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Triangle Calculator
How to use this Triangle Calculator 🤔 There are input fields for side length \((a)\) , side length \((b)\) , side length \((c)\) , angle \((∠A)\) , angle \((∠B)\) , angle \((∠C)\) , Area \((Area)\) , Perimeter \((P)\) , and \((s)\) . Enter any three inputs that defines a triangle, The calculator uses the formula, substitues given values, and calcuates the missing value. The missing value is calcuated and displayed in the input field. Also, the caculation is displayed under the input section. Examples In the following examples, we cover scenarios where we find all the properties of a given triangle: all the three sides, all the three angles, perimeter, and area.
1. Example to solve triangle when three sides are given.Answer Given:
Side a = 5 Side b = 6 Side c = 7 Solving Perimeter:
The perimeter \( P \) is calculated using the formula:
\( P = a + b + c \)
\( P = 5 + 6 + 7 \)
\( P = 18 \)
Solving half perimeter:
The half perimeter \( s \) is calculated using the formula:
\( s = \frac{a + b + c}{2} \)
\( s = \frac{5 + 6 + 7}{2} \)
\( s = 9 \)
Solving area:
The area is calculated using Heron's formula:
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)
\( \text{Area} = \sqrt{9(9-5)(9-6)(9-7)} \)
\( \text{Area} = \sqrt{9 \times 4 \times 3 \times 2} \)
\( \text{Area} = \sqrt{216} \)
\( \text{Area} = 14.7 \)
Solving angle A:
\( \angle A = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right) \frac{180}{\pi} \)
\( \angle A = \cos^{-1} \left( \frac{6^2 + 7^2 - 5^2}{2 \times 6 \times 7} \right) \frac{180}{\pi} \)
\( \angle A = \cos^{-1} \left( \frac{36 + 49 - 25}{84} \right) \frac{180}{\pi} \)
\( \angle A = \cos^{-1} \left( \frac{60}{84} \right) \frac{180}{\pi} \)
\( \angle A = \cos^{-1} \left( 0.7143 \right) \frac{180}{\pi} \)
\( \angle A = 44.42° \)
Solving angle B:
\( \angle B = \cos^{-1} \left( \frac{a^2 + c^2 - b^2}{2ac} \right) \frac{180}{\pi} \)
\( \angle B = \cos^{-1} \left( \frac{5^2 + 7^2 - 6^2}{2 \times 5 \times 7} \right) \frac{180}{\pi} \)
\( \angle B = \cos^{-1} \left( \frac{25 + 49 - 36}{70} \right) \frac{180}{\pi} \)
\( \angle B = \cos^{-1} \left( \frac{38}{70} \right) \frac{180}{\pi} \)
\( \angle B = \cos^{-1} \left( 0.5429 \right) \frac{180}{\pi} \)
\( \angle B = 57.12° \)
Solving angle C:
\( \angle C = 180 - \angle A - \angle B \)
\( \angle C = 180 - 44.42 - 57.12 \)
\( \angle C = 78.46° \)
2. Example to solve triangle when two sides and included angle are given.Answer Given:
Side a = 5 Side b = 6 Included Angle ∠C = 45° Solving Side c:
The side c is calculated using the formula:
\( c = \sqrt{a^2 + b^2 - 2ab \cos \angle C} \)
\( c = \sqrt{5^2 + 6^2 - 2 \times 5 \times 6 \times \cos 45°} \)
\( c = \sqrt{25 + 36 - 60 \times 0.7071} \)
\( c = \sqrt{25 + 36 - 42.426} \)
\( c = \sqrt{18.574} \)
\( c = 4.31 \)
Solving Perimeter:
The perimeter \( P \) is calculated using the formula:
\( P = a + b + c \)
\( P = 5 + 6 + 4.31 \)
\( P = 15.31 \)
Solving angle A:
\( \angle A = \sin^{-1} \left( \frac{a \sin \angle C}{c} \right) \frac{180}{\pi} \)
\( \angle A = \sin^{-1} \left( \frac{5 \times \sin 45°}{4.31} \right) \frac{180}{\pi} \)
\( \angle A = \sin^{-1} \left( \frac{5 \times 0.7071}{4.31} \right) \frac{180}{\pi} \)
\( \angle A = \sin^{-1} \left( \frac{3.5355}{4.31} \right) \frac{180}{\pi} \)
