Question

Find the lateral area, total area, and volume of the solid figure. Leave answer in exact form.

ill post the pic :)

Answer 1

but like.. step by step

Answer 2

so i can do the math?

Answer 3

quick question, have you learned about hero's or heron's formula ?

Answer 4

nope

Answer 5

alright, just thought I'd check

Answer 6

:)

Answer 7

Start by taking one face and dividing it in half like so

Answer 8

okay

Answer 9

The height h is what we want. This will allow us to find the area of the face.


We can use the pythagorean theorem to find h

a^2 + b^2 = c^2

2^2 + h^2 = 4^2

4 + h^2 = 16

h^2 = 16 - 4

h^2 = 12

h = sqrt(12)

So the height is exactly sqrt(12) units

Answer 10

okay gotcha!

Answer 11

The area of the triangle is then...

A = (bh)/2

A = (4*sqrt(12))/2

A = 2*sqrt(12)

A = 2*sqrt(4*3)

A = 2*sqrt(4)*sqrt(3)

A = 2*2*sqrt(3)

A = 4*sqrt(3)

Answer 12

this is one face, multiply this by 3 to get the total lateral surface area

3*(4*sqrt(3)) = 12*sqrt(3)


So the lateral surface area is 12*sqrt(3) cubic units

Answer 13

oops meant to say "square units"

Answer 14

ohhhhhhh okay okay!

Answer 15

awesome.. got it!

Answer 16

The total area is equal to 4 times one triangular face

Total surface area = 4*(4*sqrt(3)) = 16*sqrt(3)


So the total surface area is 16*sqrt(3) square units

Answer 17

k.

Answer 18

how do i get the volume?

Answer 19

Do you know how to integrate?

Answer 20

nope.

Answer 21

one sec while I try to draw it

Answer 22

okay:)

Answer 23

If you cut a cross section of the tetahedron, you'll get something like





The goal is to solve for H


a^2 + b^2 = c^2

(2/3h)^2 + H^2 = 4^2

(4h^2)/9 + H^2 = 16

H^2 = 16-(4h^2)/9

H^2 = 16 - (4*(sqrt(12))^2)/9

H^2 = 16 - (4*12)/9

H^2 = 16 - 48/9

H^2 = 32/3

H = sqrt(32/3)