Area of Triangle

Area of a Triangle

The area of a triangle is the amount of two-dimensional space enclosed by its three sides. It is measured in square units, such as square centimetres (cm²), square metres (m²), or square inches (in²).

The formula used to calculate the area depends on the measurements provided. A problem may give the base and perpendicular height, three side lengths, two sides and their included angle, or a side together with two angles.

Triangle Area Formulas Based on the Given Measurements

The following formulas cover the most common ways to find the area of a triangle:

Base and perpendicular height: \(A=\frac{1}{2}bh\)
Three side lengths: use Heron’s formula.
Two sides and the included angle: \(A=\frac{1}{2}ab\sin C\)
One side and the two adjacent angles: use the sine-rule area formula.
Three angles and perimeter: first calculate the sides using the sine rule, and then find the area.

Special triangles may also have simplified formulas. See the tutorials for an equilateral triangle, an isosceles triangle, and a right-angled triangle.

Area of a Triangle Given Base and Perpendicular Height

When a base and its corresponding perpendicular height are known, use the standard triangle area formula:

\[A=\frac{1}{2}\times Base\times Height\]

The height must be measured at a right angle to the selected base. It may lie inside the triangle, outside an obtuse triangle, or along one side of a right-angled triangle.

Example: Triangle with Base 10 cm and Height 7 cm

Substitute \(b=10\) cm and \(h=7\) cm:

\[A=\frac{1}{2}\times10\times7=35\thinspace cm^2\]

Therefore, the area of the triangle is 35 cm².

Area of a Triangle Given Three Side Lengths

When the lengths of all three sides are known but the height is not given, use Heron’s formula.

Consider triangle ABC with side lengths \(a\), \(b\), and \(c\).

[figure]

First calculate the semiperimeter \(s\), which is half the perimeter:

\[s=\frac{a+b+c}{2}\]

Then apply Heron’s formula:

\[Area\ of\ \triangle ABC=\sqrt{s(s-a)(s-b)(s-c)}\]

The original formula may also use \(p\) instead of \(s\) for the semiperimeter. Both notations represent half the perimeter.

Example: Triangle with Sides 5 cm, 8 cm, and 6 cm

Find the area of triangle ABC whose side lengths are 5 cm, 8 cm, and 6 cm.

[figure]

Solution

Given a = 5cm
      b = 8cm
      c = 6cm

First calculate the semiperimeter:

\[s=\frac{5+8+6}{2}=\frac{19}{2}=9.5\thinspace cm\]

Now substitute the values into Heron’s formula:

\[A=\sqrt{9.5(9.5-5)(9.5-8)(9.5-6)}\]

\[A=\sqrt{9.5\times4.5\times1.5\times3.5}\]

\[A=\sqrt{224.4375}\approx14.98\thinspace cm^2\]

Therefore, the area of the triangle is approximately 14.98 cm².

Checking Three Sides Before Applying Heron’s Formula

Three positive lengths form a triangle only when they satisfy the triangle inequality. The sum of every pair of sides must be greater than the remaining side:

\(a+b>c\)
\(b+c>a\)
\(c+a>b\)

For example, lengths 2 cm, 3 cm, and 6 cm cannot form a triangle because \(2+3\) is not greater than 6. Heron’s formula should not be applied to such measurements.

Area of a Triangle Given Two Sides and the Included Angle

If two side lengths and the angle between those sides are known, calculate the area with the sine formula:

\[A=\frac{1}{2}ab\sin C\]

Here, \(C\) must be the included angle between sides \(a\) and \(b\). Equivalent forms are:

\(A=\frac{1}{2}bc\sin A\)
\(A=\frac{1}{2}ca\sin B\)
\(A=\frac{1}{2}ab\sin C\)

Example: Two Sides of 8 cm and 11 cm with a 30° Included Angle

Use \(a=8\) cm, \(b=11\) cm, and \(C=30^\circ\):

\[A=\frac{1}{2}\times8\times11\times\sin30^\circ\]

Since \(\sin30^\circ=\frac{1}{2}\):

\[A=44\times\frac{1}{2}=22\thinspace cm^2\]

Therefore, the area is 22 cm².