\( \angle A = \sin^{-1} (0.8202) \frac{180}{\pi} \)
\( \angle A = 55.32° \)
Solving angle B:
\( \angle B = 180 - \angle A - \angle C \)
\( \angle B = 180 - 55.32 - 45 \)
\( \angle B = 79.68° \)
Solving Area:
The area is calculated using the formula:
\( \text{Area} = \frac{1}{2} ab \sin \angle C \)
\( \text{Area} = \frac{1}{2} \times 5 \times 6 \times \sin 45° \)
\( \text{Area} = \frac{1}{2} \times 5 \times 6 \times 0.7071 \)
\( \text{Area} = \frac{1}{2} \times 30 \times 0.7071 \)
\( \text{Area} = 15 \times 0.7071 \)
\( \text{Area} = 10.61 \)
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"input_pre_msg": "Enter any three inputs that defines a triangle,<br> and click on Calculate button",
"type": "Calculate",
"title": "Triangle Calculator",
"description": "Use our Triangle Calculator to calculate lengths of three sides, three angles, Area, and Permimeter of a given triangle, with detailed formuals and solution.",
"category": "Geometry",
"shape": "Triangle",
"template": "mathematics_multiple_outputs",
"formulas": [
{
"parameters": [
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"outputs": [
{
"label": "P",
"formula": " a + b + c ",
"formula_mathjax": "\\( a + b + c \\)"
},
{
"label": "s",
"formula": "( a + b + c )/2",
"formula_mathjax": "\\( ( a + b + c )/2\\)"
},
{
"label": "Area",
"formula": " Math.sqrt( s * ( s - a ) * ( s - b ) * ( s - c ) ) ",
"formula_mathjax": "\\( \\sqrt{s(s-a)(s-b)(s-c)} \\)"
},
{
"label": "∠A",
"formula": " Math.acos( ( b * b + c * c - a * a ) / ( 2 * b * c ) ) * ( 180 / Math.PI ) ",
"formula_mathjax": "\\( \\cos^{-1} \\left( \\frac{b^2 + c^2 - a^2}{2bc} \\right) \\frac{180}{\\pi} \\)"
},
{
"label": "∠B",
"formula": " Math.acos( ( a * a + c * c - b * b ) / ( 2 * a * c ) ) * ( 180 / Math.PI ) ",
"formula_mathjax": "\\( \\cos^{-1} \\left( \\frac{a^2 + c^2 - b^2}{2ac} \\right) \\frac{180}{\\pi} \\)"
},
{
"label": "∠C",
"formula": " 180 - ∠A - ∠B ",
"formula_mathjax": "\\( \\cos^{-1} \\left( \\frac{a^2 + b^2 - c^2}{2ab} \\right) \\frac{180}{\\pi} \\)"
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"parameters": [
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"outputs": [
{
"label": "c",
"formula": " Math.sqrt( a * a + b * b - 2 * a * b * Math.cos( ∠C * Math.PI / 180 ) ) ",
"formula_mathjax": "\\( \\sqrt{a^2 + b^2 - 2ab \\cos \\angle C} \\)"
},
{
"label": "P",
"formula": " a + b + c ",
"formula_mathjax": "\\( a + b + c \\)"
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{
"label": "∠A",
"formula": " Math.asin( a * Math.sin( ∠C * Math.PI / 180 ) / c ) * ( 180 / Math.PI ) ",
"formula_mathjax": "\\( \\sin^{-1} \\left( \\frac{a \\sin \\angle C}{c} \\right) \\frac{180}{\\pi} \\)"
},
{
"label": "∠B",
"formula": " 180 - ∠A - ∠C ",
"formula_mathjax": "\\( 180 - \\angle A - \\angle C \\)"
},
{
"label": "Area",
"formula": " (1 / 2) * a * b * Math.sin( ∠C * Math.PI / 180 ) ",
"formula_mathjax": "\\( \\frac{1}{2} ab \\sin \\angle C \\)"
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"parameters": [
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"outputs": [
{
"label": "b",
"formula": " Math.sqrt( a * a + c * c - 2 * a * c * Math.cos( ∠B * Math.PI / 180 ) ) ",
"formula_mathjax": "\\( \\sqrt{a^2 + c^2 - 2ac \\cos \\angle B} \\)"
},
{
"label": "P",
"formula": " a + b + c ",
"formula_mathjax": "\\( a + b + c \\)"
