What the formula for the surface area of a pyramid that is not equalateral?

Question

it's isosceles with one side length 7 and a rectangular base with sides 6 and 2.

Answer 1

You don't need a formula, just geometry. The area of the rectangular base is just length*width = 6*2 = 12. The area of a triangle is 0.5*base*height, and you've been given the lengths of the bases of the triangular faces (6 and 2), and the lengths of the other legs (all 7). You can use right triangle geometry to get the height, because the other leg is the hypotenuse of a right triangle that has half the base for a leg. By the Pythagorean theorem, the heights are sqrt(7^2 - 3^2) = sqrt(49 - 9) = sqrt(40) = 2*sqrt(10) and sqrt(7^2 - 1^2) = sqrt(49 - 1) = sqrt(48) = 4*sqrt(3). The areas of the triangular faces are therefore 0.5*6*2*sqrt(10) = 6*sqrt(10) and 0.5*2*4*sqrt(3) = 4*sqrt(3). There are two of each of those, so the total surface area is 12 + 12*sqrt(10) + 8*sqrt(3).

Answer 2

mathsmanretired (answerer 2) shows the simplest method, but you don't need Pythagoras' formula to calculate the area of the triangles with sides (6, 7, 7) and (2, 7, 7). You can use "Heron's formula" for the area of a triangle with sides (a, b, c), which is first s = (a+b+c)/2, then A = sqrt(s * (s-a) * (s-b) * (s-c)).

Let's see this working for (6, 7, 7). We get s = (6 + 7 + 7)/2 = 10, A = sqrt(10 * 4 * 3 * 3) = 6 * sqrt(10), the same as by Pythagoras but a little bit quicker. Similarly for (2, 7, 7) we get A = 4 * sqrt(3). So two triangles of each size, plus the base, gives us a total surface area of 12 * sqrt(10) + 8 * sqrt(3) + 12.

Answer 3

A little Pythagorus will help here:

What we have to do is calculate the area of the triangular planes, so two of the triangles have a base of 2, so:

Height (h) = sqrt(49-1) = 4*sqrt3
B*h/2 = 4*sqrt3

When base=6:
h=sqrt(49-9)=2*sqrt10
B*h/2 = 6*sqrt10

Since each triangle occurs twice:
2(4*sqrt3+6*sqrt10) = 4(2*sqrt3+3*sqrt10)

Base area = 6*2 = 12, so total suface area =

4(2*sqrt3+3*sqrt10+3)

Answer 4

The surface has two sides which are triangles with sides 6,7,7 and two triangles with sides 2,7,7. You split the isosceles triangles down the middle to make right-angled triangles and then do pythagoras calculations on each to find its height.

Answer 5

find the area of the 4 triangles + area of the rectangular base