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 area of the pyramid by first drawing the net of the pyramid.
To find the height of the pyramid, first, let's draw the net of the pyramid. The net of a pyramid is a two-dimensional representation of all the faces of the pyramid unfolded onto a flat surface.
Since OPQRS is a right pyramid, that means it has a square base. So draw a square with sides of 12cm. Label the vertices of the square as O, P, Q, and R, in a clockwise direction.
Next, draw lines from each vertex of the square to a common point S that is above the center of the square. These lines represent the slant height of the pyramid. Label the point where each line intersects the square as A, B, C, and D, respectively.
Now, we have a square base OPQR and four triangular faces: OSA, PSB, QSC, and RSD.
To find the height of the pyramid, we can use the Pythagorean theorem. In triangle OSA, the base OA has a length of 12 cm, and the slant height OS has a length of 15 cm. Let's use the variable h to represent the height of the pyramid.
Using the Pythagorean theorem, we can write:
OA^2 + h^2 = OS^2
Substituting the values, we get:
12^2 + h^2 = 15^2
Simplifying the equation:
144 + h^2 = 225
Rearranging the equation:
h^2 = 225 - 144
h^2 = 81
Taking the square root of both sides:
h = √81
h = 9 cm
So, the height of the pyramid is 9 cm.
Now, let's find the volume of the pyramid. The volume of a pyramid can be calculated using the formula:
Volume = (1/3) * base area * height
The base area of the pyramid is the area of the square base OPQR, which is equal to length * width. Since the sides of the square have a length of 12 cm each, the base area is:
base area = 12 cm * 12 cm
base area = 144 cm^2
Substituting the values into the volume formula, we get:
Volume = (1/3) * 144 cm^2 * 9 cm
Volume = 432 cm^3
So, the volume of the pyramid is 432 cm^3.
Finally, let's find the total surface area of the pyramid. The total surface area is the sum of the areas of all the faces.
The base area is the same as before, 144 cm^2.
The area of each triangular face can be calculated as half the product of the base length and the slant height. Since the base length of each face is 12 cm and the slant height is 15 cm, the area of each triangular face is:
Area of each triangular face = (1/2) * base length * slant height
Area of each triangular face = (1/2) * 12 cm * 15 cm
Area of each triangular face = 90 cm^2
Since there are four triangular faces, the total area of the four faces is:
Total area of the four faces = 4 * 90 cm^2
Total area of the four faces = 360 cm^2
Now, add the base area and the total area of the faces together to find the total surface area:
Total surface area = base area + total area of the faces
Total surface area = 144 cm^2 + 360 cm^2
Total surface area = 504 cm^2
So, the total surface area of the pyramid is 504 cm^2.
In conclusion:
- The height of the pyramid is 9 cm.
- The volume of the pyramid is 432 cm^3.
- The total surface area of the pyramid is 504 cm^2.
sqrt ( 15 ^ 2 - 12 ^ 2 ) = sqrt ( 225 -144 ) = sqrt ( 81 ) = 9 cm
A right pyramid has a square base 6cm by 6cm and each face has a slant height of 5 cm. Calculate the height of pyramid
Draw one of the side triangles, and consider its altitude from the base of 12, call that x
x^2 + 6^2 = 15^2
x^2 = 19
x =√189
so the surface area = the base + 4triangles
= 144 + 4(1/2)(12)(√189) = ....
for the volume , we need the height of the pyramid,
call it h
h^2 + 6^2 = (√189)^2
h^2 = 153
h = √153
volume = (1/3)(base)(height)
= (1/3)(144)(√153
= 48√153