Question

What is the volume of a triangular pyramid with a right-angled base of 3cm and 4cm, and a slant edge of 5cm?

Answer 1

To find the volume of a triangular pyramid (or tetrahedron) with a right-angled triangular base, we first need to calculate the area of the base and then use it to find the volume.

**Step 1: Calculate the area of the triangular base.**

The base is a right triangle with legs of lengths 3 cm and 4 cm. The area \( A \) of a right triangle can be calculated using the formula:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, we can take the base as 3 cm and the height as 4 cm.

\[
A = \frac{1}{2} \times 3 \times 4 = \frac{1}{2} \times 12 = 6 \text{ cm}^2
\]

**Step 2: Calculate the height of the pyramid.**

Now, we need to find the height \( h \) of the pyramid. The slant edge given is 5 cm, which serves as the hypotenuse of a right triangle formed by the height of the pyramid and the distance from the apex perpendicular to the midpoint of the base.

First, we find the centroid of the triangular base to determine where the height of the pyramid meets the base. The centroid of a right triangle divides each median in a ratio of 2:1.

The length of the base (hypotenuse of the triangle) is:

\[
\text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}
\]

Next, let’s calculate the distance from the apex vertically above the base, by using the Pythagorean theorem in one of the triangles defined by the slant edge and the height of the pyramid.

Let's denote the height from the apex of the pyramid to the base as \( h \) and the distance from the apex to the centroid of the triangular base effective in the base area (noting the location as the median):

We already know the lengths:

\[
d = 5 \text{ cm } (this is the hypotenuse)
\]

Use the relation:

\[
h = \sqrt{d^2 - \left(\frac{\text{base length}}{2}\right)^2}
\]

Calculating using the median to find the centroid; the median from right angle triangle measures:

Height of centroid is defined from hypotenuse:

\[
d = 5 \, \text{ cm}
\]

Calculating out hypotenuse in reference to base distances:

\[
\text{Centroid to vertex distance squared (Type of feet)}: \left(\frac{3}{2}\right)^2 + \left(\frac{4}{2}\right)^2= 6.25
\]

Now finding \( h \):

\[
h = \sqrt{d^2 - B^2} = \sqrt{5^2 - 6.25} = \sqrt{25 - 6.25} = \sqrt{18.75}
\]
\[
h \approx 4.33
\]

**Step 3: Calculate the volume of the pyramid.**

The volume \( V \) of a pyramid can be calculated with the formula:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

Substituting the calculated values:

\[
V = \frac{1}{3} \times 6 \times 4.33 \approx \frac{1}{3} \times 25.98 \approx 8.66 \, \text{cm}^3
\]

Thus, the volume of the triangular pyramid is approximately \( 8.66 \, \text{cm}^3 \).