Question
A right pyramid with a square base is shown. The length of the base is 8 inches and the slant height is 5 inches.
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
✔ 64
cubic inches.
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
✔ 64
cubic inches.
Answers
Answered by
GPT-4o mini
To solve for the height of the pyramid and ultimately the volume, we will follow these steps:
1. **Determine the leg of the right triangle (half of the base length)**:
The length of the base of the pyramid is 8 inches. The leg of the triangle that lies on the same plane as the base (half the base length) is:
\[
\text{leg} = \frac{8 \, \text{inches}}{2} = 4 \, \text{inches}
\]
2. **Determine the slant height**:
The slant height of the pyramid is given as 5 inches.
3. **Use the Pythagorean theorem to find the height of the pyramid**:
In a right triangle formed by the height \( h \), the leg of \( 4 \text{ inches} \) (half the base), and the slant height \( 5 \text{ inches} \), we can apply the Pythagorean theorem:
\[
h^2 + 4^2 = 5^2
\]
\[
h^2 + 16 = 25
\]
\[
h^2 = 25 - 16
\]
\[
h^2 = 9
\]
\[
h = 3 \, \text{inches}
\]
4. **Calculate the volume of the pyramid**:
The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base area of a square with side length 8 inches is:
\[
\text{Base Area} = 8 \times 8 = 64 \, \text{square inches}
\]
Plugging in the base area and the height:
\[
V = \frac{1}{3} \times 64 \times 3
\]
\[
V = \frac{192}{3} = 64 \, \text{cubic inches}
\]
Now summarizing the answers in the required form:
- The hypotenuse of the right triangle used to determine the height is **5 inches**.
- The leg of the right triangle that lies on the same plane as the base is **4 inches**.
- The height of the pyramid is **3 inches**.
- The volume of the solid pyramid is **64 cubic inches**.
1. **Determine the leg of the right triangle (half of the base length)**:
The length of the base of the pyramid is 8 inches. The leg of the triangle that lies on the same plane as the base (half the base length) is:
\[
\text{leg} = \frac{8 \, \text{inches}}{2} = 4 \, \text{inches}
\]
2. **Determine the slant height**:
The slant height of the pyramid is given as 5 inches.
3. **Use the Pythagorean theorem to find the height of the pyramid**:
In a right triangle formed by the height \( h \), the leg of \( 4 \text{ inches} \) (half the base), and the slant height \( 5 \text{ inches} \), we can apply the Pythagorean theorem:
\[
h^2 + 4^2 = 5^2
\]
\[
h^2 + 16 = 25
\]
\[
h^2 = 25 - 16
\]
\[
h^2 = 9
\]
\[
h = 3 \, \text{inches}
\]
4. **Calculate the volume of the pyramid**:
The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The base area of a square with side length 8 inches is:
\[
\text{Base Area} = 8 \times 8 = 64 \, \text{square inches}
\]
Plugging in the base area and the height:
\[
V = \frac{1}{3} \times 64 \times 3
\]
\[
V = \frac{192}{3} = 64 \, \text{cubic inches}
\]
Now summarizing the answers in the required form:
- The hypotenuse of the right triangle used to determine the height is **5 inches**.
- The leg of the right triangle that lies on the same plane as the base is **4 inches**.
- The height of the pyramid is **3 inches**.
- The volume of the solid pyramid is **64 cubic inches**.