Properties of Rectangles

Learn the defining properties of rectangles including equal opposite sides, right angles, and diagonal relationships.

Intermediate20 minLesson
Printable worksheet

Definition

A rectangle is a quadrilateral (4-sided polygon) with four right angles (90°).

Key Properties of Rectangles

1. Four Right Angles Every interior angle measures exactly 90°90°.
∠A=∠B=∠C=∠D=90°\angle A = \angle B = \angle C = \angle D = 90°
2. Opposite Sides Are Equal and Parallel The two pairs of opposite sides have equal length:
AB‾=CD‾andBC‾=AD‾\overline{AB} = \overline{CD} \quad \text{and} \quad \overline{BC} = \overline{AD}
3. Diagonals Are Congruent The two diagonals have the same length:
AC‾=BD‾\overline{AC} = \overline{BD}
4. Diagonals Bisect Each Other Each diagonal cuts the other exactly in half at their intersection point.

Try it now

How many right angles does a rectangle have?

Worked Examples

A rectangle has a length of 1212 cm and a width of 77 cm. Find its perimeter.
1

Recall the perimeter formula

Perimeter =2×(length+width)= 2 \times (\text{length} + \text{width}) → P=2(l+w)P = 2(l + w)
2

Substitute the values

P=2×(12+7)P = 2 \times (12 + 7) → P=2×19P = 2 \times 19
3

Calculate

P=38P = 38 → 3838 cm

Common Mistakes

Confusing rectangles with squares

Why it's wrong: A square IS a special type of rectangle (all angles are 90°), but not all rectangles are squares.

Correct: A rectangle has opposite sides equal. A square has ALL four sides equal. Every square is a rectangle, but not every rectangle is a square.

Forgetting that both diagonals are equal

Why it's wrong: In some quadrilaterals (like parallelograms), diagonals are NOT equal.

Correct: In a rectangle, both diagonals are always congruent (same length). This is a special property!

Using P=l×wP = l \times w instead of P=2(l+w)P = 2(l + w)

Why it's wrong: Students confuse perimeter (distance around) with area (space inside).

Correct: Perimeter = sum of all sides = 2l+2w=2(l+w)2l + 2w = 2(l + w). Area = l×wl \times w.

Interactive Visual

Area and perimeter

l = 8w = 5

Area40

A=l⋅w=8⋅5=40A = l \cdot w = 8 \cdot 5 = 40

Perimeter26

P=2(l+w)=2(8+5)=26P = 2(l + w) = 2(8 + 5) = 26

Diagonal9.43

d=l2+w2=89≈9.43d = \sqrt{l^2 + w^2} = \sqrt{89} \approx 9.43

The two diagonals of a rectangle are always equal.

Interactive Sandbox

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Practice Problems

15 problems
Problem 1 of 15
Easy
How many right angles does a rectangle have?

Why It Matters

Rectangles are one of the most common shapes in the world around us:
  • Architecture: Doors, windows, rooms, and buildings are typically rectangular
  • Technology: Screens, phones, tablets, and monitors are all rectangles
  • Sports: Soccer fields, basketball courts, and swimming pools use rectangular shapes
  • Design: Books, papers, cards, and photographs are rectangular
Understanding rectangle properties helps with construction, design, and solving real-world measurement problems!

Real World Applications

Home Improvement: Flooring

Calculating how much flooring material you need for a rectangular room.

Example:

A room is 55 m by 44 m. You need 5×4=205 \times 4 = 20 m² of flooring.

1Try It Yourself

You want to put a fence around a rectangular backyard that is 2020 meters long and 1515 meters wide.

How many meters of fencing do you need?

Step 1: Write the mathematical expression

Use the perimeter formula:

Screen Technology

TV and monitor screens are measured diagonally, but the actual viewing area uses rectangle properties.

Example:

A screen that is 4848 cm wide and 2727 cm tall has an area of 48×27=129648 \times 27 = 1296 cm².

2Try It Yourself

A tablet screen is 2424 cm long and 1818 cm wide.

What is the diagonal measurement of the screen?

Step 1: Write the mathematical expression

Use the Pythagorean theorem:

Key Takeaways

  • 1A rectangle is a quadrilateral with four right angles (90°90°)
  • 2Opposite sides of a rectangle are equal and parallel
  • 3Both diagonals of a rectangle are congruent (equal in length)
  • 4The diagonals bisect each other (cut each other in half)
  • 5Perimeter: P=2(l+w)P = 2(l + w), Area: A=l×wA = l \times w
  • 6A square is a special rectangle where all four sides are equal

Frequently Asked Questions

Yes! A square meets all the requirements of a rectangle (four right angles, opposite sides equal). It's a special rectangle where all four sides happen to be equal.
Yes! A square meets all the requirements of a rectangle (four right angles, opposite sides equal). It's a special rectangle where all four sides happen to be equal.
Use the Pythagorean theorem: d=l2+w2d = \sqrt{l^2 + w^2}. The diagonal forms a right triangle with the length and width.
Both have opposite sides equal and parallel. However, a rectangle must have four 90°90° angles, while a parallelogram's angles can vary (as long as opposite angles are equal).

Glossary

Rectangle
A quadrilateral with four right angles
Diagonal
A line segment connecting two non-adjacent vertices
Congruent
Having the same size and shape; equal in measure
Bisect
To divide into two equal parts
Perimeter
The total distance around a shape
Area
The amount of space inside a 2D shape, measured in square units
Today's challengeOne problem, three tries

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