# Calculate some things about a tetrahedron

• Harmony
In summary, the conversation is about finding the length of the perpendicular from point A to the plane BCD, the angle between the surfaces ACD and BCD, and the angle between AB and the plane BCD. The speaker has solved part A and B, but is unsure about their calculation for part C, which is to find the angle between AB and the plane BCD. Their answer is slightly larger than the given answer of 53.1 degrees, possibly due to rounding error.

#### Harmony

Question Statement:
Each surface of a tetrahedron ABCD is an equilateral triangle with each side 2 units long. The midpoint of AB and CD are L and M respectively. Calculate, by giving your answers correct to 3 s.f. or to the nearest 0.1 degree,

a) The length of the perpendicular from A to the plane BCD
b) The angle between the surface ACD and BCD
c) Angle between AB and the plane BCD

My Attempt So Far :
I have solved part A and B. The only part that confuse me is part C.
My calculation :
let the angle between AB and the plane BCD be x.
The perpendicular distance from A to the plane BCD be p.
So sin x = p/AB = 1.63/2

But the answer given is x = 53.1 degree, which is slightly smaller than the answer I found.

Is the method I used wrong?

Just looking quickly the answer you have may be less due to rounding error. How much less is your answer than the answer given in the book?

1 degree. The answer given is 53.1 degree, wheres mine is about 54 degrees.

## 1. What is a tetrahedron?

A tetrahedron is a three-dimensional shape with four triangular faces, six edges, and four vertices. It is often referred to as a pyramid with a triangular base.

## 2. How do you calculate the volume of a tetrahedron?

The formula for calculating the volume of a tetrahedron is V = (√2/12) x s^3, where s is the length of the edges. Alternatively, you can also use the formula V = (1/3) x Bh, where B is the area of the base and h is the height from the base to the top vertex.

## 3. What is the surface area of a tetrahedron?

The surface area of a tetrahedron can be calculated using the formula A = √3 x s^2, where s is the length of the edges. Alternatively, you can also use the formula A = √3/4 x a^2, where a is the length of one of the sides of the base triangle.

## 4. How many symmetries does a tetrahedron have?

A tetrahedron has 12 symmetries, which include rotations, reflections, and combinations of both. These symmetries are important in understanding the properties and relationships of tetrahedrons.

## 5. What real-life objects can be represented by a tetrahedron?

Tetrahedrons can be found in many real-life objects, such as pyramids, dice, and certain types of crystals. They can also be used in engineering and architecture as a stable and efficient shape for structures like bridges and scaffolding.