The formula to calculate the height of a triangle is:
\( h = \dfrac{2A}{b} \)
Where:
To calculate the area of a triangle, use the following formula:
\( \text{Area} = \dfrac{1}{2} \times b \times h \)
Where:
Alternatively, you can calculate the area using Heron’s Formula, which only requires the lengths of the three sides:
\( s = \dfrac{a + b + c}{2} \)
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)
Where:
Triangles are polygons with three vertices. You can calculate their sides and angles using the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.
Using the Law of Sines, you can find the missing side or angle of a triangle:
\( \dfrac{a}{\sin(A)} = \dfrac{b}{\sin(B)} = \dfrac{c}{\sin(C)} \)
For a right triangle with one angle and the hypotenuse given:
a = c \cdot \sin(\alpha) or a = c \cdot \cos(\beta)
b = c \cdot \sin(\beta) or b = c \cdot \cos(\alpha)
For right triangles, the Pythagorean theorem helps find the third side:
\( a^2 + b^2 = c^2 \)
If one side is missing, rearrange the formula:
To find side a:
\( a = \sqrt{c^2 - b^2} \)
To find the hypotenuse c:
\( c = \sqrt{a^2 + b^2} \)
The Law of Cosines is used to find a side or angle when other values are known:
\( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \)
Solving for \( \cos(A): \)
\( \cos(A) = \dfrac{b^2 + c^2 - a^2}{2bc} \)
Similarly:
\( b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \)
\( \cos(B) = \dfrac{a^2 + c^2 - b^2}{2ac} \)
\( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)
\( \cos(C) = \dfrac{a^2 + b^2 - c^2}{2ab} \)
If one angle and one side are given:
a = b \cdot \tan(\alpha)
b = a \cdot \tan(\beta)
If the area and one side are given:
\( \text{Area} = \dfrac{a \cdot b}{2} \)
To find side b:
\( b = \dfrac{2 \cdot \text{Area}}{a} \)
To find the hypotenuse c:
\( c = \sqrt{a^2 + \left( \dfrac{2 \cdot \text{Area}}{a} \right)^2} \)
It is the radius of the inscribed circle. In general, it's the circle that can fit inside the triangle and is perpendicular to each side of the polygon.
Use the below-mentioned formula to calculate the inradius of a triangle:
\(\ r=\dfrac{Area}{semiperimeter}\)
As the name shows it is the radius of the circumscribed circle. It is the minimum size of the circle that can fit inside the triangle.
Let's see the following formula that is used to calculate the circumradius of the triangle:
\(\ R=\dfrac{a}{2\sin(A)}\)
Predict a triangle along with its terms with the following given information:
Solution:
As we are given two angles and one side, let us start!
Step # 01 (Find Angle of Triangle):
\(\ m∠C = 180° - A - B\)
\(\ m∠C = 180^{o} - 50^{o} - 40^{o}\)
\(\ m∠C = 90^{o}\)
Now converting all angles into radians as follows:
\(\ m∠A = 50^{o}\times \frac{\pi}{180}\)
\(\ m∠A = 0.87266\ rad\)
Similarly:
\(\ m∠B = 40^{o}\times \frac{\pi}{180}\)
\(\ m∠B = 0.69813\ rad\)
Likewise:
\(\ m∠C = 90^{o}\times \frac{\pi}{180}\)
\(\ m∠C = 1.5708\ rad\)
Step # 02 (How To Find The Side of A Triangle?):
\(\ b = \frac{a\times \sin(B)}{\sin(A)}\)
\(\ b = \frac{4 \times \sin(0.69813)}{\sin(0.87266)}\)
\(\ b = 3.23728\)
Similarly:
\(\ c = \frac{a\times \sin(C)}{\sin(A)}\)
\(\ c = \frac{4 \times \sin(1.5708)}{\sin(0.87266)}\)
\(\ c = 4.58257\)
Step # 03 (Calculating The Triangle Area):
\(\ A = \frac{ab\sin(C)}{2}\)
\(\ A = \frac{4\times3.23728\sin(1.5708)}{2}\)
\(\ A = 6.47456\)
Step # 04 (Calculating Perimeter And Semiperimeter):
\(\ Perimeter=\ p = a + b + c\)
\(\ Perimeter=\ p = 4 + 3.23728 + 4.58257\)
\(\ Perimeter=\ p = 11.81985\)
Similarly:
\(\ Semiperimeter=\ s = \frac{a + b + c}{2}\)
\(\ Semiperimeter=\ s = \frac{4 + 3.23728 + 4.58257}{2}\)
\(\ Semiperimeter=\ s = 5.90993\)
Step # 05 (Calculation of Heights of Triangle Sides):
\(\ Height=\ h_{a}=\frac{2 \times { Area}}{a}\)
\(\ Height=\ h_{a}=\frac{2 \times 6.47456}{4}\)
\(\ Height=\ h_{a} = 3.23728\)
Similarly:
\(\ Height=\ h_{b}=\frac{2 \times { Area}}{b}\)
\(\ Height=\ h_{b}=\frac{2 \times 6.47456}{3.23728}\)
\(\ Height=\ h_{b} = 4.00000\)
And:
\(\ Height=\ h_{c}=\frac{2 \times { Area}}{c}\)
\(\ Height=\ h_{c}=\frac{2 \times 6.47456}{4.58257}\)
\(\ Height=\ h_{c} = 2.82355\)
Step # 06 (Determining Medians Of Each Side):
\(\ Median=\ m_{a}=\sqrt{(\frac{a}{2})^2 + c^2 - ac\cos(B)}\)
\(\ Median=\ m_{a}=\sqrt{(\frac{4}{2})^2 + 4.58257^2 - 4\times4.58257\cos(0.69813)}\)
\(\ Median=\ m_{a} = 2.97665\)
Similarly:
\(\ Median=\ m_{b}=\sqrt{(\frac{b}{2})^2 + a^2 - ab\cos(C)}\)
\(\ Median=\ m_{b}=\sqrt{(\frac{3.23728}{2})^2 + 4^2 - 4\times3.23728\cos(1.5708)}\)
\(\ Median=\ m_{b} = 3.63730\)
Now:
\(\ Median=\ m_{c}=\sqrt{(\frac{c}{2})^2 + b^2 - bc\cos(A)}\)
