In the adjoining figure,the slant height and the total surface area of square based pyramid are 12cm and340sq.cm respectively.find the perimeter of the base and volume

To find the perimeter of the base and volume of a square-based pyramid, we can use the given information about the slant height and total surface area.

Let's start by finding the perimeter of the base.

In a square-based pyramid, the base is a square. To find the perimeter of a square, we need to know the length of one side of the square.

To find the length of one side, we can use the formula for the total surface area of a pyramid:

Total Surface Area = Base Area + Lateral Surface Area

Given that the total surface area is 340 sq.cm, and the lateral surface area of a pyramid is the sum of the areas of the four triangular faces, we can set up the following equation:

340 = Base Area + 4 * (1/2 * Base * Slant Height)

Since the base is a square, the base area is equal to the square of the side length (squared because it's a square). Let's call the side length of the square base "x."

340 = x^2 + 4 * (1/2 * x * 12)

Now, we have an equation with one variable (x) that we can solve for the side length.

To find the x value, we can rearrange the equation:

340 - 4 * (1/2 * x * 12) = x^2

340 - 24x = x^2

Rearranging further:

x^2 + 24x - 340 = 0

Now we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. By factoring, we find that:

(x + 34)(x - 10) = 0

This gives two possible values for x: x = -34 or x = 10.

Since the side length cannot be negative, we take x = 10 as the length of one side of the square base.

Now we can find the perimeter of the base:

Perimeter of the base = 4 * side length

Perimeter of the base = 4 * 10

Perimeter of the base = 40 cm

Next, let's move on to finding the volume.

The volume of a square-based pyramid can be calculated using the formula:

Volume = (1/3) * Base Area * Height

Given that the slant height is 12 cm and the base side length is 10 cm, we need to find the height.

To find the height, we can use the Pythagorean Theorem, which states that in a right triangle (where one side is the slant height, the other side is half the base length, and the hypotenuse is the height), the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, using the Pythagorean Theorem:

Height^2 = Slant Height^2 - (1/2 * Base Length)^2

Height^2 = 12^2 - (1/2 * 10)^2

Height^2 = 144 - 25

Height^2 = 119

Taking the square root of both sides:

Height = √119 cm

Now, we can calculate the volume:

Volume = (1/3) * Base Area * Height

Volume = (1/3) * (10^2) * √119

Volume = (1/3) * 100 * √119

Volume = (100/3) * √119 cm^3

Therefore, the perimeter of the base is 40 cm, and the volume is (100/3) * √119 cm^3.

40cm,363.62cm^3

answers

If the base has side length s, the total area is a square and four triangles with height 12 (the slant height):

s^2 + 4 * (1/2)(12)s = s^2 + 24s

Now just solve for s and recall that the volume is

1/3 Bh = 1/2 s^2h

where h^2 = 12^2 - (s/2)^2