Calculate the volume of a square pyramid if it has a base area of 64 cm^2 and the distance from the apex to a corner of the base is 15 cm.
Thank you so much!!!
Base length = a = sqrt64 = 8 cm
Distance from corner to center of base
= a*sqrt2 = 11.314 cm
Pyramid height =
h = sqrt[15^2 -(8sqrt2)^2]
= sqrt[225 - 128]= 9.85 cm
Volume = (base area)*(height)/3
= 210 cm^3
Side of base = √64 = 8 cm
Distance from centre to corner of base
=8(√2)/2=4√2
Let h=height,
h²=15²-(4√2)²
=225-32=193
h=√193 = 13.89 cm approx.
Volume
=(64)h/3
=64*(√193)/3
=296.4 cm³ approx.
To calculate the volume of a square pyramid, you need to know the base area and the height. In this case, the base area is given as 64 cm^2 and the height is the distance from the apex (top) to a corner of the base, which is 15 cm.
First, let's find the height of the pyramid. To do this, we can use the Pythagorean theorem, as the height, base, and diagonal of the base form a right triangle.
The diagonal of the base, also known as the side length of the base square, can be found using the formula: diagonal = side × √2.
Let's substitute the value of the base area into the formula to find the length of the side of the base:
64 cm^2 = side × side
Square rooting both sides, we get:
√64 cm^2 = side
8 cm = side
Now, we can find the length of the diagonal:
diagonal = side × √2
diagonal = 8 cm × √2
diagonal ≈ 11.3 cm
Finally, we can find the height using the Pythagorean theorem:
height = √(diagonal^2 - side^2)
height = √(11.3 cm^2 - 8 cm^2)
height ≈ √(127.69 cm^2 - 64 cm^2)
height ≈ √63.69 cm^2
height ≈ 7.98 cm
Now, calculate the volume of the square pyramid using the formula: volume = (base area × height) / 3
volume = (64 cm^2 × 7.98 cm) / 3
volume ≈ 170.72 cm^3
Therefore, the volume of the square pyramid is approximately 170.72 cm^3.