Area of a Right-Angled Triangle

The area of a right-angled triangle is the amount of space enclosed by its three sides. Because the two legs that meet at the right angle are perpendicular, one leg can be treated as the base and the other as the height.

The basic formula is:

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

If the perpendicular base and height are not both given, use the Pythagorean theorem or trigonometry to calculate the missing side before applying the area formula.

Right-Angled Triangle Area Formulas for Different Given Measurements

You can calculate the area of a right-angled triangle when you know any of the following sets of measurements:

  • Base and perpendicular height
  • Base and hypotenuse
  • Height and hypotenuse
  • Base and the angle between the base and hypotenuse
  • Height and the angle between the height and hypotenuse

In every case, the hypotenuse is the side opposite the right angle and is always the longest side.

Area of a Right-Angled Triangle Given Base and Height

When the two perpendicular sides are known, multiply the base by the height and divide the result by 2. The base and height are marked in the following diagram.

Area of Right Angle Triangle given Base and Height

\[Area\ of\ \triangle\thinspace ABC = {{Base \cdot Height} \over 2}\]

Example: Area When the Base Is 4 cm and Height Is 6 cm

Find the area of a right-angled triangle whose base is 4 cm and height is 6 cm.
Example - Area of Right Angle Triangle - Given Base and Height
Solution:
Given Base = 4cm, Height = 6cm.

Area of Right Angle Triangle = (Base x Height)/2
                             = (4cm x 6cm)/2
                             = 12 Square cm

Therefore, the area of the triangle is 12 cm².

Area of a Right-Angled Triangle Given Base and Hypotenuse

When the base and hypotenuse are given, first calculate the perpendicular height using the Pythagorean theorem. Then use the base-height area formula.

Area of Right Angle Triangle - Given Base and Hypotenuse

In the diagram:

  • AC is the hypotenuse.
  • AB is the base.
  • BC is the perpendicular height.

According to the Pythagorean theorem:

\[AC^2 = AB^2+BC^2\]

Rearrange the equation to calculate the height:

\[Height,\ BC = \sqrt{AC^2-AB^2}\ —–(1)\]

The area formula is:

\[Area\ of\ \triangle\thinspace ABC = {{Base \cdot Height} \over 2}\]

\[= {{AB \cdot BC} \over 2}\ —–(2)\]

Substitute the value of \(BC\) from equation \((1)\) into equation \((2)\):

\[\therefore\ Area\ of\ \triangle\thinspace ABC = {{AB \cdot \sqrt{AC^2-AB^2}} \over 2}\]

In general, if the base is \(b\) and the hypotenuse is \(c\), then:

\[Area = \frac{b}{2}\sqrt{c^2-b^2}\]

Example: Area When the Base Is 5 cm and Hypotenuse Is 10 cm

Find the area of a right-angled triangle whose base is 5 cm and hypotenuse is 10 cm.
Example - Area of Right Angle Triangle - Given Base and Hypotenuse
Solution:
Given

 Base, AB = 5 cm
 Hypotenuse, AC = 10 cm

\[Area\ of\ \triangle\thinspace ABC = {{AB \cdot \sqrt{AC^2-AB^2}} \over 2}\]

Substitute \(AB=5\) and \(AC=10\):

\[Area = {{5 \cdot \sqrt{10^2-5^2}} \over 2}\]

\[Area = \frac{5}{2}\sqrt{100-25}\]

\[Area = \frac{5}{2}\sqrt{75} = \frac{25\sqrt{3}}{2}\]

\[Area \approx 21.65\thinspace cm^2\]

Therefore, the area is approximately 21.65 cm².

Area of a Right-Angled Triangle Given Height and Hypotenuse

If the perpendicular height and hypotenuse are given, calculate the missing base with the Pythagorean theorem:

\[Base = \sqrt{Hypotenuse^2-Height^2}\]

Substitute this expression into the usual area formula:

\[Area = \frac{Height}{2}\sqrt{Hypotenuse^2-Height^2}\]

If the height is \(h\) and the hypotenuse is \(c\), the formula can be written as:

\[Area = \frac{h}{2}\sqrt{c^2-h^2}\]

Example: Area When the Height Is 8 cm and Hypotenuse Is 10 cm

First calculate the base:

\[Base = \sqrt{10^2-8^2} = \sqrt{100-64} = \sqrt{36} = 6\thinspace cm\]

Now calculate the area:

\[Area = \frac{1}{2}\times 6\times 8 = 24\thinspace cm^2\]

Therefore, the area is 24 cm².

