The lateral edge of a regular tetrahedron is 6cm find total surface area, lateral surface area and volume ?

To find the apothem (AT) of a regular tetrahedron, we can use the formula:

AT = (sqrt(6)/3) * s

where s is the length of the lateral edge. Plugging in s = 6cm, we get:

AT = (sqrt(6)/3) * 6
AT = 2sqrt(6) cm

To find the apothem length (AL), we can use the formula:

AL = (sqrt(2)/3) * s

Plugging in s = 6cm, we get:

AL = (sqrt(2)/3) * 6
AL = 2sqrt(2) cm

To find the volume (V) of a regular tetrahedron, we can use the formula:

V = (sqrt(2)/12) * s^3

Plugging in s = 6cm, we get:

V = (sqrt(2)/12) * 6^3
V = (sqrt(2)/12) * 216
V = 36sqrt(2) cm^3

Therefore, the apothem (AT) is 2sqrt(6) cm, the apothem length (AL) is 2sqrt(2) cm, and the volume (V) is 36sqrt(2) cm^3.

Given:

Lateral edge of a regular tetrahedron (AB) = 6 cm

To Find:

- Altitude (AT)
- Slant height (AL)
- Volume (V)

Solution:

Altitude (AT):

In a regular tetrahedron, the ratio of the altitude to the lateral edge is 3:4. Therefore,

AT = (3/4) - AB
AT = (3/4) - 6 cm
AT = 4.5 cm

Slant Height (AL):

The slant height is the distance from a vertex to the center of the opposite face. In a regular tetrahedron, the slant height can be calculated using the formula:

AL = √(AB^2 + (2/3) - AT^2)

AL = √(6^2 + (2/3) - 4.5^2)
AL = 5.4 cm

Volume (V):

The volume of a regular tetrahedron can be calculated using the formula:

V = (1/12) - √2 - a^3

where a is the lateral edge.

V = (1/12) - √2 - 6^3
V = 18.92 cm³

Therefore, the altitude (AT) is 4.5 cm, the slant height (AL) is 5.4 cm, and the volume (V) is 18.92 cm³.