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If all the faces of pyramid P (including the base)

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If all the faces of pyramid P (including the base)

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If all the faces of pyramid P (including the base) are equilateral triangles of side 2, what is the surface area of pyramid P?

A. 4+(3√3)/4
B. 4√3
C. 4+3√3
D. 6√3
E. 12

The OA is B.

How can I calculate the height of the triangles? Can any expert give me some help? Thanks in advanced.

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Hello Vjesus12.

Let's take a look at your question.

If we have a equilateral triangle of side 2, then we can build a right triangle with one cathetus equal to 1, and the hypotenuse equal to 2. Then, using the Pythagoras we have that the other cathetus (the height) is equal to $$\sqrt{2^2-1^2}=\sqrt{4-1}=\sqrt{3}.$$ Now, we have 4 equilateral triangles with area equal to $$A=\frac{2\cdot\sqrt{3}}{2}=\sqrt{3}.$$ So, the surface area is equal to $$S_a=4\sqrt{3}.$$ So, the correct answer is B.

I hope this explanation may help you.

I'm available if you'd like a follow up.

Regards.

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VJesus12 wrote:
If all the faces of pyramid P (including the base) are equilateral triangles of side 2, what is the surface area of pyramid P?

A. 4+(3√3)/4
B. 4√3
C. 4+3√3
D. 6√3
E. 12

The OA is B.

How can I calculate the height of the triangles? Can any expert give me some help? Thanks in advanced.
in this case you need not calculate the height. you may apply the formula
Area of one equilateral triangle =(√3 /4 ) * a*a where a is the side of triangle
Area of one triangle=2X2X√3/4=√3
Therefore area of 4 triangles = 4√3
hence B is the correct answer

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Quote:
If all the faces of pyramid P (including the base) are equilateral triangles of side 2, what is the surface area of pyramid P?

A. 4+(3√3)/4
B. 4√3
C. 4+3√3
D. 6√3
E. 12

The OA is B.

How can I calculate the height of the triangles? Can any expert give me some help? Thanks in advanced.
Hi VJesus12,
Let's take a look at your question.

All the faces of pyramid P (including the base) are equilateral triangles of side 2, we need to find the surface area of the pyramid. For that purpose, we will first find the area of one face of the pyramid.

Each face is equilateral triangle of side 2.
We know that:
Area of triangle = (1/2)*(Base)*(Height)
To find area we will first calculate the height. Height of the triangle is altitude from the top vertex to the base of length 2. The altitude will bisect the base. It means the altitude will divide the triangular face into two congruent right triangles with base 1 and hypotenuse 2.

$$\left(2\right)^2=\left(1\right)^2+\left(Height\right)^2$$
$$4=1+\left(Height\right)^2$$
$$\left(Height\right)^2=4-1$$
$$\left(Height\right)^2=3$$
$$Height=\sqrt{3}$$

Now we can find the area of triangular face of the pyramid.
$$=\left(\frac{1}{2}\right)\left(Base\right)\left(Height\right)$$
$$=\left(\frac{1}{2}\right)\left(2\right)\left(\sqrt{3}\right)$$
$$=\sqrt{3}$$

This is the surface area of one face, We know that there are 4 triangular faces of pyramid including base.
Surface area of pyramid = 4 * Surface area of one triangular face of pyramid
$$=4\sqrt{3}$$

Therefore, Option B is correct.

Hope it helps.
I am available if you'd like any follow up.

_________________
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