OPQRS is a right pyramid whose base is a square of sides 12cm each. Given that the slant height of the pyramid is 15cm. Find the height of the pyramid. The volume of the pyramid and the total surface of the pyramid.
To find the height of the pyramid, we can use the Pythagorean theorem.
We know that the slant height (l) is 15 cm, and the base of the pyramid (s) is a square with sides of 12 cm each.
The height (h) of the pyramid can be found by using the formula:
h = √(l^2 - (s/2)^2)
where l is the slant height and s is the length of the side of the base.
Plugging in the values, we have:
h = √(15^2 - (12/2)^2)
= √(225 - 36)
= √189
≈ 13.79 cm (rounded to two decimal places)
So, the height of the pyramid is approximately 13.79 cm.
To find the volume of the pyramid, we can use the formula:
Volume = (1/3) * base area * height
The base area (A) of the pyramid is the area of the square base. Since the sides of the square base are each 12 cm, the base area is:
A = 12^2 = 144 cm^2
Plugging in the values, we have:
Volume = (1/3) * 144 * 13.79
= 57.6 cm^3 (rounded to one decimal place)
So, the volume of the pyramid is approximately 57.6 cm^3.
Finally, to find the total surface area of the pyramid, we need to calculate the areas of all the faces and add them together.
The base area (A) is already calculated as 144 cm^2.
The area of each triangular face can be found using the formula:
TSA of triangle = 0.5 * base * height
Since the base and height of each triangular face are equal to 12 cm and the slant height is 15 cm, the area of each triangular face is:
TSA of triangle = 0.5 * 12 * 15 = 90 cm^2
Since there are 4 triangular faces on the pyramid, we multiply the area of each triangular face by 4 to get the total surface area of the triangular faces, which is:
4 * 90 = 360 cm^2
Adding the base area and the surface area of the triangular faces, we have:
Total Surface Area = Base Area + Surface Area of Triangular Faces
= 144 + 360
= 504 cm^2
So, the total surface area of the pyramid is 504 cm^2.
h=sqrt{H²-(a/2)²} =
=sqrt{15²-6²} = 13.75 cm,
V=ha²/3 =13.75•12²/3=660 cm³,
S= 4•(ha/2) =2•15•12 =360 cm².
Since we cannot draw the figure here, you can draw the figure yourself.
Note that a right pyramid is a pyramid with a square base and the apex (the tip of the pyramid) is aligned directly at the center of the base.
Now, draw an line from the apex to the center of the base, which serves as the height of the pyramid. Then connect the center of the base to one of the midpoints of the side of the square. The slant height is from that midpoint to the apex.
Note that you have formed a right triangle, where the height is unknown, and the hypotenuse (which is also the slant height) is 15 cm. You can solve for the base of the right triangle by getting the half of the side of the square, which is 6 cm.
Solving for height, recall pythagorean theorem. For any right triangle,
a^2 + b^2 = c^2
where
a = base of right triangle
b = height of right triangle
c = hypotenuse
Substituting,
6^2 + b^2 = 15^2
36 + b^2 = 225
b^2 = 189
b = 3*sqrt(21) cm [height]
Recall that the volume of ANY pyramid is just
V = (Area of the base)*(height)/3
*Area of the base = (length of side of square)^2
V = (12^2)*[3*sqrt(21)]/3
V = 144*sqrt(21) cm^3
Recall that the SA of a SQUARE pyramid is just
SA = 2bs + b^2
where
s = slant height
b = length of side of square
SA = 2*12*15 + 12^2
SA = 504 cm^2
Hope this helps~ :)