和两个金字塔。每个金字塔体积为 $\frac{1}{3}(3)(8)(12) = 96$，三角柱体积为 $\frac{1}{2}(8)(12)(7) = 336$。因此总体积 $336+96 \cdot 2 = \boxed{\textbf{(C) } 528}$

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