PLEASE HELP Find the Lateral Area, Total Area, and Volume of regular pyramid with a base edge of 18cm and a slant height of 15cm round to the nearest tenth
Assuming that by "regular pyramid" you mean that the base is a square.
Area of one of the triangles = (1/2)(18)(15) = 135 cm^2
lateral area = 4(135) = 540 cm^2
total area, I assume you want to add the base area, which is 18^2 or 324 cm^2 to the lateral area
for the volume you need the height of the pyramid
h^2 + 9^2 = 15^2
h = 12 cm
vol = (1/3) base x height
= (1/3)(324)(12) = .... cm^3
To find the lateral area, total area, and volume of a regular pyramid, we'll need to use some formulas. Let's break it down step by step:
1. Lateral Area of a Pyramid:
The lateral area is the sum of the areas of the triangular faces that make up the sides of the pyramid. The formula to find the lateral area (LA) of a regular pyramid is LA = (1/2) × Perimeter of Base × Slant Height.
In this case, the base edge is given as 18cm and the slant height as 15cm.
Perimeter of the Base = 4 × Base Edge
= 4 × 18 cm
= 72 cm
So, the Lateral Area (LA) = (1/2) × 72 cm × 15 cm
2. Total Area of a Pyramid:
The total area (TA) includes both the lateral area and the area of the base. For a regular pyramid, the base is a regular polygon, in this case, a square. The formula to find the total area is TA = LA + Base Area.
To find the Base Area, we need to calculate the area of a square with a side length equal to the base edge.
Base Area = Base Edge^2
= 18 cm × 18 cm
= 324 cm^2
So, the Total Area (TA) = Lateral Area (LA) + Base Area
3. Volume of a Pyramid:
The volume (V) of a pyramid can be found using the formula V = (1/3) × Base Area × Height.
The height of the pyramid is the perpendicular distance from the apex (top vertex) to the base.
In this case, as it is a regular pyramid, the height will be the distance from the apex to the center of the base. To find the height, we can use the Pythagorean Theorem.
Height^2 = Slant Height^2 - (Base Edge/2)^2
= 15 cm^2 - (18 cm/2)^2
= 225 cm^2 - 81 cm^2
= 144 cm^2
Height = √(144 cm^2)
= 12 cm
So, the Volume (V) = (1/3) × Base Area × Height
= (1/3) × 324 cm^2 × 12 cm
Now that we have all the formulas and values, let's calculate the results:
- Lateral Area = (1/2) × 72 cm × 15 cm
- Total Area = Lateral Area + Base Area
- Volume = (1/3) × 324 cm^2 × 12 cm
Plugging in the values and performing the calculations, rounding to the nearest tenth, we get:
- Lateral Area ≈ 540.0 cm^2
- Total Area ≈ 864.0 cm^2
- Volume ≈ 864.0 cm^3
Therefore, the rounded values to the nearest tenth are:
Lateral Area ≈ 540.0 cm^2,
Total Area ≈ 864.0 cm^2, and
Volume ≈ 864.0 cm^3.