Question

calculating the volume of an oblique pyramid
the area of the base of the oblique pentagonal pyramid is 50 cm² and the distance from the apex to the center of the pentagon is 6√2 cm. the measure of ∠acb is 45°.
the height, ab, is cm.
the volume of the pyramid is cm³.

Explanation:

Step1: Find height AB using trigonometry

In right triangle ABC, $\sin(\angle ACB) = \frac{AB}{AC}$. Substitute $\angle ACB=45^\circ$, $AC=6\sqrt{2}$ cm:
$\sin(45^\circ) = \frac{AB}{6\sqrt{2}}$
$AB = 6\sqrt{2} \times \sin(45^\circ) = 6\sqrt{2} \times \frac{\sqrt{2}}{2}$

Step2: Simplify to get AB

$AB = 6\sqrt{2} \times \frac{\sqrt{2}}{2} = 6 \times \frac{2}{2} = 6$ cm

Step3: Calculate pyramid volume

Use volume formula $V = \frac{1}{3} \times \text{base area} \times \text{height}$. Substitute base area $=50$ cm², height $=6$ cm:
$V = \frac{1}{3} \times 50 \times 6$

Step4: Simplify to get volume

$V = \frac{1}{3} \times 300 = 100$ cm³

Answer:

The height, AB, is 6 cm.
The volume of the pyramid is 100 cm³.