Area and Perimeter of Quadrilaterals

1. Rectangle

The figure has four right angles and opposite edges. Area is the space inside; Perimeter is the total length of the boundary.

$$P = 2 \times (\text{length} + \text{width})$$; $$A = \text{length} \times \text{width}$$

l is the length, w is the width. The unit of area is always square.

Example

A rectangle has a length of 8 cm and a width of 5 cm. Find its perimeter and area.

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Step 1: Identify the components: length = 8, width = 5

Step 2: Calculate the perimeter: $$P = 2 \times (8 + 5) = 2 \times 13 = 26$$ cm

Step 3: Calculate the area: $$A = 8 \times 5 = 40$$ cm²

Practice problems

Problem 1
A rectangular garden plots measures 12 meters long and 7 meters wide. What is the total area of the garden plot?

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Answer: 84 square meters

Step 1: Identify the length (12 m) and width (7 m)
Step 2: Use the area formula: $$A = \text{length} \times \text{width}$$
Step 3: Multiply the numbers: $$12 \times 7 = 84$$ square meters
Problem 2
A metal sheet has a width of 6 inches and a length of 15 inches. Find the perimeter of the sheet.

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Answer: 42 inches

Step 1: Identify the dimensions: length = 15, width = 6
Step 2: Use the perimeter formula: $$P = 2 \times (\text{length} + \text{width})$$
Step 3: Add and multiply: $$2 \times (15 + 6) = 2 \times 21 = 42$$ inches
Problem 3
The area of a rectangular poster is 48 square feet. If the width of the poster is 6 feet, what is its length?

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Answer: 8 feet

Step 1: Write down the area formula: Area = length × width, so length = Area / width
Step 2: Set up the division: $$\frac{48}{6}$$
Step 3: Compute the length: 8 feet

Remember

Ensure both length and width are converted to the same unit before performing calculations.

2. Square

A quadrilateral with four equal sides and four right angles. Area is the space inside; Perimeter is the total boundary.

$$P = 4 \times s$$; $$A = s^2$$

s is the length of one side of the square. All sides are congruent.

Example

A square tiles has a side length of 6 cm. Find its perimeter and area.

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Step 1: Identify the side length: $$s = 6$$

Step 2: Calculate the perimeter: $$P = 4 \times 6 = 24$$ cm

Step 3: Calculate the area: $$A = 6^2 = 36$$ cm²

Practice problems

Problem 4
A square courtyard has a side length of 15 meters. What is the total area of the courtyard?

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Answer: 225 square meters

Step 1: Identify the side length: $$s = 15\text{ m}$$
Step 2: Use the area formula for a square: $$A = s^2$$
Step 3: Calculate the value: $$15 \times 15 = 225$$ square meters
Problem 5
Find the perimeter of a square window frame that has a side length of 9 inches.

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Answer: 36 inches

Step 1: Identify the side length: $$s = 9$$ inches
Step 2: Use the perimeter formula for a square: $$P = 4 \times s$$
Step 3: Multiply the numbers: $$4 \times 9 = 36$$ inches
Problem 6
The area of a square concrete slab is 64 square feet. What is the perimeter of this slab?

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Answer: 32 feet

Step 1: Find the side length by taking the square root of the area: $$s = \sqrt{64} = 8$$ feet
Step 2: Use the perimeter formula with the side length found: $$P = 4 \times s$$
Step 3: Calculate the total perimeter: $$4 \times 8 = 32$$ feet

Remember

Since all sides are equal, you only need the measurement of one side to find both area and perimeter.

3.Parallelogram

A quadrilateral with two pairs of parallel and equal opposite sides. Area measures the internal space.

$$P = 2 \times (\text{base} + \text{side})$$; $$A = \text{base} \times \text{height}$$

base is the bottom edge, side is the slanted edge, and height is the perpendicular distance between bases.

Example

A parallelogram has a base of 10 cm, a slanted side of 6 cm, and a vertical height of 5 cm. Find its area.

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Step 1: Identify the base (10) and the vertical height (5)

Step 2: Disregard the slanted side for the area calculation

Step 3: Multiply base by height: $$A = 10 \times 5 = 50$$ cm²

Practice problems

Problem 7
A banner shaped like a parallelogram has a base of 12 inches and a perpendicular height of 8 inches. What is its area?

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Answer: 96 square inches

Step 1: Identify the base (12 in) and perpendicular height (8 in)
Step 2: Apply the parallelogram area formula: $$A = \text{base} \times \text{height}$$
Step 3: Multiply the measurements: $$12 \times 8 = 96$$ square inches
Problem 8
Calculate the perimeter of a parallelogram with a base of 14 cm and an adjacent slanted side of 9 cm.

