Area and Perimeter

Perimeter is the walk around a shape. Area is the floor space inside it. A triangle makes the second idea trickier because its height is a straight perpendicular distance, not whichever sloping side looks tallest.

Parents: let your child point to the requested measure and the actual height before opening each solution.

By the end you will be able to find perimeter and area for squares and rectangles, identify a triangle base and its matching perpendicular height, find triangle area, and split or subtract composite figures made from rectangles, squares and triangles.

Perimeter and area of rectangles

The perimeter is the sum of every outside side. A rectangle uses two lengths and two widths, so its perimeter is 2 × (length + width). The area is length × width.

For the rectangle shown, the perimeter is a length and the area is a square measure.

Keep the measures separate

$$\text{Perimeter}=2\times(8+3)=22\,\text{cm}$$

$$\text{Area}=8\times3=24\,\text{cm}^2$$

Squares and a useful manipulative

A square is a special rectangle: it has four equal sides and four right angles. Its perimeter is 4 × side and its area is side × side. A square is therefore also a rectangle, so the rectangle rules still make sense.

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Change a rectangle, watch both measures

🤔 Predict first: If the length doubles while the breadth stays fixed, which usually grows faster?

AreaPerimeterBoth are always the same

Square measures

$$\text{Perimeter}=4\times5=20\,\text{cm}$$

$$\text{Area}=5\times5=25\,\text{cm}^2$$

Choose a base and the actual perpendicular height

For a triangle, choose one side as the base. The matching perpendicular height is the distance from the opposite vertex to that base line. It must meet the base at a right angle. A sloping side can be longer, but it is not automatically the height.

Show the worked base and height

Name the matching pair

$$\text{Chosen base}=8\,\text{cm}$$

$$\text{Perpendicular height}=5\,\text{cm}$$

$$\text{The sloping sides are not the height for base AB}$$

Area of a triangle

Every triangle is half of a rectangle with the same base and perpendicular height. That gives the rule $\frac{1}{2}$ × base × perpendicular height.

For the shown 8 cm base and 5 cm actual height, the area is 20 square cm.

Triangle area

$$\text{Area}=\frac{1}{2}\times8\,\text{cm}\times5\,\text{cm}$$

$$=20\,\text{cm}^2$$

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Slide an apex, keep the height

🤔 Predict first: If the apex slides sideways along the same parallel line, what happens to its area?

It stays the sameIt gets biggerIt gets smaller

Composite figures: joined rectangles, squares and triangles

A composite figure can be split into familiar pieces. Find each piece using its own correct rule, then add the areas. A joined shape has no overlap between the pieces.

Show the joined-shape solution

Add the two areas

$$\text{Rectangle}=8\times4=32\,\text{cm}^2$$

$$\text{Triangle}=\frac{1}{2}\times8\times3=12\,\text{cm}^2$$

$$\text{Total}=32+12=44\,\text{cm}^2$$

Composite figures: subtract a cut-out

For a cut-out, find the large outer area first. Then subtract the removed shape. The triangular height still has to be perpendicular to its chosen base.

Show the cut-out solution

Subtract the triangular cut-out

$$\text{Outer rectangle}=10\times8=80\,\text{cm}^2$$

$$\text{Notch}=\frac{1}{2}\times4\times3=6\,\text{cm}^2$$

$$\text{Remaining area}=80-6=74\,\text{cm}^2$$

Square pieces count as area too

Do not leave a square component out just because the outside outline looks like a rectangle. A 3 cm by 3 cm square is 9 square cm. The same add-or-subtract method works for square pieces.

Show the square cut-out solution

Subtract the square

$$\text{Outer rectangle}=10\times8=80\,\text{cm}^2$$

$$\text{Square cut-out}=3\times3=9\,\text{cm}^2$$

$$\text{Remaining area}=80-9=71\,\text{cm}^2$$

Three graduated practices

Try each problem with all the givens visible. Open the solution only after deciding which pieces and which units you need.

Practice 1 — one triangle. A triangle has base 12 cm and perpendicular height 4 cm. Find its area.

Reveal solution: Practice 1

Triangle area

$$\text{Area}=\frac{1}{2}\times12\times4=24\,\text{cm}^2$$

The area is 24 square cm.

Practice 2 — joined rectangle and square. A 6 cm by 4 cm rectangle is joined to a 4 cm by 4 cm square along one complete side. Find the total area.

Reveal solution: Practice 2

Add rectangle and square

$$\text{Rectangle}=6\times4=24\,\text{cm}^2$$

$$\text{Square}=4\times4=16\,\text{cm}^2$$

$$\text{Total}=24+16=40\,\text{cm}^2$$

The total area is 40 square cm.

Practice 3 — a triangular cut-out. A 9 cm by 7 cm rectangle has a triangular notch removed. The notch has base 3 cm and perpendicular height 2 cm. Find the remaining area.

Reveal solution: Practice 3

Rectangle minus triangle

$$\text{Outer rectangle}=9\times7=63\,\text{cm}^2$$

$$\text{Notch}=\frac{1}{2}\times3\times2=3\,\text{cm}^2$$

$$\text{Remaining area}=63-3=60\,\text{cm}^2$$

The remaining area is 60 square cm.

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

1. What is the perimeter of a rectangle 10 cm long and 4 cm wide?

   14 cm28 cm40 cm

2. What is the area of a square with side 6 cm?

   12 square cm24 square cm36 square cm

3. A triangle has base 9 cm and perpendicular height 4 cm. What is its area?

   18 square cm36 square cm13 square cm

4. A triangle uses an 8 cm base. A dashed segment meets that base at a right angle and measures 5 cm. Which length is its height?

   5 cm8 cmThe sloping side

5. An 8 cm by 4 cm rectangle is joined to a triangle with base 8 cm and perpendicular height 3 cm. What is the total area?

   44 square cm52 square cm32 square cm

6. A 10 cm by 8 cm rectangle has a triangular cut-out with base 4 cm and perpendicular height 3 cm. What area remains?

   74 square cm86 square cm71 square cm

7. A 10 cm by 8 cm rectangle has a 3 cm by 3 cm square cut out. What area remains?

   71 square cm77 square cm89 square cm

8. A 6 cm by 4 cm rectangle is joined to a 4 cm by 4 cm square. What is their total area?

   40 square cm32 square cm24 square cm

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