E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Perimeter and Area of Composite Shapes

E2.3 A1About 50 minutes

Learning goals

I can break a composite shape into rectangles, triangles, circles and parts of circles.
I can find the perimeter of a composite shape, including curved parts and slanted sides.
I can find the area of a composite shape by adding or subtracting areas.
I can choose the right formulas and use the Pythagorean theorem when a side length is missing.

Words to know

composite shape
A shape made of two or more simple shapes, like a rectangle with a semicircle on top.

perimeter
The distance around the outside of a shape. Inside lines where pieces join don't count.

circumference
The perimeter of a circle: C = π × d = 2 × π × r.

semicircle
Half of a circle. Its curved edge is half the circumference.

area
The amount of surface a shape covers, in square units.

Remember from earlier grades

You already know these formulas (use π ≈ 3.14, as in Grade 7):

rectangle: A = l × w; parallelogram: A = b × h; triangle: A = b × h ÷ 2; trapezoid: A = (a + b) × h ÷ 2
circle: C = π × d = 2 × π × r and A = π × r × r

In Grade 6 you found areas of composite polygons. In Grade 8 you'll mix straight and curved pieces, find perimeters as well as areas, and use the Pythagorean theorem to find missing sides.

Let's start with a window

A family is replacing an arched window: a rectangle with a half circle on top. They need the area to order the glass and the perimeter to order the trim around the edge. Neither is a single simple shape, but each is easy once you break the window into pieces.

A large window shaped like a rectangle with a half circle on top, with white trim on beige siding. — This window is a composite shape: a rectangle plus a semicircle. Find the area of each piece and add them; for the trim, add the three straight sides to half the circle's circumference. AI-generated image

Two composite shapes, each drawn to scale. Left: a window made of an 80 cm by 100 cm rectangle with a semicircle (diameter 80 cm) on top. Right: a running track field made of a 100 m by 60 m rectangle with a semicircle at each end. The dashed lines show where the pieces join; they are not part of the perimeter.

Finding the area

Break the shape into simple pieces (or think of it as a big shape with a piece cut out).
Find each piece's area.
Add the pieces, or subtract the cut-out part.

Finding the perimeter

Trace around the outside only. Skip the dashed lines where pieces join.
Add the straight sides and the curved parts. A semicircle's curved edge is half of π × d. A quarter circle's is a quarter.
If a slanted side isn't labelled, look for a right triangle and use the Pythagorean theorem.

Worked example

Example 1: The window

Find the area of glass and the length of trim for the window: a rectangle 80 cm wide and 100 cm tall with a semicircle on top.

Area of the rectangle: 80 × 100 = 8 000 cm².
The semicircle has diameter 80 cm, so radius 40 cm. Area: 3.14 × 40 × 40 ÷ 2 = 5 024 ÷ 2 = 2 512 cm².
Total area: 8 000 + 2 512 = 10 512 cm² (about 1.05 m²).
Perimeter: the two sides and the bottom are 100 + 100 + 80 = 280 cm. The curved top is half the circumference: 3.14 × 80 ÷ 2 = 125.6 cm.
Trim: 280 + 125.6 = 405.6 cm.

About 10 512 cm² of glass and 405.6 cm (about 4.1 m) of trim.

Worked example

Example 2: The track field

Find the perimeter and the area of the track field: a 100 m by 60 m rectangle with a semicircle on each 60 m end.

The two semicircles together make one whole circle with diameter 60 m (radius 30 m).
Perimeter: two straights 2 × 100 = 200 m, plus one full circumference 3.14 × 60 = 188.4 m. Total 388.4 m.
Area: rectangle 100 × 60 = 6 000 m², plus the circle 3.14 × 30 × 30 = 2 826 m². Total 8 826 m².

Perimeter about 388.4 m; area about 8 826 m².

Worked example

Example 3: Area with a piece cut out

A circle with a 10 cm diameter is cut out of a 10 cm by 10 cm square of felt. What area of felt is left?

Square: 10 × 10 = 100 cm².
Circle: radius 5 cm. 3.14 × 5 × 5 = 78.5 cm².
Left over: 100 − 78.5 = 21.5 cm².

21.5 cm² of felt is left (in the four corners).

A garden plot for Example 4, drawn to scale: a 10 m by 8 m rectangle with a right triangle (legs 8 m and 6 m) on the right. The slanted side is unknown.

Worked example

Example 4: A missing slanted side

How much fencing does the garden plot need, and what is its area?

The slanted side is the hypotenuse of the right triangle with legs 8 m and 6 m: √(8² + 6²) = √(64 + 36) = √100 = 10 m.
Perimeter: left 8 + top 10 + slant 10 + bottom 16 = 44 m.
Area: rectangle 10 × 8 = 80 m², plus triangle 6 × 8 ÷ 2 = 24 m². Total 104 m².

