UNIT 9 • GEOMETRY & TRIGONOMETRY

Area & Perimeter
Every 2D Shape, Step by Step

Area is the space inside a flat shape, in square units; perimeter is the distance around it. Each shape has its own formula, from $l \times b$ for a rectangle to $\pi r^2$ for a circle.

THE MOST-USED FORMULAS11 SHAPES

\begin{gathered} \text{Rectangle: } A = lb,\ P = 2(l + b) \\ \text{Circle: } A = \pi r^2,\ C = 2\pi r \\ \text{Triangle: } A = \tfrac{1}{2}bh \end{gathered}

1. Pick the ShapeMatch the figure to its formula. Split odd shapes into rectangles, triangles and circles.

2. Same UnitsPut every length in the same unit (all cm or all m) before you substitute.

3. Square for AreaArea answers are in square units (cm²); perimeter answers are plain lengths (cm).

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A triangular field has sides 13 m, 14 m and 15 m. Find its perimeter and its area.

STEP 01 • PERIMETER AND HALF-PERIMETER

P = a + b + c = 13 + 14 + 15 = 42 \text{ m}

s = \frac{P}{2} = \frac{42}{2} = 21 \text{ m}

STEP 02 • HERON'S FORMULA

We know all three sides but not the height, so use Heron's formula:

A = \sqrt{s(s - a)(s - b)(s - c)}

A = \sqrt{21 \times (21 - 13) \times (21 - 14) \times (21 - 15)}

A = \sqrt{21 \times 8 \times 7 \times 6}

STEP 03 • SIMPLIFY THE ROOT

21 \times 8 \times 7 \times 6 = 7056, \qquad \sqrt{7056} = 84

Check: the height onto the 14 m side is 12 m, and $\tfrac{1}{2} \times 14 \times 12 = 84$ ✓

FINAL CONCLUSION

Perimeter = 42 m, Area = 84 m²

P = 42 \text{ m}, \qquad A = 84 \text{ m}^2

The calculator below opens on this triangle. Switch shape to try a circle, trapezium or regular polygon.

LABORATORY

Interactive Area & Perimeter Calculator

LABORATORY

Area & Perimeter Engine

ShapeUnit

Side aSide bSide c

Common Questions

What is the difference between area and perimeter?

Perimeter is the length all the way round the edge (in cm, m…). Area is the space inside (in cm², m²…). A 4 × 2 rectangle has perimeter 12 cm and area 8 cm².

What is the formula for the area of a circle?

A = \pi r^2

, and the circumference is

C = 2\pi r

. Many school books use

\pi = \tfrac{22}{7}

: with r = 7 cm the area is 154 cm².

When do you use Heron's formula?

When you know all three sides of a triangle but not its height. Find the half-perimeter

s = \frac{a + b + c}{2}

, then

A = \sqrt{s(s-a)(s-b)(s-c)}

How do you find the area of a trapezium?

Add the two parallel sides, halve, and multiply by the height between them:

A = \tfrac{1}{2}(a + b)h

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Unit 9•39 of 47

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f(x)∫dxπ

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Evaluating numerical algorithm

UNIT 9 • GEOMETRY & TRIGONOMETRY

Area & Perimeter
Every 2D Shape, Step by Step

Area is the space inside a flat shape, in square units; perimeter is the distance around it. Each shape has its own formula, from $l \times b$ for a rectangle to $\pi r^2$ for a circle.

THE MOST-USED FORMULAS11 SHAPES

\begin{gathered} \text{Rectangle: } A = lb,\ P = 2(l + b) \\ \text{Circle: } A = \pi r^2,\ C = 2\pi r \\ \text{Triangle: } A = \tfrac{1}{2}bh \end{gathered}

1. Pick the ShapeMatch the figure to its formula. Split odd shapes into rectangles, triangles and circles.

2. Same UnitsPut every length in the same unit (all cm or all m) before you substitute.

3. Square for AreaArea answers are in square units (cm²); perimeter answers are plain lengths (cm).

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A triangular field has sides 13 m, 14 m and 15 m. Find its perimeter and its area.

STEP 01 • PERIMETER AND HALF-PERIMETER

P = a + b + c = 13 + 14 + 15 = 42 \text{ m}

s = \frac{P}{2} = \frac{42}{2} = 21 \text{ m}

STEP 02 • HERON'S FORMULA

We know all three sides but not the height, so use Heron's formula:

A = \sqrt{s(s - a)(s - b)(s - c)}

A = \sqrt{21 \times (21 - 13) \times (21 - 14) \times (21 - 15)}

A = \sqrt{21 \times 8 \times 7 \times 6}

STEP 03 • SIMPLIFY THE ROOT

21 \times 8 \times 7 \times 6 = 7056, \qquad \sqrt{7056} = 84

Check: the height onto the 14 m side is 12 m, and $\tfrac{1}{2} \times 14 \times 12 = 84$ ✓

FINAL CONCLUSION

Perimeter = 42 m, Area = 84 m²

P = 42 \text{ m}, \qquad A = 84 \text{ m}^2

The calculator below opens on this triangle. Switch shape to try a circle, trapezium or regular polygon.

LABORATORY

Interactive Area & Perimeter Calculator

LABORATORY

Area & Perimeter Engine

ShapeUnit

Side aSide bSide c

Common Questions

What is the difference between area and perimeter?

Perimeter is the length all the way round the edge (in cm, m…). Area is the space inside (in cm², m²…). A 4 × 2 rectangle has perimeter 12 cm and area 8 cm².

What is the formula for the area of a circle?

A = \pi r^2

, and the circumference is

C = 2\pi r

. Many school books use

\pi = \tfrac{22}{7}

: with r = 7 cm the area is 154 cm².

When do you use Heron's formula?

When you know all three sides of a triangle but not its height. Find the half-perimeter

s = \frac{a + b + c}{2}

, then

A = \sqrt{s(s-a)(s-b)(s-c)}

How do you find the area of a trapezium?

Add the two parallel sides, halve, and multiply by the height between them:

A = \tfrac{1}{2}(a + b)h

PreviousMatrix Calculator

Unit 9•39 of 47

NextVolume & Surface Area

Area & Perimeter Every 2D Shape, Step by Step

Step-by-Step Worked Example

Perimeter = 42 m, Area = 84 m²

Interactive Area & Perimeter Calculator

Area & Perimeter Engine

Common Questions

NETZ Engine

Area & Perimeter Every 2D Shape, Step by Step

Step-by-Step Worked Example

Perimeter = 42 m, Area = 84 m²

Interactive Area & Perimeter Calculator

Area & Perimeter Engine

Common Questions

Area & Perimeter
Every 2D Shape, Step by Step

Area & Perimeter
Every 2D Shape, Step by Step