Calculate Triangle - Calculation of Area, Sides, Angles, Perimeter and Height

Contents on the topic "Calculate Triangle"

What are the three corners of a triangle called?

In a triangle, the three corners are usually labelled with the capital letters A, B and C. The labelling with A, B and C is usually done counterclockwise and starts with A at the bottom-left corner.

What are the three sides of a triangle called?

The three sides of a triangle are labelled with the lower-case letters a, b and c. The side opposite corner A is labelled a, the side opposite corner B is labelled b and the side opposite corner C is labelled c.

What are the three angles of a triangle called?

The three inner angles of a triangle are designated by the Greek letters α (alpha), β (beta) and γ (gamma). They are located at the corresponding corners, i.e., at corner A, B and C there is an angle α, β and γ, respectively.

What is the height of a triangle?

The height of a base side corresponds to the perpendicular distance from the opposite corner point to the base side or its extension. Thus the height of a (h_a) corresponds to the distance between the corner A and the opposite side a to which h_a is perpendicular. Accordingly, the heights of b (h_b) and c (h_c) are defined.

Which values are given?

Please select which values of the triangle are available for calculating the area or other properties of the triangle. Although the area of a triangle can be calculated quite easily using the first selection "One side and corresponding height h", the two entries are not sufficient to calculate an entire triangle exactly from them.

In order to calculate a triangle uniquely, other or further given values are required: If the values of one of the other options that can be selected here are given in each case, a unique triangle can be constructed for it. With these options, the usual matching abbreviations are also displayed. Here, "S" stands for the correspondence of a side length and "A" for the correspondence of an angle. A triangle can only be calculated unambiguously using these options. For example, a triangle cannot be unambiguously determined on the basis of only three given angles (AAA).

In the following, all selectable options, i.e., combinations of given values, are described in detail.

One side and corresponding height h

Please select this option if you know the length of a side a, b, or c of the triangle and the corresponding height. The height of a triangle at a base side g always corresponds to the perpendicular distance from the opposite corner point to the side g or its extension. Using the length of a side of the triangle and the corresponding height h, the area F of the triangle can be calculated. However, the calculation of the other sides and heights as well as the angles is not possible using these two given values.

All three sides a, b and c (SSS)

Please select this option if you know the lengths of all three sides of the triangle. When calculating triangles, a constellation with three given sides is often also abbreviated as 'SSS'. Using this information, the entire triangle can be constructed. Thus, the area and perimeter of the triangle, as well as the heights of a, b, and c, and the angles α, β and γ, can be calculated.

One side of an equilateral triangle (SSS)

Please select this option if the triangle is equilateral, i.e., it has three sides of equal length with known length a. This case for the equilateral triangle is a special case for the SSS calculation in the general triangle, since all three sides are known für a given side. One could also choose the previous option 'All three sides a, b and c (SSS)' to calculate the triangle, but the simplified formulas allow the calculation of the equilateral triangle.

To calculate the area and all other properties of the triangle, only the length of one side is required, since the length of all three sides is given at the same time. With this, all the other properties of the triangle can be calculated and thus the entire triangle can be constructed.

Two sides with an included angle (SAS)

Please select this option if two sides of the triangle are known together with the angle enclosed by them. When calculating triangles, a constellation in which an angle and its enclosing sides are known is often abbreviated as "SAS". This can be used, for example, to calculate the length of the third side, so that all other properties of the triangle can subsequently be calculated.

Two cathetus sides of a right triangle (SAS)

Please select this option if you know the lengths of these two sides of the right triangle. The cathetus are the two sides that lie against the right angle of the triangle, while the so-called hypotenuse lies opposite the right angle of the triangle.

Thus the case with two given cathethi forms a special case for the SAS calculation in the general triangle, since the intervening angle of 90 degrees is known anyway. One could therefore also choose the previous option "Two sides with an included angle (SAS)" to calculate the triangle, but the simplified formulas enable the calculation of the right triangle.

S So, for a right triangle, these two sides, namely the values of the two catheti, are sufficient to calculate all the other properties of the triangle and thus construct the entire triangle.

One side and two angles (SAA, AAS or ASA)

Please select this option if any side of the triangle and any two angles are known. When calculating triangles, a constellation where one side and two angles are known is often abbreviated as 'AAS', 'SAA' or 'ASA'. For example, the third angle can be calculated using the angle sum theorem and then all other properties of the triangle.

Two sides and an angle opposite the longer side (SsA or AsS)

Please select this option if two sides of the triangle as well as the angle opposite to the longer given side are known. When calculating triangles, a constellation where two sides and the opposite angle of the longer side are known is often also abbreviated as 'SsA' or 'AsS'.

Thus, with the help of the sine theorem, the angle opposite the smaller given side can be calculated. Subsequently, the third angle can be determined using the angle sum theorem and finally the entire triangle can be uniquely calculated and constructed. However, if only the angle opposite the shorter given side is given, the triangle cannot be calculated unambiguously.