Area of Triangle

Area of Triangle
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Triangles are three-sided closed figures.

They can be acute, obtuse, or right triangles when they are classified based on their angles.

They can be scalene, isosceles, or equilateral when classified based on their sides.

Drag the vertices in the simulation below to see the types of triangles and its elements.

Click on the type of triangles to explore their properties.

In this lesson, we will explore the area of a right-angled triangle, scalene triangle, and an obtuse angle.

Lesson Plan

What Do You Mean by Area of A Triangle?

The area of a triangle is the amount of space enclosed between the sides of the triangle.

Consider the triangle ABC.

The shaded interior region enclosed within AB, BC, and AC is the area of the given triangle.

Area of a triangle


What Is the Area of A Triangle Formula?

There are several ways to find the area of a triangle.

Area of Triangle with Base and Height

Area of triangle formula is shown using a triangle with height h.

In the given triangle \(ABC\),

Area of \(\Delta ABC  = \frac{1}{2} h\times \text{BC}\)

BC is the base and h is the height of the triangle.

Area = \(\dfrac{1}{2}\) × Base × Height

Area of Triangle with 3 Sides

Heron's formula is used to find the area of a triangle when the length of the 3 sides of the triangle are known.

Area of triangle with 3 sides a, b, and c

Consider the triangle ABC with sides a, b and c.

Heron's formula to find the area of the triangle is:

Area = \(\sqrt {s(s - a)(s - b)(s - c)}\)

Note that (a + b + c) is the perimeter of the triangle.

s is the semi-perimeter which is given by:

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

Area of Triangle with 2 Sides and Included Angle (SAS)

There are three variations to the same formula based on which sides and included angle are given.

Consider the triangle,

Area of a triangle when two sides of a triangle and included angle are given.

When sides 'b' and 'c' and included angle A is known, the area of the triangle is:

\(\dfrac{1}{2}\) × bc × sin(A)

When sides 'b' and 'a' and included angle B is known, the area of the triangle is:

\(\dfrac{1}{2}\) × ab × sin(C)

When sides 'a' and 'c' and included angle C is known, the area of the triangle is:

\(\dfrac{1}{2}\) × ac × sin(B)

How Do You Calculate the Area Of a Triangle?

The area of triangle can be calculated using the formulas as discussed above or using the area of triangle calculators.

With Area of Triangle Formulas

Area of a Triangle Formula

Base and height of a triangle are given

\( \frac{1}{2} \times \text{base} \times \text{height}\)
Sides of a triangle \((a,b,c)\) are given \(\sqrt {s(s - a)(s - b)(s - c)} \)

where \(a, \, b, \, c\) are the sides and \(s\) is the semi-perimeter \(s = \frac{a+b+c}{2}\)
Two sides and the included angle are given \(\frac{1}{2}\:{side_1} \times{side_2} \times sin(θ)\)

where '\(\theta\)' is the angle between the given two sides
Area of an equilateral triangle \(\frac{\sqrt{3}}{4}\text{a}^2\)

where \(a\) is the side of the triangle
Area of an Isosceles triangle \(\frac{1}{4}b\sqrt {4{a^2} - {b^2}}\)

where \(b\) is the base and \(a\) is the measure of equal side.

Use the Formulas listed above to calculate the area of a triangle based on the given parameters.

The area of a triangle calculator will help you find them easily.

With Area Of Triangle Calculator

Use this Area of a triangle calculator to find the area when the base and height of a triangle are given.

Try this Area of a triangle calculator to find the area of a triangle using Heron's formula.

Use this Area of a triangle calculator to find the area when two sides and included angle is known (SAS)

 
tips and tricks
Tips and Tricks
  1. An easy way to recall the formula for the area of a triangle when two sides and the included angle are given is to remember: 'abc'
    Area of a triangle = \(\frac{1}{2}\)ab \(sin\)(C)
  2. Area of an isosceles triangle and equilateral triangle can be derived from Heron's formula. 
  3. Area of a triangle is equal to half the product of the altitude of a triangle and the base of a triangle.
  4. The median of a triangle in an isosceles triangle is perpendicular to the base.

Solved Examples

Example 1

 

 

The signboard at a cross road needs to be repainted.

The measurements are as given in the image.

Find the area to be painted.

A signboard in the shape of a triangle shows measurements for two sides and an included angle.

Solution

The area of a triangle when 2 sides and included angle are given is:

\[\begin{align}
&\frac{1}{2}\:{side_1} \times{side_2} \times sin(θ) \\
&=\frac{1}{2} \times 7 \times 8 \times \frac{\sqrt{3}}{2} (\because sin \:60 = \frac{\sqrt{3}}{2}) \\
&=24.24   
\end{align}\]

\(\therefore\) Area to be painted = 24.24 inch2
Example 2

 

 

The sides of a triangular park are in the ratio \(12:17:25,\) and its perimeter is \(1080 \; m\)

What is its area?

