Mathematics notes, chapter 10

Perimeter and area

Textbook Mathematics Form 1 (DLP), pages 224–244

What this chapter covers

Determine the perimeter of shapes when the side lengths are given or need to be measured, including finding missing sides (10.1)
Estimate the perimeter of a shape on a grid and judge the accuracy with the percentage error (10.1)
Solve problems involving perimeter, such as fencing and its cost (10.1)
Estimate the area of irregular shapes by counting squares on a grid (10.2)
Explain how the area formulae of triangles, parallelograms, kites and trapeziums come from the area of a rectangle (10.2)
Calculate the areas of triangles, parallelograms, kites, trapeziums and combined shapes, including shaded regions (10.2)
Make and verify conjectures about the relationship between perimeter and area of rectangles (10.3)
Solve problems involving both perimeter and area (10.3)

Photo of a green paddy field with palm trees and a small house — Kedah, the "rice bowl" (Jelapang Padi) of Malaysia. How long is the bund around a paddy field, and how much land does it cover?

A farmer who wants to fence a paddy field needs to know the perimeter: the distance all the way round. A farmer who wants to know how much rice he can plant needs the area: how much surface the field covers. In this chapter you will find perimeters, use formulae for the areas of triangles, parallelograms, kites and trapeziums, and discover how perimeter and area are related.

The word "perimeter" comes from Greek: peri means "around" and meter means "measure". Ideas about area were written in the Rhind Papyrus in Egypt, around 1650 B.C.

English	Bahasa Melayu
perimeter	perimeter
area	luas
length, width, height	panjang, lebar, tinggi
square, rectangle	segi empat sama, segi empat tepat
triangle, parallelogram	segi tiga, segi empat selari
kite, trapezium	lelayang, trapezium
square unit, formula, conjecture	unit persegi, rumus, konjektur

10.1Perimeter

What is perimeter?

Perimeter is the total length around the boundary of a closed shape. To find it, add the lengths of all the sides. Perimeter is a length, so its unit is cm, m or km (not cm²).

Two swimming pools: a rectangular pool and a pool with curved sides — To find the perimeter of the curved pool, lay a string along the edge and then measure the string (textbook p. 226).

Finding missing sides

In a shape made of horizontal and vertical lines, some sides are not labelled. Use the sides that are labelled: the total of the top pieces equals the bottom, and the total of the right pieces equals the left.

Example 1 (like textbook p. 226)

L-shaped figure ABCDEF: AB = 6 cm, BC = 3 cm, CD = x cm, DE = 4 cm, EF = 10 cm, FA = 7 cm. All corners are right angles.

Find the perimeter of the shape ABCDEF.

Find the missing side: AB + CD = FE, so 6 + x = 10, and x = 10 − 6 = 4 cm.
Check the vertical sides: BC + DE = 3 + 4 = 7 cm = FA. ✓
Perimeter = AB + BC + CD + DE + EF + FA = 6 + 3 + 4 + 4 + 10 + 7 = 34 cm

Shortcut for shapes with only right-angled corners: the perimeter is the same as the rectangle around it. Here 2 × (10 + 7) = 34 cm. (This works only when every corner is a right angle.)

Go round the shape in order and tick each side as you add it. Students often forget the short sides that are not labelled.

Example 2: a slanted side

Shape made of a rectangle and a triangle: top 8 cm, right 5 cm, slanted side 5 cm, bottom 12 cm, left 8 cm

The sides are 8 cm, 5 cm, 5 cm, 12 cm and 8 cm.
Perimeter = 8 + 5 + 5 + 12 + 8 = 38 cm

A rectangle is 9 cm long and 4 cm wide. A square has sides of 6 cm. Which shape has the longer perimeter?

Show answerHide answer

Rectangle: 9 + 4 + 9 + 4 = 26 cm. Square: 4 × 6 = 24 cm. The rectangle has the longer perimeter.

Estimating perimeter and percentage error

A shape drawn on a grid of 1 cm squares can be estimated: count the squares along the straight sides and estimate the slanted or curved sides. Then measure with a ruler (or a thread for curves) to see how good the estimate was.

