Before getting to the actual formula of prism’s surface area, let’s see the definitions of prism and triangular prism.

Prism is the geometrical figure that is built with two identical top and bottom polygonal, which are called the bases. The remaining faces are called lateral. The planes of bases are parallel. The side faces are parallelograms.

If the base of the prism is a triangle, then this is a triangular prism. If the base of the prism is rectangular, then this is a rectangular prism, and so on. There are also decagonal and icosagonal prisms.

The triangular prism is a prism with three lateral faces in geometry. This polyhedron has triangular base as faces, its copy obtained by parallel transition, and three faces joining the corresponding sides.

Direct triangular prism has rectangular sides, otherwise the prism is called a skew prism.

Homogeneous triangular prism is a direct triangular prism with equilateral base and square sides.

Prism is a pentahedron, when its two faces are parallel, while the normal of other three lie in one plane (which is not necessarily parallel to the bases). These three faces are parallelograms. All sections parallel to the bases are identical triangles.

Computing the surface area of a triangular prism can be a daunting task if you've never done it before. However, the formulas for calculating the surface areas of many of these figures are quite understandable. When finding the surface area, it is necessary to calculate the area of each side, the bases, and summarize the obtained values.

There are four methods for calculating the surface area of a triangular prism, so let’s discuss each of them below.

As we already found out above, the triangular prism is a three-dimensional figure consisting of two triangular bases and three square or rectangular sides. When finding the surface area of a triangular prism you need to summarize the values of three sides and two bases.

Note that the three-dimensional figure with four triangular sides and a square base is called pyramid, rather than a triangular prism.

The main formula for calculating the surface area of a triangular prism: SA = L + 2 * B, where SA is the surface area, L is the side surface (the side surface is the area of all three rectangular sides (the sum of squares), and B is the area of the base. Since there are two bases and they are the same, you need to multiply this area by 2.

Expanded Formula

A more detailed version of this same formula can be written as: SA = ah + bh + ch + 2 * (1/2 * A * b), where

- A is the height of the triangle (which lies at the base of the prism).
- B is the base of the triangle (a triangle side, on which the height is lowered on).
- H is the height of the prism.
- a, b, and c are the triangle sides (lying in the base of the prism). Note that a and A in the formula have two different values.
- The formula for finding the lateral surface: ah + bh + ch.
- The formula for finding the surface area of the two triangular bases: 2 * (1/2 * A * b).

Usually, a formula for calculating the surface area of a triangular prism is written as: SA = h * (a + b + c) + (A * b), where h is takeoff the brackets in the record ah + bh + ch. Numbers 2 and 1/2 in the expression 2 * (1/2 * A * b) are simply cut and then there is only A * b.

As it was shown above, the formula for the surface area of a triangular prism is: (A * b), where A is the height of the triangle; b is the base of the triangle. For example: A = 2 cm and b = 4 cm.

Multiply the height and the base of the triangle, for example: A * b = 2 * 4 = 8 cm2.

This formula is not the same as the standard formula for calculating the surface area of a triangular prism, because this is the formula for finding the area of two triangles. In order to find the area of a triangle, we use the following formula: 1/2 * A * b. However, in our case it is necessary to summarize the areas of the two triangular prism bases.

As it was shown above, the formula for the side surface area of a triangular prism can be written as: h * (a + b + c), where h is the height of the prism (the longest side of the rectangle); a, b, and c are the triangle sides (lying on the base of the prism). For example: h = 7 cm, b = 4 cm, a = 6 cm, and c = 5 cm.

Summarize the values of the three sides of the triangle: a + b + c = 6 + 4 + 5 = 15 cm.

Then multiply this value by the height of the prism. As a result, we obtain the surface area of a triangular prism. As in an example: h * (a + b + c) = 7 * (6 + 4 + 5) = 7 * 105 = 15 cm2.

Thus, the lateral surface area of a triangular prism is the sum of three sides of a triangular prism.

- The standard formula for calculating the area of a rectangle is the length multiplied by the width.
- Here each rectangle has a total length. In the formula for calculating the lateral surface area, the rectangle’s length becomes the prism’s height, i.e. h.
- Each rectangle in the triangular prism has a width that corresponds to one side of the triangle a, b or c. Thus, here the triangle side replaces the width of the rectangle.

Remember that the surface area of a triangular prism is the sum of the areas of two triangular bases and lateral surface of the prism: SA = L + 2 * B. The detailed formula is the following: SA = h * (a + b + c) + (A * b).

Now add up the values found for the prism’s lateral surface area and the area of the base. For example: SA = L + 2 * B = 105 + 8 or SA = h * (a + b + c) + (A * b) = 7 * (6 + 4 + 5) + 2 = 7 * 4 * 15 * 2 + 4 + 8 = 105 + 8. Thus, you have successfully found the surface area of a triangular prism: SA = L + 2 * B = 105 + 8 = 113 cm2.

The volume of any prism equals to the product of the base area by the distance between the bases. In our case, when the base is triangular, you just need to calculate the area of a triangle and multiply it by the length of the prism: V = 1 / 2* bhl, where b is the length of the base, h is the height of the triangle, and l is the distance between the triangles.

The first geometrical concepts originated in prehistoric times, when a man observed different forms of material bodies in the nature: the forms of plants and animals, mountains and river meanders, circles and rectangular, etc. People not only passively observed the nature, but practically mastered and used its wealth. People accumulated geometric information. Material needs encouraged people to produce tools, hew stones and build houses, sculpt pottery and pull the string on the bow. Practical human activities served as the basis of a long process of development of abstract concepts, founding the simplest geometric constraints and relationships.

The basis of geometry was laid in ancient times when dealing with purely practical tasks. Eventually, when a large number of geometric facts accumulated, people had a need to summarize some elements and establish logical connections and evidence. The geometric science was gradually created.

A great deal of geometric knowledge was set almost 2,200 years ago in the «Elements» by Euclid. Of course, it is well known that Euclid in his work applied the the works of dozens of his predecessors, among whom were Thales and Pythagoras, Democritus, Hippocrates, Archytas, Theaetetus, Eudoxus, and others. At the cost of great efforts, based on the selected geometric information accumulated for thousands of years in the practice of people, these great scientists were able to lead the geometric science to high level of perfection for 3 – 4 centuries. The historical merit of Euclid is that he while creating his «Elements», combined the results of his predecessors, streamlined, and brought into a single system the main geometric knowledge of the time. For two millennia, the geometry has been studied to that extent, manner, and style, as it was outlined in the «Elements» by Euclid. Many textbooks of elementary geometry around the world were (and many still are) only processing information contained in Euclid’s book.

In the 17th century, Descartes due to the method of coordinates made it possible to study the properties of geometric figures using algebra. Since that time, the analytic geometry began developing. In the 17th – 18th centuries, the differential geometry that studies the properties of figures by means of mathematical analysis methods was conceived and developed. In the 18th – 19th centuries, the development of military art and architecture has led to the development of methods of accurate displaying of spatial figures on a flat figure, in connection with which the descriptive geometry was founded, the scientific foundations of which was laid by the French mathematician G. Monge. At the same time, the projective geometry was founded, the foundations of which were created in the writings of French mathematicians D. Dezarga and B. Pascal.

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