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Area of Triangles

Calculating the area of a triangle depends on the kind of triangle you have.

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Last Updated: 01.12.2023
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Formulas for calculating the area of a triangle

The area of a triangle can be found in two formulas:

For all triangles, you can use the formula: $a r e a = \frac{b a s e \times h e i g h t}{2}$

Area of triangles Showing a triangle's base and height StudySmarter Showing a triangle's base and height - StudySmarter Originals

Triangle A is shown below (all lengths are in cm):

Area of triangles Worked example of area of a triangle StudySmarter

The area of the triangle = $\frac{10 \times 5}{2} = 25 c m^{2}$

For most triangles, the base and height are used as shown above.

For all non-angled triangles, the formula is $a r e a = \frac{a \times b \times \sin (c)}{2}$

Area of triangles a non angled triangle StudySmarter A non-angled triangle - StudySmarter Orignals

To use this formula, angle C needs to be between the two sides. You can remember this through the acronym SAS (Side, Angle, Side).

A triangle is shown below (all lengths are in cm)

Area of triangles Worked example for finding area of a triangle StudySmarter

What is the area of the triangle?

First, label the sides of the triangles according to the formula.
This is not a right-angled triangle. We can use the formula below.
Area of the triangle = $\frac{a \times b \times \sin c}{2} = \frac{12 \times 28 \times \sin 40^{0}}{2} = 107.99 c m^{2}$

Right-angled triangles

For right-angled triangles, the height for the right-angled triangle in the formula $a r e a = \frac{b a s e \times h e i g h t}{2}$ is equivalent to the vertical side.

Area of triangles Showing the base and height of a right angle triangle StudySmarter A right angle triangle

When using the formula you might need to work out one of the sides to get two sides next to the angle. To do so, you need to use Pythagoras theorem, whereby $a^{2} + b^{2} = c^{2}$

Area of triangles Showing the sides of a right angle triangle StudySmarter

An equilateral triangle can be seen below (all lengths are in cm):

Area of triangles Worked example for finding area of a triangle StudySmarter

The formula for the area of the triangle is $\frac{b a s e x h e i g h t}{2}$ but the height is unknown. To work out the height, you need to rearrange and use Pythagoras theorem.

To use Pythagoras theorem, you need to find a and c: c is the hypotenuse and therefore, c = 5; a is half the base and therefore a = 4.
Substitute the values into Pythagoras theorem: $\sqrt{c^{2} - a^{2}}$ = $\sqrt{25 - 16} = \sqrt{9} = 3$ . Therefore, the height is 3.
Substitute the values into the Area of a Triangle formula: $\frac{b \times h}{2} = \frac{8 \times 3}{2} = 12 c m^{2}$

Area of Triangles - Key takeaways

The area of any triangle can be calculated using the formula $A r e a = \frac{b a s e \times h e i g h t}{2}$
For all non-right-angled triangles, you can also use the formula $A r e a = \frac{1}{2} \times a \times b \times S i n (C)$
The acronym SAS is used to figure out what values should be substituted into $A r e a = \frac{1}{2} \times a \times b \times \sin (C)$

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Frequently Asked Questions about Area of Triangles

What is the area of a triangle?

The area of the triangle is the amount of space that the triangle takes up.

How do you work out the area of the triangle?

The area of any triangle can be found using the formula, area=1/2(absin(c)) or you can also use the formula area= (base × height)/2 if the triangle is right-angled.

How do you measure the area of the triangle?

The unit of the triangle’s area is the unit the side is measured in ^2 so if the sides were measured in metres the unit of the area would be m^2.

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