Question

using slant height to compute volume
kabir wants to know the volume of a solid right pyramid with a square base. he uses a ruler to measure the length of the base as 8 inches. he then measures the slant height to be 5 inches.
the hypotenuse of the right triangle used to determine the height is ▼ inches.
the leg of the right triangle that lies on the same plane as the base is ▼ inches.
the height of the pyramid is ▼ inches.
the volume of the solid pyramid is ▼ cubic inches.

Explanation:

Step1: Hypotenuse of the right triangle

The slant height of the pyramid is the hypotenuse of the right triangle formed with the height of the pyramid and the apothem (or the distance from the center of the base to the midpoint of a side). Here, the slant height is given as 5 inches, so the hypotenuse is 5 inches.

Step2: Leg on the base plane

The base is a square with side length 8 inches. The leg of the right triangle on the base plane is half of the side length of the square base (since it's the distance from the center to the midpoint of a side). So, $\frac{8}{2} = 4$ inches.

Step3: Height of the pyramid

Using the Pythagorean theorem $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (slant height = 5 inches) and $b$ is the leg on the base plane (4 inches). Let $a$ be the height $h$. Then $h^2 + 4^2 = 5^2$. So $h^2 = 25 - 16 = 9$, and $h = \sqrt{9} = 3$ inches.

Step4: Volume of the pyramid

The formula for the volume of a pyramid is $V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height. The base is a square with side 8 inches, so $B = 8 \times 8 = 64$ square inches. The height $h = 3$ inches. Then $V = \frac{1}{3} \times 64 \times 3 = 64$ cubic inches.

Answer:

• The hypotenuse of the right triangle used to determine the height is $\boldsymbol{5}$ inches.
• The leg of the right triangle that lies on the same plane as the base is $\boldsymbol{4}$ inches.
• The height of the pyramid is $\boldsymbol{3}$ inches.
• The volume of the solid pyramid is $\boldsymbol{64}$ cubic inches.