Example 3 (like textbook pp. 227–228)

Quadrilateral on a 1 cm grid: top 2 squares, bottom 4 squares, left side 3 squares, slanted right side

Top = 2 cm, bottom = 4 cm, left = 3 cm. The slanted side is a little longer than 3.5 squares, so estimate 3.5 cm.
Estimate = 2 + 3.5 + 4 + 3 = 12.5 cm
Measured with a ruler: 12.6 cm
Percentage error = 12.6 − 12.512.6 × 100% = 0.79% (2 decimal places)

Percentage error = difference between estimated value and actual valueactual value × 100%. The smaller the percentage error, the more accurate the estimate.

Divide by the actual (measured) value, not by the estimate.

Solving problems involving perimeter

Example 4: fencing (like textbook pp. 228–229)

L-shaped field: top 18 m, step down 8 m, right side 12 m, bottom 30 m, left side 20 m. A 4 m gate is on the bottom side.

Encik Hafiz fences the whole field except a 4 m gate. Fencing costs RM15 per metre. Find the cost.

The unlabelled horizontal side = 30 − 18 = 12 m. Check the vertical sides: 8 + 12 = 20 m ✓
Perimeter = 18 + 8 + 12 + 12 + 30 + 20 = 100 m
Length of fence = 100 − 4 = 96 m
Cost = 96 × RM15 = RM1 440

Read the question: if there is a gate or an entrance, subtract it from the perimeter before you multiply by the price.

The perimeter of a rectangular laboratory is 64 m. Its length is 23 m. Find its width. (Textbook p. 230)

Show answerHide answer

Length + width = 64 ÷ 2 = 32 m
Width = 32 − 23 = 9 m

10.2Area of triangles, parallelograms, kites and trapeziums

Estimating area

Area is the amount of surface a shape covers. It is measured in square units: cm², m², unit².

For an irregular shape on a grid of 1 unit squares: count the full squares, then count the squares that are half full or more as 1 each, and ignore squares that are less than half full.

Example 5 (like textbook p. 231)

Irregular curved shape on a grid of 1 unit squares. Ticks mark 32 full squares; dots mark 14 squares that are half full or more.

Full squares (✓) = 32
Squares half full or more (●) = 14
Area ≈ 32 + 14 = 46 unit²

Another method: draw lines 1 unit apart across the shape, measure each line and add the lengths (each line stands for a strip 1 unit wide).

Where the formulae come from

Every formula comes from the area of a rectangle, length × width.

Triangle with base a and height b inside a rectangle a by b; the height splits it into parts 1 and 2 — **Triangle:** The height splits the rectangle into two smaller rectangles. The triangle is exactly half of each one, so it is half of the whole rectangle. **Area = ½ × base × height = ½ab**

Parallelogram with base a and height b; the triangle cut from the left side moves to the right to make a rectangle — **Parallelogram:** Cut off the triangle on one side and move it to the other side. You get a rectangle with the same base and height. **Area = base × height = ab**

Kite inside a rectangle; the diagonals a and b are the width and height of the rectangle — **Kite:** The diagonals split the rectangle into 4 small rectangles. The kite is half of each, so half of the rectangle a × b. **Area = ½ × product of the diagonals = ½ab**

Two identical trapeziums with parallel sides a and b and height t joined to make a parallelogram with base a + b — **Trapezium:** Two identical trapeziums make a parallelogram with base (a + b) and height t. One trapezium is half of it. **Area = ½ × (sum of parallel sides) × height = ½(a + b)t**

The height is always perpendicular (at 90°) to the base. Do not use a slanted side as the height.

A rhombus is a parallelogram and also a kite, so you can use either base × height or ½ × product of the diagonals.

Calculating areas

Example 6 (like textbook pp. 236–237)

(a) Triangle base 8 cm, height 5 cm. (b) Parallelogram base 7 cm, height 4 cm, slanted side 5 cm. (c) Trapezium parallel sides 4 cm and 9 cm, height 6 cm. (d) Kite with diagonals 6 cm and 10 cm.

(a) Triangle = ½ × 8 × 5 = 20 cm²
(b) Parallelogram = 7 × 4 = 28 cm² (the 5 cm side is not the height)
(c) Trapezium = ½ × (4 + 9) × 6 = ½ × 13 × 6 = 39 cm²
(d) Kite = ½ × 10 × 6 = 30 cm²

Find the area of: a) a triangle with base 12 cm and height 7 cm; b) a trapezium with parallel sides 5 cm and 11 cm and height 4 cm.

