How to Find Area of Triangle Using Coordinate Geometry Formula with Solved Examples
In Geometry, a triangle is a three-sided polygon that has three edges and three vertices. The area of triangle in coordinate geometry is a measure of the space covered by the triangle in the two-dimensional plane. If the three vertices of the triangle are given in the coordinate plane, the area can be determined. In this article, let us discuss what the area of a triangle is, various methods to find the area of the triangle according to the given parameters and formula of area of triangle in coordinate geometry along with solved examples.
Formula of Area of Triangle in Coordinate Geometry
There are three methods for calculating the area of a triangle. The three approaches are described in detail below.
Method 1
When the triangle's base and altitude are known.
Area of the triangle, A=\[\frac{bh}{2}\]square units
Where b and h are the triangle's base and altitude, respectively.
Method 2
When the length of three sides of the triangle is given, then Heron’s Formula is used to calculate the area of a triangle.
Therefore, the equation used to calculate the triangle's area is given below.
A=\[\sqrt{s(s-a)(s-b)(s-c)}\]
The side lengths of the triangle are a, b, and c, and the semi perimeter is s.
The s value is found by using the formula S=\[\frac{a+b+c}{2}\]
Area of Triangle with Coordinates
Method 3
In this method, we will learn a formula to calculate the area of triangle with coordinates.
If coordinates points are A(x1,y1),B(x2,y2),C(x3,y3) then the area of triangle using coordinates is $\frac{1}{2}$ [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]
Derivation of the Formula is given below:
Assume a triangle PQR with the coordinates P, Q, and R given as (x1,y1),(x2,y2),(x3,y3)
respectively.
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From the figure, the area of a triangle PQR, lines such as \[\underset{QA}{\leftrightarrow},\underset{PB}{\leftrightarrow}And\underset{RC}{\leftrightarrow}\] are drawn from Q, P and R, respectively perpendicular to the x-axis.
In the coordinate plane, three separate trapeziums, such as PQAB, PBCR, and QACR, are now created.
Calculate the region of each trapezium now.
Therefore, the area of ∆PQR can be calculated as Area of ∆PQR=[\text{Area of trapezium PQAB + Area of trapezium PBCR}] -[\text{Area of trapezium QACR}] —-(1)
Finding Area of a Trapezium PQAB
The formula for calculating the region of a trapezium is as follows:
Since Area of a trapezium = $\frac{1}{2}$ (sum of the parallel sides)×(distance between them)
Area of trapezium PQAB = $\frac{1}{2}$ (QA + PB) × AB
QA = y2
PB = y1
AB = OB – OA = x1-x2
Area of trapezium PQAB = $\frac{1}{2}$ (y1+y2)(x1-x2) —-(2)
Finding Area of a Trapezium PBCR
Area of trapezium PBCR = $\frac{1}{2}$ (PB + CR) × BC
PB = y1
CR =y3
BC = OC – OB = x3-x1
Area of trapezium PBCR = $\frac{1}{2}$ (y1+y3)(x3-x1) (1/2) —-(3)
Finding Area of a Trapezium QACR
Area of trapezium QACR = $\frac{1}{2}$ (QA + CR) × AC
QA = y2
CR = y3
AC = OC – OA = x3-x2
Area of trapezium QACR = $\frac{1}{2}$ (y2+y3)(x3-x2)-(4)
Substituting (2), (3) and (4) in (1),
Area of ∆PQR = $\frac{1}{2}$ [(y1+y2)(x1-x2)+(y1+y3)(x3-x1)-(y2+y3)(x3-x2)
A=$\frac{1}{2}$ [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
Special Case:
If one of the triangle's vertices is the origin, the triangle's area can be determined using the formula below.
Calculate area of a triangle with vertices are (0,0), P(a, b), and Q(c, d)
A = $\frac{1}{2}$ [0(b – d) + a(d – 0) + c(0 – b)]
A = (ad – bc)/2
Area of Triangle with Vertices
When the area of a triangle with vertices P(x1,y1),Q(x2,y2) and R(x3,y3) is zero, then $\frac{1}{2}$ [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)=0 and the points P(x1,y1),Q(x2,y2) and R(x3,y3)are collinear. They do not form a triangle since they are in a straight line.
Area of triangle with 3 points
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The formula for calculating area can be expressed using determinants, as shown below.
\[\alpha =\frac{1}{2}\begin{vmatrix}
X_{1} &Y_{1} &1 \\
X_{2}&Y_{2} & 1\\
X_{3} &Y_{3} & 1
\end{vmatrix}\]
Where denotes the area of a triangle and A(x1,y1),B(x2,y2) and C(x3,y3)represent the vertices of the triangle.
We know the determinant's value can be either negative or positive. Since we are finding an area and it can never be taken as a negative, therefore we should take the absolute value of the determinant.