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{
"label": "∠A",
"formula": " Math.asin( a * Math.sin( ∠B * Math.PI / 180 ) / b ) * ( 180 / Math.PI ) ",
"formula_mathjax": "\\( \\sin^{-1} \\left( \\frac{a \\sin \\angle B}{b} \\right) \\frac{180}{\\pi} \\)"
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{
"label": "∠C",
"formula": " 180 - ∠A - ∠B ",
"formula_mathjax": "\\( 180 - \\angle A - \\angle B \\)"
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{
"label": "Area",
"formula": " (1 / 2) * a * c * Math.sin( ∠B * Math.PI / 180 ) ",
"formula_mathjax": "\\( \\frac{1}{2} ac \\sin \\angle B \\)"
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"parameters": [
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"outputs": [
{
"label": "a",
"formula": " Math.sqrt( b * b + c * c - 2 * b * c * Math.cos( ∠A * Math.PI / 180 ) ) ",
"formula_mathjax": "\\( \\sqrt{b^2 + c^2 - 2bc \\cos \\angle A} \\)"
},
{
"label": "P",
"formula": " a + b + c ",
"formula_mathjax": "\\( a + b + c \\)"
},
{
"label": "∠B",
"formula": " Math.asin( b * Math.sin( ∠A * Math.PI / 180 ) / a ) * ( 180 / Math.PI ) ",
"formula_mathjax": "\\( \\sin^{-1} \\left( \\frac{b \\sin \\angle A}{a} \\right) \\frac{180}{\\pi} \\)"
},
{
"label": "∠C",
"formula": " 180 - ∠A - ∠B ",
"formula_mathjax": "\\( 180 - \\angle A - \\angle B \\)"
},
{
"label": "Area",
"formula": " (1 / 2) * b * c * Math.sin( ∠A * Math.PI / 180 ) ",
"formula_mathjax": "\\( \\frac{1}{2} bc \\sin \\angle A \\)"
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"content": "<h2>Examples</h2><p>In the following examples, we cover scenarios where we find all the properties of a given triangle: all the three sides, all the three angles, perimeter, and area.</p><h3><span class=\"example_n\">1.</span> Example to solve triangle when three sides are given.</h3><h4 class=\"answer\">Answer</h4><div class=\"solution\"><p><strong>Given:</strong></p><ul><li>Side a = 5</li><li>Side b = 6</li><li>Side c = 7</li></ul><p><strong>Solving Perimeter:</strong></p><p>The perimeter \\( P \\) is calculated using the formula:</p><p class=\"step\">\\( P = a + b + c \\)</p><p class=\"step\">\\( P = 5 + 6 + 7 \\)</p><p class=\"step\">\\( P = 18 \\)</p><p><strong>Solving half perimeter:</strong></p><p>The half perimeter \\( s \\) is calculated using the formula:</p><p class=\"step\">\\( s = \\frac{a + b + c}{2} \\)</p><p class=\"step\">\\( s = \\frac{5 + 6 + 7}{2} \\)</p><p class=\"step\">\\( s = 9 \\)</p><p><strong>Solving area:</strong></p><p>The area is calculated using Heron's formula:</p><p class=\"step\">\\( \\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)} \\)</p><p class=\"step\">\\( \\text{Area} = \\sqrt{9(9-5)(9-6)(9-7)} \\)</p><p class=\"step\">\\( \\text{Area} = \\sqrt{9 \\times 4 \\times 3 \\times 2} \\)</p><p class=\"step\">\\( \\text{Area} = \\sqrt{216} \\)</p><p class=\"step\">\\( \\text{Area} = 14.7 \\)</p><p><strong>Solving angle A:</strong></p><p class=\"step\">\\( \\angle A = \\cos^{-1} \\left( \\frac{b^2 + c^2 - a^2}{2bc} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\cos^{-1} \\left( \\frac{6^2 + 7^2 - 5^2}{2 \\times 6 \\times 7} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\cos^{-1} \\left( \\frac{36 + 49 - 25}{84} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\cos^{-1} \\left( \\frac{60}{84} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\cos^{-1} \\left( 