Trig formula for area of a triangle

Here you will learn about the trig formula for area of a triangle, including how to generate the formula, use the formula to find the area of a triangle, and apply this formula to other polygons.

Students will first learn about the trig formula for the area of a triangle as part of geometry in high school.

What is the trig formula for area of a triangle?

The trig formula for area of a triangle is a formula used to calculate the area of any triangle.

Area of a triangle =\cfrac{1}{2} \, ab\sin{C}

Previously, you have calculated the area of a triangle using the area formula

Area of a triangle =\cfrac{\text{base }\times\text{ height}}{2}

To use this, you need to know the vertical height of the triangle (perpendicular height to the base) and the base of the triangle.

You can adapt this formula using the trigonometric ratio \sin(\theta)=\cfrac{O}{H} to work out the area of a triangle when you do not know its vertical height. The formula you get is:

Area of a triangle =\cfrac{1}{2} \, ab\sin{C}

The triangle should be labeled as follows, with the lower case letter for each side opposite the corresponding upper case letter for the angle.

You need to know

1. the length of at least 2 sides of a triangle
2. the included angle between these two sides.

For example, triangle ABC has been labeled where C is the included angle between the two sides of the triangle a and b.

What is the trig formula for area of a triangle?

Which area of a triangle formula should I use?

If you know or can work out the vertical height of a triangle, it can be easier to use

Area of a triangle =\cfrac{\text{base }\times\text{ height}}{2}.

For example,

However, if the vertical height is not labeled and there are two given sides and the angle in between, you would need to use

Area of a triangle =\cfrac{1}{2} \, ab\sin{C}

Once you know which formula to use, you need to substitute the correct values into it and then solve the equation to calculate the area.

The area is always written with square units.

Remember: other polygons can be split into triangles to find the interior angles, so the formula for the area of a triangle \left(A=\cfrac{1}{2} \, ab\sin{C}\right) can be applied to find the area of a rectangle, the area of an equilateral triangle, the area of a pentagon, or the area of a parallelogram etc.

Common Core State Standards

How does this relate to high school math?

• High School – Geometry – Similarity, Right Triangles, & Trigonometry (HS.G.SRT.D.9)
  Derive the formula A=\cfrac{1}{2} \, ab\sin{C} for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

How to use the trig formula for area of a triangle

In order to use the trig formula for area of a triangle:

1. Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
   Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .
2. Substitute the given values into the formula, Area =\bf{\cfrac{1}{2}} \, \textbf{ab} \, \bf{\sin} \, \textbf{{C}}.
3. Solve the equation.

Trig formula for area of a triangle examples

Example 1: with two sides and the angle in between

Calculate the area of the triangle ABC. Write your answer to the nearest hundredth.

1. Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
   Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Remember that angles are capitalized and the side opposite each angle is the respective lower case letter (for example, side a is opposite angle A ).

2Substitute the given values into the formula.

3Solve the equation.

Example 2: with two sides and the angle in between

Calculate the area of the triangle. Write your answer to the nearest hundredth.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Here, you label each side a, \, b, and c and each vertex A, \, B and C .

Example 3: with three sides and one angle

Calculate the area of the scalene triangle PQR. Write your answer to 3 significant figures.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Here, you label each side and each angle. You could do this in an alternative color pen.

Example 4: area of an isosceles triangle with a known angle

Triangle XYZ is an isosceles triangle. Find the area of the triangle to the nearest hundredth.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Here, you label each side and each angle.

Example 5: with two sides and two angles

Calculate the area of the triangle ABC. Write your answer to the nearest hundredth.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Here, you label 25.4^{\circ} as C as this angle is between two known sides.

Example 6: with two sides and two angles

Calculate the area of the triangle ABC. Write your answer to 4 significant figures.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Here, you have to think carefully before Step 2 because a, \, b, and C do not correspond to a, \, b and C in A=\cfrac{1}{2} \, ab\sin{C}.

As the known sides b and c have the included angle at A with all three values known, you can adjust the area of a triangle using sine to make A=\cfrac{1}{2} \, bc\sin{A} .

Example 7: finding a length given the area

The area of this triangle is 30 \, cm^2. Find the length labeled x.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Example 8: finding an angle given the area

The area of this triangle is 42 \, cm^2. Find the angle labeled x.

Label the angle you are going to use angle \textbf{C} and its opposite side \textbf{c} .
Label the other two angles \textbf{A} and \textbf{B} and their corresponding sides \textbf{a} and \textbf{b} .

Teaching tips for trig formula for area of a triangle

• Begin with the basic formula for the area of a triangle (halving the base times height): \text{Area }=\cfrac{1}{2}\times\text{ base }\times{height}. Explain that when the height is not directly given, we can use trigonometric functions, specifically \sin(C), to calculate it.

• Introduce Heron’s formula for the area of a triangle when all three sides are known:
  \text{Area }=\sqrt{s(s-a)(s-b)(s-c)}
  where s=\cfrac{a+b+c}{2} is the semi-perimeter.
  Discuss when each formula is most appropriate:
  • Use Heron’s formula when the lengths of all three sides are known.
  • Use \cfrac{1}{2} absinC when two sides and the included angle are given.

Easy mistakes to make

• Forgetting to use sine for the angle
  Some students may substitute the angle C directly into the formula instead of finding \sin(C). Remember that \sin(C) is the sine function of the angle, not the angle itself. Use a calculator or table to compute it.

• Confusing different formulas
  Understand when to use each formula. Use \cfrac{1}{2} absinC specifically when you have two sides and the included angle.

• Using degrees when the angle is in radians (or vice versa)
  Check the calculator mode (DEG or RAD) and ensure it matches the given angle’s unit.

Related trigonometry lessons

• Trigonometry
• How to find the exact value of a trig function
• Sin cos tan
• Law of sines
• SOHCAHTOA
• Trig formulas
• Trig identities
• Law of cosines
• Trig table

The next lessons are

• Circle math
• Angles of a circle
• Circle theorems

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