We use both the positive and negative values of the determinant if the triangle's area is already known.
Area of triangle with 3 points formula is \[\alpha =\frac{1}{2}\][x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]
Another form
The following pictorial representation makes writing the above formula very simple.
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Area of ABC=12{(x1.y2+x2y3+x3y1)-(x2y1+x3y2+x1y3) sq. units.
Note: As we know a triangle's area can never be negative. So we must take the absolute value, in case the area happens to be negative.
Area of a Triangle in Coordinate Geometry Example
1. Calculate the area of the ∆ABC whose vertices are A(1, 2), B(4, 2) and C(3, 5)?
Sol: Using the formula,
12[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
A=12[1(2-5)+4(5-2)+3(2-2)]
A=12[-3+12]
A=92 square units.
Hence, the area of a triangle ABC is 92 square units.
2. Find the area of the triangle using vertices A(3,4), B(4,7) and C(3,5).
Sol: Here x1=3,y1=4,x2=4,y2=7,x3=3 and y3=5
We know the formula of area of a triangle is given by
12[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)
=12[3(7-3)+4((-3)-(-4))+6(4-(-7))]
=1230+4-18
=8 sq. units
Hence, the area of a triangle is 8 unit square
Conclusion:
We have discussed the area of triangle formula in coordinate geometry. When three points are collinear, the triangle's area is zero. The area of a triangle cannot be negative. When we get the answer in negative terms, we should consider the numerical value of the area, without the negative sign as the area is always positive.
FAQs on Area of Triangle in Coordinate Geometry Explained with Formula and Concept
1. What is the formula for the area of a triangle in coordinate geometry?
The area of a triangle in coordinate geometry with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by Area = 1/2 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
- This formula is also called the determinant method.
- The vertical bars indicate the absolute value, ensuring the area is always positive.
- It works for all triangles in the Cartesian plane.
2. How do you find the area of a triangle using coordinates step by step?
To find the area using coordinates, substitute the given points into the coordinate geometry area formula and simplify.
- Step 1: Write the vertices as (x₁, y₁), (x₂, y₂), (x₃, y₃).
- Step 2: Apply 1/2 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
- Step 3: Simplify and take the absolute value.
- Step 4: Multiply by 1/2 to get the final area.
3. Why do we take the absolute value in the area of a triangle formula?
We take the absolute value because area is always positive, even if the determinant gives a negative value.
- The sign depends on the order of the vertices (clockwise or anticlockwise).
- A negative result only indicates orientation, not negative area.
- Using absolute value ensures the final area of triangle is non-negative.
4. What is the area of a triangle with vertices (1, 2), (4, 6), and (6, 3)?
The area of the triangle with vertices (1,2), (4,6), and (6,3) is 7 square units.
- Apply the formula: 1/2 |1(6−3) + 4(3−2) + 6(2−6)|
- = 1/2 |1×3 + 4×1 + 6×(−4)|
- = 1/2 |3 + 4 − 24| = 1/2 |−17|
- = 17/2 = 8.5 square units
5. How do you know if three points form a triangle in coordinate geometry?
Three points form a triangle if their calculated area using the coordinate formula is not equal to zero.
- If Area ≠ 0, the points are non-collinear and form a triangle.
- If Area = 0, the points are collinear and lie on a straight line.
- This test is commonly called the collinearity condition.
6. What is the condition for collinearity using the area of a triangle formula?
The condition for collinearity is that the area of the triangle formed by the three points is zero.
- Use the formula: 1/2 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
- If the result equals 0, the points are collinear.
- If the result is non-zero, they form a triangle.
7. Can the area of a triangle in coordinate geometry be negative?
No, the area of a triangle cannot be negative because we take the absolute value of the determinant.
- The intermediate value may be negative due to vertex order.
- After applying absolute value, the final area is always positive or zero.
- Area equals zero only when points are collinear.
8. How is the determinant method related to the area of a triangle?
The determinant method calculates the area of a triangle by evaluating a 3×3 determinant formed using the coordinates.
- The area equals 1/2 of the absolute value of the determinant:
- |x₁ y₁ 1|
- |x₂ y₂ 1|
- |x₃ y₃ 1|
- This simplifies to the standard coordinate geometry area formula.
9. What are common mistakes when finding the area of a triangle using coordinates?
Common mistakes include sign errors, incorrect substitution, and forgetting the 1/2 factor in the formula.
- Not applying the absolute value.
- Mixing up x and y coordinates.
- Forgetting to multiply by 1/2.
- Arithmetic mistakes during simplification.
10. What is the geometric meaning of the area formula in coordinate geometry?
Geometrically, the area formula represents half the area of a parallelogram formed by two side vectors of the triangle.
- The determinant gives the signed area of a parallelogram.
- Dividing by 2 gives the triangle’s area.
- This connects coordinate geometry with vector cross product concepts.