Area of a Triangle Given One Side and Two Adjacent Angles

Suppose side \(c\) is known, together with the angles \(A\) and \(B\) at its two endpoints. The third angle is:

\[C=180^\circ-(A+B)\]

The area can then be calculated directly using:

\[A_{triangle}=\frac{c^2\sin A\sin B}{2\sin(A+B)}\]

Because \(\sin(180^\circ-C)=\sin C\), this formula may also be written as:

\[A_{triangle}=\frac{c^2\sin A\sin B}{2\sin C}\]

Example: Side 10 cm with Adjacent Angles 45° and 60°

The third angle is:

\[C=180^\circ-(45^\circ+60^\circ)=75^\circ\]

Substitute the known values:

\[A_{triangle}=\frac{10^2\sin45^\circ\sin60^\circ}{2\sin75^\circ}\]

\[A_{triangle}\approx31.70\thinspace cm^2\]

Therefore, the area is approximately 31.70 cm².

Area of a Triangle Given Three Angles and the Perimeter

Three angles determine the shape of a triangle but not its size. The perimeter supplies the additional measurement needed to determine the side lengths.

For angles \(A\), \(B\), and \(C\), the sine rule shows that the sides are proportional to their opposite sines:

\[a:b:c=\sin A:\sin B:\sin C\]

If the perimeter is \(P\), define:

\[k=\frac{P}{\sin A+\sin B+\sin C}\]

The side lengths are then:

\(a=k\sin A\)
\(b=k\sin B\)
\(c=k\sin C\)

After finding two sides, calculate the area using the included-angle formula. For example:

\[A_{triangle}=\frac{1}{2}ab\sin C\]

Combining the steps gives the direct formula:

\[A_{triangle}=\frac{P^2\sin A\sin B\sin C}{2(\sin A+\sin B+\sin C)^2}\]

Triangle Area Formula Reference Table

Measurements available	Formula or method
Base \(b\) and perpendicular height \(h\)	\(A=\frac{1}{2}bh\)
Three sides \(a\), \(b\), and \(c\)	\(A=\sqrt{s(s-a)(s-b)(s-c)}\), where \(s=\frac{a+b+c}{2}\)
Two sides \(a\), \(b\) and included angle \(C\)	\(A=\frac{1}{2}ab\sin C\)
Side \(c\) and adjacent angles \(A\), \(B\)	\(A=\frac{c^2\sin A\sin B}{2\sin(A+B)}\)
Angles \(A\), \(B\), \(C\) and perimeter \(P\)	Find the sides using the sine rule, or use the perimeter-angle formula.

Choosing the Correct Triangle Area Formula

Use \(\frac{1}{2}bh\) when a perpendicular base and height are known.
Use Heron’s formula when all three side lengths are known.
Use \(\frac{1}{2}ab\sin C\) when two sides and the angle between them are known.
Use the sine rule when angles are known but one or more side lengths must first be calculated.
For a right-angled triangle, the two perpendicular legs can be used directly as the base and height.

Units and Rounding in Triangle Area Calculations

All lengths must use the same unit before they are substituted into a formula. Convert mixed measurements, such as metres and centimetres, to one common unit first.

The final answer must be expressed in square units. For example, side lengths measured in centimetres produce an area in square centimetres.

When a calculation contains square roots or trigonometric values, retain several decimal places during the working and round only the final area. This reduces accumulated rounding error.

Common Mistakes in Finding the Area of a Triangle

Using a sloping side as the height: The height must be perpendicular to the selected base.
Forgetting the factor \(\frac{1}{2}\): Base multiplied by height gives the area of a parallelogram or rectangle, not a triangle.
Using the wrong angle: In \(\frac{1}{2}ab\sin C\), angle \(C\) must be between sides \(a\) and \(b\).
Using the full perimeter in Heron’s formula: The formula requires the semiperimeter.
Ignoring the triangle inequality: Verify that three supplied side lengths can form a triangle.
Mixing degrees and radians: Set the calculator to the angle unit used in the problem.
Writing linear units: Area must be stated in square units.

Frequently Asked Questions About the Area of a Triangle

What is the basic formula for the area of a triangle?

The basic formula is \(A=\frac{1}{2}bh\), where \(b\) is the base and \(h\) is the perpendicular height measured from the opposite vertex.

How can the area be found without knowing the height?

Use Heron’s formula when all three sides are known. When two sides and their included angle are known, use \(A=\frac{1}{2}ab\sin C\).

Can any side of a triangle be chosen as the base?

Yes. Any side may be selected as the base, but the corresponding height must be the perpendicular distance from that side to the opposite vertex.

Can three angles alone determine the area of a triangle?

No. Three angles determine only the shape of a triangle. At least one length measurement, such as a side or the perimeter, is required to determine its size and area.

Why is triangle area written in square units?

Area measures a two-dimensional surface. Multiplying two lengths produces squared units, such as cm², m², or in².

TutorialKart.com