Area Given the Base and Angle Between the Base and Hypotenuse

Suppose the base length and the acute angle between the base and hypotenuse are known. The perpendicular height can be found using the tangent ratio.

[figure]

For angle \(\theta\) at vertex A:

\[Area\ of\ \triangle\thinspace ABC = {{Base \cdot Height} \over 2}\]

\[= {{AB \cdot BC} \over 2}\ \ \ —–(1)\]

Using the tangent ratio:

\[\tan \theta = {BC \over AB}\]

Therefore:

\[BC = AB \tan \theta\ \ \ —–(2)\]

Substitute the value of \(BC\) from equation \((2)\) into equation \((1)\):

\[Area\ of\ \triangle\thinspace ABC = {{AB \cdot AB \cdot \tan \theta} \over 2}\]

\[Area = {{AB^2\tan \theta} \over 2}\]

If the base is \(b\), the area formula becomes:

\[Area = \frac{b^2\tan\theta}{2}\]

Example: Area When the Base Is 6 cm and the Angle Is 45°

Use \(b=6\) cm and \(\theta=45^\circ\):

\[Area = \frac{6^2\tan45^\circ}{2}\]

Since \(\tan45^\circ=1\):

\[Area = \frac{36\times1}{2}=18\thinspace cm^2\]

Therefore, the area is 18 cm².

Area Given the Height and Angle Between the Height and Hypotenuse

If the height \(h\) and the acute angle \(\theta\) between the height and hypotenuse are known, the base is opposite that angle. Therefore:

\[\tan\theta = \frac{Base}{Height}\]

\[Base = Height\tan\theta\]

Substituting this into the area formula gives:

\[Area = \frac{h^2\tan\theta}{2}\]

Right-Angled Triangle Area Formula Summary

Measurements givenArea formula
Base \(b\) and height \(h\)\(\frac{1}{2}bh\)
Base \(b\) and hypotenuse \(c\)\(\frac{b}{2}\sqrt{c^2-b^2}\)
Height \(h\) and hypotenuse \(c\)\(\frac{h}{2}\sqrt{c^2-h^2}\)
Base \(b\) and angle with hypotenuse \(\theta\)\(\frac{b^2\tan\theta}{2}\)
Height \(h\) and angle with hypotenuse \(\theta\)\(\frac{h^2\tan\theta}{2}\)

Units Used for the Area of a Right-Angled Triangle

Area is always written in square units. Use the same unit as the given side lengths, but square it in the final answer.

  • Side lengths in centimetres produce an area in square centimetres, written as cm².
  • Side lengths in metres produce an area in square metres, written as m².
  • Side lengths in millimetres produce an area in square millimetres, written as mm².

Convert all side lengths to the same unit before performing the calculation.

Common Errors When Calculating Right-Triangle Area

  • Using the hypotenuse as the height: The base and height must meet at the 90-degree angle.
  • Forgetting to divide by 2: Multiplying the base by the height gives the area of a rectangle, not the triangle.
  • Subtracting Pythagorean terms in the wrong order: When finding a leg, calculate \(\sqrt{c^2-a^2}\), where \(c\) is the hypotenuse.
  • Using inconsistent units: Convert measurements such as metres and centimetres to one common unit first.
  • Writing linear units: The final area must use square units such as cm² or m².

Frequently Asked Questions About Right-Angled Triangle Area

What is the formula for the area of a right-angled triangle?

The formula is \(Area=\frac{1}{2}\times Base\times Height\), where the base and height are the two perpendicular sides.

Can the hypotenuse be used directly as the base or height?

No. The base and height used in the standard formula must be perpendicular. The hypotenuse does not form a right angle with either leg, so first use it to calculate a missing leg.

How do you find the area when only the hypotenuse and one leg are known?

Use the Pythagorean theorem to calculate the missing leg, and then apply \(Area=\frac{1}{2}bh\).

Can either perpendicular side be chosen as the base?

Yes. Either leg can be called the base, provided the other perpendicular leg is used as the corresponding height. The calculated area will be the same.

Why is the area divided by 2?

Two identical right-angled triangles can be joined to form a rectangle with area \(Base\times Height\). Each triangle occupies half of that rectangle, so its area is \(\frac{1}{2}\times Base\times Height\).