Determine the volume of a triangular prism with sides of the triangular base measuring 10cm, 14 cm, and 21 cm and having height 30 cm.
That will not work because you do not know the altitude of the triangle. You must get the area of the triangular base from the lengths of the three sides.
To find the altitude from the 21 cm base:
call the part next to the 10 cm leg x
call the part next to the 14 cm leg (21-x)
call the altitude h
then
x^2 + h^2 = 100
(21-x)^2 + h^2 = 196
so
(21-x)^2 -x^2 =96
441 - 42x = 96
42 x = 345
x = 8.214
x^2 = 67.47
100-x^2 = 32.5 = h^2
h = 5.7
so
(1/2)h *21 = 59.9 = area of base triangle
multiply by 30 to get volume
To determine the volume of a triangular prism, you can use the formula:
Volume = Area of the base x Height
Let's break this down step by step:
Step 1: Calculate the area of the triangular base.
The formula to calculate the area of a triangle is given by:
Area = (base x height) / 2,
where the base is the length of one side of the triangle, and the height is the perpendicular distance between the base and the opposite vertex.
In this case, we have the lengths of all three sides of the triangle. We can use Heron's formula to calculate the area of the triangle:
s = (a + b + c) / 2,
where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter (half the perimeter).
Using the given side lengths:
a = 10 cm
b = 14 cm
c = 21 cm
Calculating the semi-perimeter:
s = (10 + 14 + 21) / 2 = 45 / 2 = 22.5.
Now, we can calculate the area using Heron's formula:
Area = √(s(s-a)(s-b)(s-c)),
where √ denotes the square root.
Substituting the values:
Area = √(22.5(22.5-10)(22.5-14)(22.5-21)).
Calculating inside the square root:
Area = √(22.5(12.5)(8.5)(1.5)).
Evaluating the square root using a calculator:
Area ≈ √(2531.25) ≈ 50.312 cm² (rounded to three decimal places).
Step 2: Calculate the volume of the prism.
Now that we have the area of the triangular base, we can calculate the volume of the prism by multiplying it with the height:
Volume = Area x Height,
Volume = 50.312 cm² x 30 cm,
Volume ≈ 1509.36 cm³ (rounded to two decimal places).
Therefore, the volume of the triangular prism is approximately 1509.36 cm³.