Question

a solid oblique pyramid has a square base with an edge length of 2 cm. angle bac measures 45° and ac measures 3.6cm.

what is the volume of the pyramid?

2.4 cm³
3.6 cm³
4.8 cm³
7.2 cm³

Explanation:

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<reasoning>

Calculate the base area of the pyramid

Using the Pyramid Base Geometry knowledge point
\[

$$\begin{aligned} s &= 2\text{ cm}\\ B &= s^2 = 2^2 = 4\text{ cm}^2 \end{aligned}$$

\]

Determine the height of the oblique pyramid

We find the perpendicular height \(h\) from the apex \(B\) to the plane containing the base. In the right triangle \(\triangle BCA\) with the right angle at \(C\):
\[

$$\begin{aligned} \tan(\angle BAC) &= \frac{BC}{AC}\\ \tan(45^\circ) &= \frac{h}{3.6}\\ 1 &= \frac{h}{3.6}\\ h &= 3.6\text{ cm} \end{aligned}$$

\]

Calculate the volume of the oblique pyramid

Using the Volume of Oblique Pyramids knowledge point
\[

$$\begin{aligned} V &= \frac{1}{3} \cdot B \cdot h\\ V &= \frac{1}{3} \cdot 4 \cdot 3.6\\ V &= 4.8\text{ cm}^3 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(2.4\text{ cm}^3\)</mcq-option>
<mcq-option>(B) \(3.6\text{ cm}^3\)</mcq-option>
<mcq-correct>(C) \(4.8\text{ cm}^3\)</mcq-correct>
<mcq-option>(D) \(7.2\text{ cm}^3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Calculate the base area of the pyramid

Using the Pyramid Base Geometry knowledge point
\[

$$\begin{aligned} s &= 2\text{ cm}\\ B &= s^2 = 2^2 = 4\text{ cm}^2 \end{aligned}$$

\]

Determine the height of the oblique pyramid

We find the perpendicular height \(h\) from the apex \(B\) to the plane containing the base. In the right triangle \(\triangle BCA\) with the right angle at \(C\):
\[

$$\begin{aligned} \tan(\angle BAC) &= \frac{BC}{AC}\\ \tan(45^\circ) &= \frac{h}{3.6}\\ 1 &= \frac{h}{3.6}\\ h &= 3.6\text{ cm} \end{aligned}$$

\]

Calculate the volume of the oblique pyramid

Using the Volume of Oblique Pyramids knowledge point
\[

$$\begin{aligned} V &= \frac{1}{3} \cdot B \cdot h\\ V &= \frac{1}{3} \cdot 4 \cdot 3.6\\ V &= 4.8\text{ cm}^3 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(2.4\text{ cm}^3\)</mcq-option>
<mcq-option>(B) \(3.6\text{ cm}^3\)</mcq-option>
<mcq-correct>(C) \(4.8\text{ cm}^3\)</mcq-correct>
<mcq-option>(D) \(7.2\text{ cm}^3\)</mcq-option>
</answer>

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