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Answer: 46 cm

Step 1: Identify the base (14 cm) and the adjacent side (9 cm)
Step 2: Use the perimeter formula: $$P = 2 \times (\text{base} + \text{side})$$
Step 3: Add and multiply: $$2 \times (14 + 9) = 2 \times 23 = 46$$ cm
Problem 9
The area of a parallelogram structure is 72 square meters. If the base is 9 meters, what is its vertical height?

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Answer: 8 meters

Step 1: Write down the area formula: Area = base × height, so height = Area / base
Step 2: Divide the total area by the given base: $$\frac{72}{9}$$
Step 3: Calculate the vertical height: 8 meters

Remember

Never use the slanted side length as the height when calculating the area.

4.Rhombus

A special parallelogram with four equal sides. Opposite angles are equal, and diagonals intersect at 90 degrees.

$$P = 4 \times s$$; $$A = 0.5 \times d_1 \times d_2$$

s is the side length. d1 and d2 are the lengths of the two intersecting diagonals.

Example

A rhombus has diagonals measuring 10 cm and 8 cm. Find its area.

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Step 1: Identify the diagonal lengths: $$d_1 = 10$$, $$d_2 = 8$$

Step 2: Use the diagonal area formula: $$A = 0.5 \times d_1 \times d_2$$

Step 3: Multiply the components: $$0.5 \times 10 \times 8 = 40$$ cm²

Practice problems

Problem 10
A rhombus shaped kite has diagonals of 16 inches and 12 inches. What is the area of the kite?

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Answer: 96 square inches

Step 1: Identify diagonal measurements: $$d_1 = 16$$, $$d_2 = 12$$
Step 2: Apply the rhombus area formula: $$A = 0.5 \times d_1 \times d_2$$
Step 3: Perform the calculation: $$0.5 \times 16 \times 12 = 8 \times 12 = 96$$ square inches
Problem 11
Find the perimeter of a decorative rhombus tile that has a single side length of 7 cm.

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Answer: 28 cm

Step 1: Identify the side length: $$s = 7$$ cm
Step 2: Since all four sides are equal, use the formula: $$P = 4 \times s$$
Step 3: Multiply the numbers: $$4 \times 7 = 28$$ cm
Problem 12
A rhombus has a perimeter of 40 inches and a vertical height of 6 inches. What is its total area?

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Answer: 60 square inches

Step 1: Find the base side length using the perimeter: side = $$\frac{40}{4} = 10$$ inches
Step 2: Treat the rhombus as a parallelogram using the formula: Area = base × height
Step 3: Multiply the base side by the height: $$10 \times 6 = 60$$ square inches

Remember

The area can be calculated either by base and height or by half the product of its two diagonals.

5.Trapezoid

A quadrilateral with exactly one pair of parallel sides (called bases). Area measures the region enclosed within.

$$P = \text{base}_1 + \text{base}_2 + \text{side}_1 + \text{side}_2$$; $$A = 0.5 \times (\text{base}_1 + \text{base}_2) \times \text{height}$$

base1 and base2 are the parallel sides. height is the perpendicular line between the two bases.

Example

A trapezoid has parallel bases of 8 cm and 6 cm, and a height of 4 cm. Find its area.

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Step 1: Identify the components: base1 = 8, base2 = 6, height = 4

Step 2: Add the two bases together: $$8 + 6 = 14$$

Step 3: Apply the formula: $$A = 0.5 \times 14 \times 4 = 7 \times 4 = 28$$ cm²

Practice problems

Problem 13
A window shaped like a trapezoid has parallel bases of 10 inches and 14 inches. The height is 5 inches. Find its area.

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Answer: 60 square inches

Step 1: Identify dimensions: base1 = 10, base2 = 14, height = 5
Step 2: Sum up the parallel bases: $$10 + 14 = 24$$
Step 3: Apply the trapezoid formula: $$A = 0.5 \times 24 \times 5 = 12 \times 5 = 60$$ square inches
Problem 14
Find the perimeter of a trapezoid patch where the bases are 12 cm and 15 cm, and the non-parallel sides are 6 cm and 7 cm.

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Answer: 40 cm

Step 1: Identify all four bounding side lengths: 12, 15, 6, 7
Step 2: Use the perimeter formula by adding all sides together: $$P = b_1 + b_2 + s_1 + s_2$$
Step 3: Calculate the sum: $$12 + 15 + 6 + 7 = 40$$ cm
Problem 15
The area of a trapezoid layout is 45 square meters. The parallel bases are 6 meters and 12 meters. What is its vertical height?

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Answer: 5 meters

Step 1: Sum the bases: $$6 + 12 = 18$$. The formula is Area = 0.5 × (sum of bases) × height
Step 2: Substitute the known values: $$45 = 0.5 \times 18 \times \text{height} \rightarrow 45 = 9 \times \text{height}$$
Step 3: Solve for height: height $$= \frac{45}{9} = 5$$ meters

Remember

Always add the lengths of the two parallel bases together before multiplying by half of the height.