VIDEO: How do you calculate the base area of ​​a prism?

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This is how you calculate base areas with 3 sides

A prism with 3 sides has a triangular base. As you will probably remember, there are several ways to calculate the area of ​​triangles.

  1. If you know the length of at least one side of the triangle and the heights on that side, all you have to do is calculate the length of the side times the height and divide the result by 2. The height is the vertical on one side that ends in the opposite corner point.
  2. A right angled prism has a right triangle as its base, in which case you can use the length of the sides that make up the right one angle Form, multiply and divide by 2 to find the base area.
  3. In the isosceles triangle, the height of the base side is exactly in the middle of this side. In this case you can calculate the height using the Pythagorean Theorem hc2= a2- (1/2 c)2. So the area of ​​the triangle is 1/2.c. Hc, where hc the root of hc2 is. As a reminder: c2= a2+ b2, where c is the longest side in a right triangle. Hc and c / 2 are a short side of the right triangle and a is the longest side in this case.
  4. In the case of the equilateral prism, c = a, because the triangle consists of 3 equal sides which are called a. in that case h2= a2- (1 / 2a) 2 = a2-1/4 a2= 3/4 a2.
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  5. So H is a / 2 times the root 3 and the area F = 1/2 a h = a / 2 a / 2 root 3 = a2/ 4 root 3.
  6. If you know 3 sides of the prism and this is neither equilateral, isosceles nor right-angled, you have to calculate the area according to Heron's theorem. Form the sum of the 3 sides and divide them by 2, they have half the circumference, this is called s. Now calculate three 3 values ​​that result from the difference between the circumference and one side each, so you calculate s-a, s-b and s-c. Now you have to multiply these 3 values ​​together and additionally with s. The root of the product of this 3 Counting is the area of ​​the triangle.

These are the different ways of calculating the area of ​​a triangular prism.

Area calculation for a square prism

  • For a square prism, calculate the base area using the formula a2 and in the case of a rectangular one according to the formula a.b. So you only have to multiply 2 sides that are at right angles to each other.
  • The base of the prism can also be a trapezoid or a parallelogram. Here you either have to know the distance between the parallel sides or calculate it according to the Pythagorean theorem by cleverly dividing the figures into triangles. In the case of a parallelogram, the area is the side times the side distance (also known as g.h) and in the case of the trapezoid the sum of the two parallel sides multiplied by 2 by the distance ((a + c) / 2.H).

Calculate the base of a regular prism

A regular prism can consist of any number of sides, but they are all of the same length. Regular prisms are also isosceles Triangles or square prisms. But also hexagonal, octagonal or prisms with any number of sides belong to it. In this case one speaks of n-cornered prisms. The bases are regular polygons. How to calculate the base of this prism:

  1. As you can see from the sketch, dividing the polygon into isosceles triangles is not a problem. The base of the triangles is the edge length a of the prism and the side length is the radius of the circumference R. The height of the triangles corresponds to the incircle radius r. You also know that the angle at the apex of the triangle is 360 °: n. So the angle of the right triangle formed from h, a / 2 and R is 180 °: n.
  2. The area of ​​the polygon that forms the base of the prism is n times the area of ​​the individual triangles. You now need to calculate as the area of ​​a triangle and multiply by n.
  3. Depending on which sides you are familiar with, you have to determine the values ​​h and a required for calculating the area. Note that there is the following relationship: a = 2 R sin (180 ° / n) and r = h = R cos (180 ° / n). So you can easily determine the area by simply calculating the missing sizes.

Since the base area of ​​a prism can be very different, it is always calculated according to the formula that must be used for the respective base area.

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