Quadrilateral Formulas

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7. Quadrilateral Formulas

A quadrilateral is a closed plane figure having four sides, four vertices and four angles.

The total of the four interior angles of a quadrilateral is 360°.

Quadrilateral questions in mensuration commonly involve area, perimeter, diagonals and angles.

What Is a Quadrilateral?

A B C D d₁ d₂

AB, BC, CD and DA are the four sides, while AC and BD are the diagonals.

1. Sum of Interior Angles of a Quadrilateral

Every quadrilateral has four interior angles. The sum of these four angles is:

Sum of Interior Angles = 360°

If the four angles are A, B, C and D:

A + B + C + D = 360°

Example

Three angles of a quadrilateral are 80°, 90° and 110°. Find the fourth angle.

Using:

A + B + C + D = 360°

80° + 90° + 110° + D = 360°

280° + D = 360°

D = 80°

Answer: 80°

2. Perimeter of a Quadrilateral

The perimeter of any quadrilateral is the sum of all four sides.

If the four sides are a, b, c and d:

Perimeter = a + b + c + d

Example

A quadrilateral has sides measuring 8 cm, 10 cm, 12 cm and 15 cm. Find its perimeter.

P = a + b + c + d

P = 8 + 10 + 12 + 15

P = 45 cm

Answer: 45 cm

3. Area of a Quadrilateral Using Diagonals

When the lengths of the two diagonals and the angle between them are known, the area can be calculated using the diagonal formula.

Area = ½ × d₁ × d₂ × sin θ

Where:

  • d₁ = First diagonal
  • d₂ = Second diagonal
  • θ = Angle between the diagonals

Special Case: Perpendicular Diagonals

If the two diagonals intersect at a right angle, θ = 90°. Since:

sin 90° = 1

the formula becomes:

Area = ½ × d₁ × d₂

Example

The diagonals of a quadrilateral are 12 cm and 10 cm and they intersect at right angles. Find its area.

Area = ½ × d₁ × d₂

= ½ × 12 × 10

= 6 × 10

= 60 cm²

Answer: 60 cm²

4. Area of a Quadrilateral by Dividing It into Triangles

A quadrilateral can be divided into two triangles by drawing one of its diagonals.

For example, diagonal AC divides quadrilateral ABCD into:

  • Triangle ABC
  • Triangle ACD

Therefore:

Area of Quadrilateral = Area of △ABC + Area of △ACD

Example

A diagonal divides a quadrilateral into two triangles whose areas are 45 cm² and 35 cm². Find the area of the quadrilateral.

Area = 45 + 35

Area = 80 cm²

Answer: 80 cm²

5. Diagonals of a Quadrilateral

A quadrilateral has exactly two diagonals. A diagonal joins two opposite vertices.

Number of diagonals = 2

If the vertices are A, B, C and D, the diagonals are:

AC and BD

Important Point

The properties of the diagonals depend on the type of quadrilateral. For example, the diagonal properties of a square, rectangle, rhombus and parallelogram are different.

Therefore, do not assume that the two diagonals are always equal or perpendicular.

6. Important Quadrilateral Types

Type Important Property
Square Four equal sides and four right angles
Rectangle Opposite sides equal and four right angles
Parallelogram Opposite sides are parallel and equal
Rhombus All four sides are equal
Trapezium One pair of opposite sides is parallel
Kite Two pairs of adjacent sides are equal
Exam Tip:

Many mensuration questions first identify the type of quadrilateral and then ask for its area or perimeter. Identify the shape before applying a formula.

7. General Strategy for Quadrilateral Area Questions

When a quadrilateral does not have a direct area formula, look for a diagonal.

Step 1: Draw or identify a diagonal.

Step 2: Divide the quadrilateral into two triangles.

Step 3: Find the area of each triangle.

Step 4: Add both triangular areas.

Quadrilateral Area = Triangle 1 Area + Triangle 2 Area

8. Common Mistakes in Quadrilateral Questions

Mistake 1: Forgetting the 360° angle sum

The four interior angles of a quadrilateral always add up to:

360°

Mistake 2: Confusing area and perimeter

Perimeter is measured in ordinary units such as cm or m. Area is measured in square units such as cm² or m².

Mistake 3: Assuming diagonal properties

The diagonals of every quadrilateral do not have the same properties. Always identify the particular type of quadrilateral.

Mistake 4: Forgetting the ½ in diagonal-area questions

Area = ½ × d₁ × d₂

9. Quadrilateral Formula Chart

Quantity Formula
Sum of Interior Angles 360°
Perimeter a + b + c + d
Area using Diagonals ½ × d₁ × d₂ × sin θ
Area when Diagonals are Perpendicular ½ × d₁ × d₂
Area by Triangles Area of △1 + Area of △2
Number of Diagonals 2

10. Practice Questions

Question 1: The four angles of a quadrilateral are 70°, 80°, 100° and x. Find x.

Question 2: Find the perimeter of a quadrilateral whose sides are 12 cm, 15 cm, 18 cm and 20 cm.

Question 3: The diagonals of a quadrilateral are 16 cm and 12 cm and are perpendicular. Find its area.

Question 4: A diagonal divides a quadrilateral into two triangles of areas 75 cm² and 125 cm². Find the area of the quadrilateral.

Question 5: How many diagonals does a quadrilateral have?

Question 6: The diagonals of a quadrilateral are 10 cm and 14 cm and the angle between them is 90°. Find its area.

11. Quadrilateral Formula Quick Revision

Angle Sum: 360°
Perimeter: a + b + c + d
Diagonal Area: ½ × d₁ × d₂ × sin θ
Perpendicular Diagonals: ½ × d₁ × d₂
Two-Triangle Method: Area △1 + Area △2

12. Mensuration Exam Tip

For quadrilateral questions, first identify what information is given: sides, angles, diagonals, base and height.

Then determine whether the figure is a general quadrilateral or a special quadrilateral such as a square, rectangle, parallelogram, rhombus or trapezium.

This prevents using the wrong mensuration formula.

Next Part:

In Part 8 — Rhombus Formulas, we will cover the area, perimeter, diagonals and important relationships of a rhombus, with diagrams, explanations, solved examples, practice questions and exam tips.

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