\(\ Median=\ m_{c}=\sqrt{(\frac{4.58257}{2})^2 + 3.23728^2 - 3.23728\times4.58257\cos(0.87266)}\)
\(\ Median=\ m_{c} = 3.86590\)
Step # 07 (Finding Inradius):
\(\ Inradius=\ r=\frac{Area}{s}\)
\(\ Inradius=\ r=\frac{6.47456}{5.90993}\)
\(\ Inradius\ r=1.0956\)
Step # 08 (Finding Circumradius):
\(\ Circumradius\ R=\frac{a}{2\sin(A)}\)
\(\ Circumradius\ R=\frac{4}{2 \times \sin(0.87266)}\)
\(\ Circumradius\ R=2.2942\)
| Property | Example | Formula |
|---|---|---|
| Base (b) | b = 5 | |
| Height (h) | h = 10 | |
| Area of Triangle | Area = \( \frac{1}{2} \times b \times h \) | Area = \( \frac{1}{2} \times 5 \times 10 = 25 \) |
| Side 1 (a) | a = 3 | |
| Side 2 (b) | b = 4 | |
| Side 3 (c) | c = 5 | |
| Perimeter of Triangle | Perimeter = a + b + c | Perimeter = 3 + 4 + 5 = 12 |
| Heron's Formula (Area) | s = \( \frac{3 + 4 + 5}{2} = 6 \) | Area = \( \sqrt{s(s-a)(s-b)(s-c)} \) |
| Angle between sides a and b | \( \theta \) | Cosine Rule: \( \cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab} \) |
| Right Angle Triangle Check | Pythagorean Theorem: \( a^2 + b^2 = c^2 \) | Check if 3² + 4² = 5² |
A geometric Angle-Side analyzer helps in discerning various characteristics of a triangle, for example, its angular measurements, linear edges, spatial extent, and outer boundary, upon suppliing known quantities. This instrument applies spatial concepts similar to the Principle of Sines, Rule of Cosines, and Heron's Theorem to secure unspecified information from the provided information. It streamlines rectilinear figure arithmetic and is applied in domains such as geometry, mechanics, and construction.
To activate the Triangle Measurement Device, enter the specified data concerning the triangle, for example, the dimensions of the sides, angle degrees, or alternative metrics. Considering the data given, the math device will determine the remaining characteristics of the geometric figure. This may involve rewriteing the information involving additional angles, sides, areas, and perimeters based on what is provided.
The Triangle Calculator solves different triangle types such as equal-sided, equal-sided-angle, inequal-sided, right-angle, smaller-angle, and bigger-angle triangles. Depending on what information you give (sides or angles), this calculator can figure out the hidden facts for these triangles.
The law of trigonometric sines constitutes a method used to deduce the absent sides or angular measurements in a quadrilateral, when aware of certain linear dimensions and angular measurements. "The law connects each side of a triangle with the sin of its corresponding vertiginous angle, enabling the determination of ambiguous values in scalene triangles. "Please rearticulate the previous statement using only substitIt's useful in solving oblique triangles.
His Rule helps to determine elusive sides or angles in a triangular figure if you possess knowledge of two sides and the encompassing angle or when the measurements of all three sides are present. This tool shines in solving triangles that are not a right angle and fits acute and obtuse triangles well.
Certainly, the Trigonometry Tool can calculate the triangle’s space using the provided information. It can use various ways such as using the fundamental and elevation or by leveraging the components and angles to determine the space. The calculation method will vary depending on the information provided.
If one of the angles is perpendicular, the calculator can quickly determine the missing length with the Pythagorean theorem and gauge the square footage with the length and vertical measure.
The Triangle Calculator is highly accurate when provided with accurate input values. The results rely on trusted shapes and angle rules, guaranteeing stable figure calculations. Even though it matters, the truth still needs the exact numbers we give it, so we have to make sure we write down right.
To find the area of a triangle when you know the lengths of all three sides, use Heron's formula. It is an essential approach for measuring the space without having to require the elevation of the triangle. The computer uses this equation when receiving all sides' measurements to determine the space.
Yes, the Triangle Calculator is extremely helpful in solving real-world problems. It serves multiple areas such as establishment, creation, science, and travel orientation. This instrument can help in crafting constructions, computing separations, establishing angles in engineering, and solving issues related to force vectors in mechanics.
