Find the surface area of a square pyramid with side length 6 yd and slant height 4 yd.
To find the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
The area of the base can be found by using the formula for the area of a square, which is side length squared. In this case, the side length is 6 yd, so the area of the base is 6 yd * 6 yd = 36 square yards.
To find the area of each triangular face, we can use the formula for the area of a triangle, which is 1/2 * base length * height. In this case, the base length is 6 yd and the slant height is 4 yd.
The height can be found by using the Pythagorean theorem. The height is the perpendicular distance from the apex of the pyramid to the base. The base of the triangle is half of the side length of the square, so it is 6/2 = 3 yd.
Using the Pythagorean theorem, we can find the height:
(Height)^2 + (3 yd)^2 = (4 yd)^2
(Height)^2 + 9 yd^2 = 16 yd^2
(Height)^2 = 16 yd^2 - 9 yd^2
(Height)^2 = 7 yd^2
Height = sqrt(7) yd
Now that we have the height, we can find the area of each triangular face:
Area of triangular face = 1/2 * base length * height
= 1/2 * 6 yd * sqrt(7) yd
= 3 sqrt(7) yd^2
There are four triangular faces, so the total area of the four triangular faces is 4 * 3 sqrt(7) yd^2 = 12 sqrt(7) yd^2.
The total surface area of the square pyramid is the sum of the area of the base and the area of the four triangular faces:
Surface area = area of base + area of four triangular faces
= 36 square yards + 12 sqrt(7) square yards
= 36 + 12 sqrt(7) square yards
Therefore, the surface area of the square pyramid is 36 + 12 sqrt(7) square yards.
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To find the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
The area of the base is found by squaring the side length. In this case, the base side length is 6 yd, so the area of the base is 6 yd * 6 yd = 36 square yards.
To find the area of each triangular face, we can use the formula for the area of a triangle: 1/2 * base length * height.
The base length of each triangular face is equal to the side length of the square base, which is 6 yd.
The height of each triangular face can be found using the Pythagorean theorem. The height is the slant height of the pyramid, which is given as 4 yd.
Using the Pythagorean theorem, we find:
(Height)^2 + (base length/2)^2 = (slant height)^2
(Height)^2 + 3^2 = 4^2
(Height)^2 + 9 = 16
(Height)^2 = 16 - 9
(Height)^2 = 7
Height = sqrt(7) yd
Now that we have the height, we can calculate the area of each triangular face:
Area of triangular face = 1/2 * base length * height
= 1/2 * 6 yd * sqrt(7) yd
= 3 sqrt(7) square yards
There are four triangular faces, so the total area of the four triangular faces is 4 * 3 sqrt(7) square yards = 12 sqrt(7) square yards.
The total surface area of the square pyramid is the sum of the area of the base and the area of the four triangular faces:
Surface area = area of base + area of four triangular faces
= 36 square yards + 12 sqrt(7) square yards
= 36 + 12 sqrt(7) square yards
Therefore, the surface area of the square pyramid with a side length of 6 yd and slant height of 4 yd is 36 + 12 sqrt(7) square yards.
To find the surface area of a square pyramid, you need to calculate the area of the base and the areas of the four triangular faces.
First, let's find the area of the base. Since the base of a square pyramid is a square, you can find its area by squaring the length of one of its sides. In this case, the side length of the square base is 6 yd, so the area of the base is 6 yd * 6 yd = 36 square yards.
Next, let's find the areas of the triangular faces. Each triangular face is a right triangle with one leg being the slant height and the other leg being half the length of a side of the base.
For a right triangle, the area can be calculated using the formula:
Area = (base * height) / 2
The height of each triangular face is the slant height, which is 4 yd. The base of each triangular face is half the length of a side of the base, which is 6 yd / 2 = 3 yd.
Using the formula, the area of each triangular face is:
Area = (3 yd * 4 yd) / 2 = 6 yd^2
Since there are four triangular faces in total, you need to calculate the total area of the triangular faces by multiplying the area of each triangular face by four:
Total area of triangular faces = 6 yd^2 * 4 = 24 yd^2
Finally, to find the surface area of the square pyramid, you need to add the area of the base and the total area of the triangular faces:
Surface area = Area of base + Total area of triangular faces
Surface area = 36 yd^2 + 24 yd^2 = 60 yd^2
Therefore, the surface area of the square pyramid with side length 6 yd and slant height 4 yd is 60 square yards.