How To Measure the Area of Triangles?
By Robert O
Any object occupies space, area. If an object is triangular, it will occupy a triangular space on the surface it rests. This tutorial will uncover four different ways of calculating the area of triangles. That will depend on the measurements that you have.
How to calculate the area of a triangle given base and height?
The area of a triangle with height and base dimensions is:
![\[A=\frac{1}{2} b h\]](data:image/svg+xml;nitro-empty-id=MzA0OjQ5OA==-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNjggMzYiIHdpZHRoPSI2OCIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Example 1
George has a garden in the shape of a triangle. The base dimension is 52m, and the perpendicular height is 100m. Calculate the area of his garden.
Solution
![\[A=\frac{1}{2} b h\]](data:image/svg+xml;nitro-empty-id=MzA0OjEyMjc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNjggMzYiIHdpZHRoPSI2OCIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[A=\frac{1}{2} \times 52 \times 100=2600 \mathrm{~m}^{2}\]](data:image/svg+xml;nitro-empty-id=MzA0OjE3MDM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjI1IDM2IiB3aWR0aD0iMjI1IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
How do I calculate the area of a triangle if I have dimensions of all sides?
To calculate the area of the above rectangle, we use Heron’s formula:
![\[A=\sqrt{s(s-a)(s-b)(s-c)}\]](data:image/svg+xml;nitro-empty-id=MzA0OjI1MDU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjIwIDIyIiB3aWR0aD0iMjIwIiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Where:
![\[s=\frac{1}{2}(a+b+c), \text { and } a, b, \text { and } c \text { are sides of the triangle. }\]](data:image/svg+xml;nitro-empty-id=MzA0OjMwMzM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDQ4IDM2IiB3aWR0aD0iNDQ4IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Example 2
A proposed community project sits on a triangular piece of land measuring 40m, 35m, and 25m. Calculate the area under the project.
Solution
Start by calculating s.
![\[s=\frac{1}{2}(a+b+c)\]](data:image/svg+xml;nitro-empty-id=MzA0OjM3ODA=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTI2IDM2IiB3aWR0aD0iMTI2IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[s=\frac{1}{2}(40+35+25)=50\]](data:image/svg+xml;nitro-empty-id=MzA0OjQyMjc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTk2IDM2IiB3aWR0aD0iMTk2IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Applying Heron’s formula:
![\[A=\sqrt{s(s-a)(s-b)(s-c)}\]](data:image/svg+xml;nitro-empty-id=MzA0OjQ3MDc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjIwIDIyIiB3aWR0aD0iMjIwIiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[A=\sqrt{50(50-40)(50-35)(50-25)}\]](data:image/svg+xml;nitro-empty-id=MzA0OjUxNjA=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjg2IDIyIiB3aWR0aD0iMjg2IiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[A=\sqrt{50(10)(15)(25)}=433.01 \mathrm{~m}^{2}\]](data:image/svg+xml;nitro-empty-id=MzA0OjU2Mjc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY3IDIyIiB3aWR0aD0iMjY3IiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
How do I calculate the area of a triangle using the sine rule?
Using the sine rule is one of the simplest methods for calculating the area of a triangle if you know the dimensions of two adjacent sides and the angle between them. Here are the formula and its variations depending on the dimensions that you have.
![\[\text { Area }=\frac{1}{2} b c \sin A \text { or } \text { Area }=\frac{1}{2} a c \sin B \text { or } \text { Area }=\frac{1}{2} a b \sin C\]](data:image/svg+xml;nitro-empty-id=MzA0OjY3MDc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDc5IDM2IiB3aWR0aD0iNDc5IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Example 3
Find the area of a triangle ABC where the length AB = 10cm and BC = 4cm, and angle ABC = 40o
Solution
![\[\text { Area }=\frac{1}{2} bc\sin A B C=\frac{1}{2} \times 10 \times 4 \times \sin (40)=12.86 \mathrm{~cm}^{2}\]](data:image/svg+xml;nitro-empty-id=MzA0Ojc0ODc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDQ1IDM2IiB3aWR0aD0iNDQ1IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
How to calculate the area triangle with all the three sides equal
You can use Heron’s formula or Pythagoras theorem to get the perpendicular height to calculate the area of an equilateral triangle. But since all sides are equal, we can use a shorter form of these formulas to calculate the same area.
![\[\text { Area }=\frac{s^{2} \sqrt{3}}{4}, \text { where s is length of any side of the triangle. }\]](data:image/svg+xml;nitro-empty-id=MzA0OjgzNzY=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDY0IDM5IiB3aWR0aD0iNDY0IiBoZWlnaHQ9IjM5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Example 4
A theater room is in the shape of an equilateral triangle of sides 6m. Calculate the area of the room.
Solution
![\[\text { Area }=\frac{s^{2} \sqrt{3}}{4}=\frac{6^{2} \sqrt{3}}{4}=15.59 \mathrm{~m}^{2}\]](data:image/svg+xml;nitro-empty-id=MzA0OjkxMzE=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY5IDQwIiB3aWR0aD0iMjY5IiBoZWlnaHQ9IjQwIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Remarks
You can calculate the area of any triangle by using the right formula. All you need to do is to identify a formula that applies to the statement problem that you have. Any surface in the shape of a triangle has a formula for calculating its area.
About the Author
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.