Show answerHide answer

a) ½ × 12 × 7 = 42 cm²
b) ½ × (5 + 11) × 4 = ½ × 16 × 4 = 32 cm²

Combined shapes and shaded regions

For a shaded region, find the area of the big shape and subtract the unshaded parts. For a combined shape, add the parts.

Example 7 (like textbook p. 237)

Rectangle PRTV 20 m by 12 m. Triangle PQW at corner P with PQ = 5 m and PW = 6 m. Trapezium UTSX at corner T with XS = 3 m, UT = 6 m and ST = 6 m. The rest is shaded.

PRTV is a rectangular plot. Bananas grow in triangle PQW and rambutans in trapezium UTSX. Find the shaded area.

Area of PRTV = (5 + 15) × (6 + 6) = 20 × 12 = 240 m²
Area of PQW = ½ × 5 × 6 = 15 m²
Area of UTSX = ½ × (3 + 6) × 6 = 27 m²
Shaded area = 240 − 15 − 27 = 198 m²

10.3Relationship between perimeter and area

Kassim has a fixed length of fencing for a rectangular vegetable plot. Which rectangle gives the biggest plot? Try it with a loop of string 24 cm long: every rectangle below has a perimeter of 24 cm, but their areas are different.

Four rectangles with perimeter 24 cm: 11 by 1 has area 11 cm², 9 by 3 has 27 cm², 7 by 5 has 35 cm², 6 by 6 has 36 cm²

Same perimeter (24 cm)				Same area (36 cm²)
Length	Width	Area	Difference	Length	Width	Perimeter
11	1	11	10	36	1	74
10	2	20	8	18	2	40
9	3	27	6	12	3	30
8	4	32	4	9	4	26
7	5	35	2	6	6	24
6	6	36	0

Conjectures (verified by the tables):
For rectangles with the same perimeter, the area gets smaller as the difference between length and width gets bigger. The area is largest when the rectangle is a square.
For rectangles with the same area, the perimeter gets bigger as the difference between length and width gets bigger. The perimeter is smallest when the rectangle is a square.

Two shapes with the same perimeter do not always have the same area, and two shapes with the same area do not always have the same perimeter.

Problems with perimeter and area

Example 8 (textbook p. 241)

The area of a square garden is 500 m². Find its perimeter.

Let the side be x m. Then x² = 500.
x = √500 = 22.36 m (2 decimal places)
Perimeter = 4 × 22.36 = 89.44 m

Example 9 (like textbook p. 241)

Triangle PQS drawn inside rectangle PQRT, with S on the side TR

Triangle PQS is inside rectangle PQRT. The perimeter of the rectangle is 54 cm and its length is twice its width. Find the area of triangle PQS.

Let the width be w cm. The length is 2w cm.
2w + 2(2w) = 54, so 6w = 54 and w = 9. Length = 2 × 9 = 18 cm.
Check: 2(18) + 2(9) = 36 + 18 = 54 ✓
The base of the triangle is PQ = 18 cm and its height is the width, 9 cm.
Area of PQS = ½ × 18 × 9 = 81 cm² (half the rectangle)

a) The perimeter of a square is 36 cm. Find its area. b) The length of a rectangle is 4 cm more than its width and its perimeter is 32 cm. Find its area.

Show answerHide answer

a) Side = 36 ÷ 4 = 9 cm, area = 9 × 9 = 81 cm²
b) 2w + 2(w + 4) = 32, 4w + 8 = 32, w = 6 cm, length = 10 cm, area = 10 × 6 = 60 cm²

Summary

Shape	Perimeter	Area
Rectangle (length l, width w)	2l + 2w	l × w
Square (side s)	4s	s × s
Triangle (sides a, b, c; base, height)	a + b + c	½ × base × height
Parallelogram (sides a, b)	2a + 2b	base × height
Kite (sides a, b; diagonals d₁, d₂)	2a + 2b	½ × d₁ × d₂
Trapezium (parallel sides a, b; height t)	sum of the 4 sides	½ × (a + b) × t

Perimeter = total length around a shape (cm, m). Area = surface covered (cm², m²).
Percentage error = difference ÷ actual value × 100%. Smaller means more accurate.
Estimate area on a grid: full squares + squares that are half full or more.
Height is always perpendicular to the base.
Shaded region = big shape − unshaded parts.
Same perimeter: a square gives the largest area. Same area: a square gives the smallest perimeter.