Find the lateral and surface area of the regular square pyramid. Give exact answers. The length is 6 m and the height is 5 m. I need help.
A regular square pyramid has a square on the bottom. Each side of the square is the base of a triangle, (so there are four triangles).
The base is a square. Lateral area is the surface area of a 3D figure, but excluding the area of the base.
The area of each triangle is A=1/2(b)(h)
A=1/2(6)(5)=15
Multiply by four because there are four triangles to get 60 square units as your lateral area.
For the surface area, figure the area of the square base.
A=s^2
A=6^2=36
Add that to the lateral area (60+36) to get a surface area of 96 square units.
To find the lateral area of a square pyramid, you need to calculate the area of the four triangular faces. The formula for the lateral area of a pyramid is given by:
Lateral Area = (perimeter of the base) × (slant height) / 2.
Let's start by finding the perimeter of the base of the pyramid. Since you have a regular square pyramid, it means that all four sides of the base are equal. In your case, you mentioned the length of the square base is 6 m. Since a square has all sides equal, the perimeter of the base can be calculated by multiplying the length of one side by 4:
Perimeter = 4 × (length of one side) = 4 × 6 = 24 m.
Next, let's find the slant height of the pyramid. The slant height is the perpendicular distance from the apex (top vertex) to the base, along the slanting edges. In your case, the height of the pyramid is given as 5 m.
To find the slant height, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides (height and half the base length).
In this case, the height is 5 m, and half the base length is 6/2 = 3 m. So, using the Pythagorean theorem, we can find the slant height:
Slant height = √(height^2 + (1/2 * base length)^2) = √(5^2 + 3^2) = √34 m.
Now that we have the perimeter of the base (24 m) and the slant height (√34 m), we can calculate the lateral area using the formula mentioned earlier:
Lateral Area = (Perimeter of the base) × (Slant height) / 2
= (24 m) × (√34 m) / 2
= 12√34 m^2 (exact answer)
To find the surface area of the square pyramid, we need to include the base area as well. The base area of a square can be calculated by squaring the length of one side:
Base Area = (Length of one side)^2
= 6 m^2.
So, the surface area of the square pyramid is the sum of the base area and the lateral area:
Surface Area = Base Area + Lateral Area
= 6 m^2 + 12√34 m^2 (exact answer).