44 m of fencing; area 104 m².

Think like a mathematician

Area is in square units (cm², m²); perimeter is in plain units (cm, m).
Inside joining lines are never part of the perimeter.
Radius or diameter? Areas need the radius. Halve the diameter first.
Estimate with π ≈ 3 to check: the window's semicircle is about 3 × 40 × 40 ÷ 2 = 2 400 cm², close to 2 512 ✓

Hands-on activity

Measure a composite shape at home

You need: a measuring tape or ruler, paper, a pencil and a calculator.

Find a composite shape at home: a table with rounded ends, an arched window or doorway, a kitchen counter shaped like an L, a rug, or a plate on a placemat.
Sketch it and break it into simple pieces. Mark the dashed joining lines.
Measure the lengths you need. For a curved end, measure the width across (the diameter).
Calculate the perimeter and the area.
Check the perimeter by laying a string (or the measuring tape) around the edge.

What to look for: a correct sketch with pieces labelled, only outside edges in the perimeter, and the right formula for each piece. The string check should be close to the calculation.

Funny but true

A running track is a composite shape too: two straights and two semicircles. Runners in the outer lanes run around bigger semicircles, so in races that go around a curve they start farther ahead (a staggered start). Otherwise the outside lane would be running extra metres for free.

Your turn

A semicircle has a diameter of 10 cm. Find the length of its curved edge, and the perimeter of the whole semicircle shape.

Find the area of a semicircle with a radius of 6 cm.

A 6 cm by 4 cm rectangle has a semicircle attached outside on one 4 cm side. Find the area and perimeter.

An L-shape is an 8 m by 6 m rectangle with a 3 m by 2 m rectangle cut from one corner. Find its area and perimeter.

A quarter circle with radius 20 cm is cut from a 20 cm by 20 cm square, centred at one corner. What area is left?

A circle with radius 3 cm is cut out of a 10 cm by 8 cm rectangle. What area is left?

More practice: warm-up

Find the circumference of a circle with a diameter of 10 cm.

Find the area of a circle with a radius of 10 cm.

Find the area of a 7 m by 5 m rectangle.

How long is the curved edge of a semicircle with a diameter of 20 cm?

Find the area of a triangle with a base of 6 cm and a height of 4 cm.

More practice: on your own

A track has straights of 80 m and semicircular ends with a diameter of 50 m. Find its perimeter.

A window is a 60 cm wide, 90 cm tall rectangle with a semicircle on top. Find the area of glass.

A metal washer is a circle with radius 5 mm with a hole of radius 3 mm in the middle. Find the area of metal.

A lot is a 12 m by 5 m rectangle with a right triangle attached to its 5 m end. The triangle's legs are 5 m (the shared side) and 12 m (continuing the bottom edge). Find the perimeter and area of the lot.

A round pizza with a 30 cm diameter sits in a square box with 30 cm sides. How much of the box bottom is not covered?

Challenge

A circular flower bed with a radius of 2 m sits in the middle of a 10 m by 8 m lawn. Sod costs $6 per m². How much will it cost to sod the lawn around the flower bed?

Find the perimeter and area of a quarter circle with a radius of 10 cm.

A square fits exactly inside a circle with a radius of 5 cm (its corners touch the circle). Find the area between the square and the circle.

Check your understanding

Quiz

What is the area of a semicircle with a radius of 4 cm?

What is the perimeter of a semicircle shape with a diameter of 10 cm (curved edge plus straight edge)?

A 10 cm by 10 cm square has a circle with a 10 cm diameter cut out of it. What area is left?

A field is a 100 m by 60 m rectangle with a semicircle on each 60 m end. What is its perimeter?

A right triangle in a garden plan has legs of 6 m and 8 m. How long is the slanted fence along the hypotenuse?

Which line is not part of a composite shape's perimeter?

For parents and teachers

Big idea: The Ontario curriculum (E2.3) asks Grade 8 students to solve problems involving the perimeter, circumference and area of composite two-dimensional shapes, using appropriate formulas. This lesson pulls together formulas from earlier grades and the Pythagorean theorem from this unit.

Common difficulties:

Including inside joining lines in the perimeter. Have your child trace the outside with a finger.
Using the diameter in the area formula. Areas need the radius.
Forgetting to halve (or quarter) for part of a circle.
Mixing up units: area in cm² or m², perimeter in cm or m.

How to help: Measure a composite shape at home together: a rounded table, an arched doorway, an L-shaped counter. Ask what you'd need to buy to cover it (area) or edge it (perimeter). This lesson also builds A1 skills: making a plan and checking with estimation.