Identify the lateral area and surface area of a regular square pyramid with base edge length 11 cm and slant height 15 cm.
To identify the lateral area and surface area of a regular square pyramid with the given measurements, we will break down the problem into steps.
Step 1: Find the slant height of the triangular faces using the Pythagorean theorem.
In a regular square pyramid, the base is a square, and the triangular faces are isosceles triangles. The slant height (l) connects the vertex (apex) of the pyramid to the midpoint of one of the edges of the square base.
To find the slant height, we can use the Pythagorean theorem:
l^2 = h^2 + (a/2)^2
where l is the slant height, h is the height of the pyramid, and a is the length of one side of the base (base edge).
Given the slant height (l) is 15 cm, and the base edge length (a) is 11 cm. We can substitute these values into the Pythagorean theorem:
15^2 = h^2 + (11/2)^2
225 = h^2 + 30.25
h^2 = 225 - 30.25
h^2 = 194.75
h ≈ √194.75
h ≈ 13.95 cm
Step 2: Calculate the lateral area.
The lateral area of a pyramid refers to the sum of the areas of the triangular faces. In this case, since the base is a square pyramid, there are four identical triangular faces.
The formula to calculate the area of a triangle is:
Area = (1/2) × base × height
In the case of a regular square pyramid, we can consider each triangular face as an isosceles triangle with base equal to the length of one side of the square base (a = 11 cm) and height equal to the slant height (l = 15 cm).
So, the area of each triangular face is:
Area = (1/2) × a × l
Substituting the values, we get:
Area = (1/2) × 11 × 15
Area = 82.5 cm^2
Since there are four identical triangular faces, the total lateral area is:
Lateral Area = 4 × Area
Lateral Area = 4 × 82.5
Lateral Area = 330 cm^2
Therefore, the lateral area of the given square pyramid is 330 cm^2.
Step 3: Calculate the surface area.
The surface area of a regular square pyramid consists of the sum of the lateral area and the area of the base (which is a square).
To calculate the area of the base, we use the formula:
Area = side^2
Where the side length (s) is equal to the base edge length (a).
So, the area of the base is:
Area = 11^2
Area = 121 cm^2
Now, we can calculate the surface area by adding the lateral area and the area of the base:
Surface Area = Lateral Area + Base Area
Surface Area = 330 + 121
Surface Area = 451 cm^2
Therefore, the surface area of the given square pyramid is 451 cm^2.
To find the lateral area and surface area of a regular square pyramid, we can use the formulas:
Lateral Area = perimeter of the base × slant height
Surface Area = area of the base + lateral area
Here, the base of the pyramid is a square, so the perimeter of the base is found by multiplying the base edge length by 4.
Given:
Base edge length = 11 cm
Slant height = 15 cm
Step 1: Calculate the lateral area.
Perimeter of the base = base edge length × 4
Perimeter of the base = 11 cm × 4
Perimeter of the base = 44 cm
Lateral Area = perimeter of the base × slant height
Lateral Area = 44 cm × 15 cm
Lateral Area = 660 cm²
Step 2: Calculate the surface area.
Area of the base = base edge length² = 11 cm × 11 cm = 121 cm²
Surface Area = area of the base + lateral area
Surface Area = 121 cm² + 660 cm²
Surface Area = 781 cm²
Therefore, the lateral area of the pyramid is 660 cm², and the surface area is 781 cm².
lateral area is four triangles with base 11 and altitude 15
Surface is that area plus the square base.
I assume you can find the area of triangles and squares...