what is the slantheight of the square pyramid that can be curved out from a wooden cube of side 12cm?what is the lateral surface area?
To find the slant height of the square pyramid that can be curved out from a wooden cube, we first need to understand the geometry of the pyramid.
A square pyramid is formed by connecting a square base to an apex above the center of the base. The slant height of the pyramid is the distance from the apex to any point on the edge of the base.
In this case, since the pyramid is formed by curving out a wooden cube, the base of the pyramid will be a square with sides equal to the sides of the cube. Given that the side of the wooden cube is 12 cm, the base of the pyramid will also have sides measuring 12 cm.
To calculate the slant height of the pyramid, we can use the Pythagorean theorem. The slant height, abbreviated as "l", the height of the pyramid (measured from the apex to the center of the base), and half the length of the base (measured from a corner of the base to the center) form a right triangle.
Using the Pythagorean theorem, the formula for the slant height becomes:
l^2 = h^2 + (1/2 * s)^2
Where:
l is the slant height,
h is the height of the pyramid,
and s is the side length of the base.
Since the height of the pyramid would bisect the wooden cube, it will be equal to half of the side length of the cube. Therefore, h = 12/2 = 6 cm.
Now, substituting the known values into the formula, we get:
l^2 = 6^2 + (1/2 * 12)^2
l^2 = 36 + 36
l^2 = 72
l = √72 cm
l ≈ 8.485 cm (approximated to three decimal places)
So, the slant height of the square pyramid, which can be curved out from a wooden cube with a side length of 12 cm, is approximately 8.485 cm.
To find the lateral surface area of the square pyramid, we need to calculate the area of the four triangular faces. Each face will be an isosceles triangle with a base equal to the side length of the cube (12 cm) and a height equal to the slant height we just calculated.
The formula for the area of an isosceles triangle is:
Area = (1/2) * base * height
For each triangular face of the pyramid, substituting the known values into the formula, we get:
Area = (1/2) * 12 cm * 8.485 cm
Area ≈ 50.91 cm² (approximated to two decimal places)
Since there are four triangular faces, the total lateral surface area of the square pyramid is:
4 * 50.91 cm² = 203.64 cm²
So, the lateral surface area of the square pyramid that can be curved out from a wooden cube with a side length of 12 cm is approximately 203.64 cm².