Find the surface area of a triangular pyramid with b =4 cm, h= 8.2 cm and s = 7 cm. PLEASE. How ?

GEOMETRY (SA oftriangular pyramid) - MathMate, Sunday, July 14, 2013 at 8:41am
Lateral surface area
=perimeter of base * apothem /2
=3b*s/2
(for a regular pyramid with a triangular base)

Area of triangular base with side b
=(√3)*b/2*(b/2)
=(√3)b²/4
GEOMETRY (SA oftriangular pyramid) - Anonymous, Sunday, July 14, 2013 at 10:05am
What is apothem ?
GEOMETRY (SA oftriangular pyramid) - Anonymous, Sunday, July 14, 2013 at 10:15am
why did you not include the height ?

In the calculations of surface area of a pyramid, the height is not required if the apothem (slant height) is given.

The height is required, however, to calculate the volume of a pyramid.

In fact, the height h is not consistent, because h should always be less than the slant height, s.

If you could check the question for typos, there may be a different answer.

To find the surface area of the triangular pyramid, you need to calculate the lateral surface area and add it to the area of the triangular base.

1. Calculate the lateral surface area:
The lateral surface area of a regular pyramid with a triangular base is given by the formula:
Lateral surface area = perimeter of base * apothem / 2
Since the base of the pyramid is a triangle, the perimeter of the base is equal to 3 times the side length (b).
So, the lateral surface area = 3b * apothem / 2

2. Calculate the area of the triangular base:
The area of a triangle with side length (b) is given by the formula:
Area = (√3) * b² / 4

3. Determine the apothem:
The apothem of the pyramid is the distance from the center of the base triangle to the apex (top) of the pyramid. To calculate the apothem, you need the height (h) and the slant height (s). The slant height (s) is a line segment from the apex to the center of one of the triangular faces.
There is a relationship between the height, slant height, and apothem given by the formula:
(h² + (b/2)²) = s² - apothem²
You can rearrange this formula to solve for the apothem:
apothem = √(s² - (h² + (b/2)²))

4. Substitute the given values into the formulas:
b = 4 cm
h = 8.2 cm
s = 7 cm

Calculate the apothem:
apothem = √(7² - (8.2² + (4/2)²))

Once you have the value of the apothem, substitute it into the formula for the lateral surface area and calculate it. Finally, add the lateral surface area to the area of the triangular base to get the total surface area of the triangular pyramid.

To find the surface area of a triangular pyramid, you need to calculate the lateral surface area of the pyramid and the area of the triangular base.

1. Lateral surface area:
The lateral surface area of a regular pyramid with a triangular base can be found using the formula: perimeter of base * apothem / 2. For a regular triangular pyramid, the lateral surface area is 3b * s / 2, where b is the base length and s is the slant height.

In this case, b = 4 cm and s = 7 cm. So the lateral surface area of the pyramid is:
= 3 * 4 cm * 7 cm / 2
= 84 cm²

2. Area of the triangular base:
To find the area of the triangular base, you can use the formula for the area of a triangle: base * height / 2. In this case, the base of the triangle is b = 4 cm and the height is h = 8.2 cm. So the area of the triangular base is:
= 4 cm * 8.2 cm / 2
= 16.4 cm²

3. Total surface area:
To find the total surface area, you need to add the lateral surface area and the area of the triangular base. Thus, the total surface area of the triangular pyramid is:
= 84 cm² + 16.4 cm²
= 100.4 cm²

So, the surface area of the given triangular pyramid is 100.4 cm².