0.7143 \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = 44.42° \\)</p><p><strong>Solving angle B:</strong></p><p class=\"step\">\\( \\angle B = \\cos^{-1} \\left( \\frac{a^2 + c^2 - b^2}{2ac} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle B = \\cos^{-1} \\left( \\frac{5^2 + 7^2 - 6^2}{2 \\times 5 \\times 7} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle B = \\cos^{-1} \\left( \\frac{25 + 49 - 36}{70} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle B = \\cos^{-1} \\left( \\frac{38}{70} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle B = \\cos^{-1} \\left( 0.5429 \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle B = 57.12° \\)</p><p><strong>Solving angle C:</strong></p><p class=\"step\">\\( \\angle C = 180 - \\angle A - \\angle B \\)</p><p class=\"step\">\\( \\angle C = 180 - 44.42 - 57.12 \\)</p><p class=\"step\">\\( \\angle C = 78.46° \\)</p></div><h3><span class=\"example_n\">2.</span> Example to solve triangle when two sides and included angle are given.</h3><h4 class=\"answer\">Answer</h4><div class=\"solution\"><p><strong>Given:</strong></p><ul><li>Side a = 5</li><li>Side b = 6</li><li>Included Angle ∠C = 45°</li></ul><p><strong>Solving Side c:</strong></p><p>The side c is calculated using the formula:</p><p class=\"step\">\\( c = \\sqrt{a^2 + b^2 - 2ab \\cos \\angle C} \\)</p><p class=\"step\">\\( c = \\sqrt{5^2 + 6^2 - 2 \\times 5 \\times 6 \\times \\cos 45°} \\)</p><p class=\"step\">\\( c = \\sqrt{25 + 36 - 60 \\times 0.7071} \\)</p><p class=\"step\">\\( c = \\sqrt{25 + 36 - 42.426} \\)</p><p class=\"step\">\\( c = \\sqrt{18.574} \\)</p><p class=\"step\">\\( c = 4.31 \\)</p><p><strong>Solving Perimeter:</strong></p><p>The perimeter \\( P \\) is calculated using the formula:</p><p class=\"step\">\\( P = a + b + c \\)</p><p class=\"step\">\\( P = 5 + 6 + 4.31 \\)</p><p class=\"step\">\\( P = 15.31 \\)</p><p><strong>Solving angle A:</strong></p><p class=\"step\">\\( \\angle A = \\sin^{-1} \\left( \\frac{a \\sin \\angle C}{c} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\sin^{-1} \\left( \\frac{5 \\times \\sin 45°}{4.31} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\sin^{-1} \\left( \\frac{5 \\times 0.7071}{4.31} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\sin^{-1} \\left( \\frac{3.5355}{4.31} \\right) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = \\sin^{-1} (0.8202) \\frac{180}{\\pi} \\)</p><p class=\"step\">\\( \\angle A = 55.32° \\)</p><p><strong>Solving angle B:</strong></p><p class=\"step\">\\( \\angle B = 180 - \\angle A - \\angle C \\)</p><p class=\"step\">\\( \\angle B = 180 - 55.32 - 45 \\)</p><p class=\"step\">\\( \\angle B = 79.68° \\)</p><p><strong>Solving Area:</strong></p><p>The area is calculated using the formula:</p><p class=\"step\">\\( \\text{Area} = \\frac{1}{2} ab \\sin \\angle C \\)</p><p class=\"step\">\\( \\text{Area} = \\frac{1}{2} \\times 5 \\times 6 \\times \\sin 45° \\)</p><p class=\"step\">\\( \\text{Area} = \\frac{1}{2} \\times 5 \\times 6 \\times 0.7071 \\)</p><p class=\"step\">\\( \\text{Area} = \\frac{1}{2} \\times 30 \\times 0.7071 \\)</p><p class=\"step\">\\( \\text{Area} = 15 \\times 0.7071 \\)</p><p class=\"step\">\\( \\text{Area} = 10.61 \\)</p></div>",
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