Find the area of a triangular park whose sides are in the ratio 12:17:25.

Solution

The sides of the triangle are \(12x\), \(17x\) and \(25x\)

\[\begin{align}12x + 17x + 25x &= 1080\\  \Rightarrow 54x &= 1080\\ \Rightarrow x &= 20\end{align}\]

Thus, the sides of the triangle are:

\[240 m, \; 340 m,\; 500 m\]

Now, the semi-perimeter of the triangle is:

\[s = \frac{{1080}}{2} = 540\:m\]

Using Heron’s Formula, the area of the triangle is:

\[\begin{align}&A\! =\! \sqrt {540\left( {540 \!- \!240} \right)\left( {200} \right)\left( {540\! -\! 500} \right)} \\ &\quad= \sqrt {540 \times 300 \times 200 \times 40} \\& \quad= 36,000\, \rm{m^2}\end{align}\]

\(\therefore\) Area of the triangular park =36,000 m2

Example 3

 

 

Find the area of the shaded region in the following figure.

Area of shaded region in a triangle

Solution

\(\Delta\) ABC is an isosceles triangle, therefore its area is:

\(\begin{align}Area(\Delta ABC)&=\dfrac{1}{4}BC\sqrt{4(AB)^2-(BC)^2}\\&=\dfrac{1}{4} \times 10\sqrt{4(15)^2-10^2}\\&=\dfrac{5}{2}\sqrt{4\times 225-100}\\&=\dfrac{5}{2}\sqrt{800}\\&=\dfrac{5}{2}\times 20\sqrt{2}\\&=50\sqrt{2} \end{align}\)

Also,

\[\begin{align}&area\left( {\Delta BDC} \right) = \frac{1}{2} \times CD \times BD\\&\qquad\qquad\qquad\; = \frac{1}{2} \times 6 \times 8 = 24\, \rm{cm^2}\end{align}\]

Thus, the area of the shaded region is:

\[\begin{align}
& A \;= area(\Delta ABC) - area(\Delta BDC)\\
&\quad = (50\sqrt 2 - 24)\,\rm{cm^2}\\
&\quad \approx 46.7\, \rm{cm^2}
\end{align}\]

\(\therefore\) Area of the shaded region = 46.7 cm2
 
Challenge your math skills
Challenging Question
  1. In the given figure, \(ABCD\) is a parallelogram.

    If the area of \(\Delta BFC = 40\;cm^2\), calculate the area of \(\Delta ABE\).
                                  Calculate the area of a triangle within a parallelogram.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the 'Check Answer' button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of the area of triangle. The math journey around area of triangle starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.


FAQs on Area Of Triangle

1. What is area and perimeter in maths?

Area of a figure is the region enclosed by that figure.

Perimeter of a figure is the total length of the boundary of the figure.

2. What is the height of a triangle?

The height of a triangle is the length of the perpendicular from one side of the triangle to the opposite vertex.

3. What is the formula of perimeter and area of a triangle?

The formula for the perimeter of a triangle is a + b + c, where a, b, c are the lengths of the sides of a triangle.

The formula for the area of a triangle is \(\dfrac{1}{2}\) × Base × Height

4. How do you find the base and height of a triangle?

The area of triangle can be calculated with the formula: \(\dfrac{1}{2}\) × Base × Height.

The height and base can be calculated with the help of the same formula, when the other dimensions are known.

5. How do you find the area and perimeter of a triangle?

The area of a triangle can be calculated with the help of the formula \(\dfrac{1}{2}\) × Base × Height.

The perimeter of a triangle can be calculated by adding the three sides of a triangle.

6. How do you find the area of a triangle without height?

 Heron's formula can be used to find the area of a triangle when the length of the 3 sides of the triangle are known.

Hence, the area of the triangle can be calculated using Heron's formula without height.

7. How do you find the area of a triangle given two sides and an angle?

The area of a triangle is half the product of the given two sides and sine of the included angle.

For a detailed explanation refer to Area of Triangle with 2 Sides and Included Angle (SAS)

8. How do you find the area of a triangle with 3 sides?

The area of a triangle with 3 sides can be calculated using Heron's formula, that is Area = \(\sqrt {s(s - a)(s - b)(s - c)} \).

9. How do you find the area of an irregular triangle?

The area of an irregular triangle (sometimes referred to as a scalene triangle) can be calculated using the formula:  \(\sqrt {s(s - a)(s - b)(s - c)} \).

10. How do you calculate the area of an obtuse triangle?

The area of an obtuse triangle can be calculated using the formula : \(\dfrac{1}{2}\) × Base × Height,  by determining the length of any of its sides and height.

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