077i - Composite Shapes and Sectors

Lesson 77 — Explicit Instruction: Composite Shapes and Sectors

Strand: Measurement | Descriptor: AC9M8M03 | Duration: 45 minutes

Learning Intentions

• To calculate the perimeter and area of composite shapes formed by combining circles, semicircles or sectors with rectilinear shapes.
• To calculate the arc length and area of a sector as a fraction of the full circle.

Success Criteria

I can:

1. Identify the circular and straight components that make up a composite shape.
2. Calculate the perimeter of a composite shape, including curved edges, taking care not to double-count or omit straight edges.
3. Calculate the area of a composite shape by adding or subtracting circular regions from rectilinear regions.
4. Calculate the arc length and area of a sector, using the sector angle as a fraction of .

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. State the formulas for the circumference and area of a full circle.
2. A semicircle is exactly half of a circle. If a full circle has area , what is the area of a semicircle with the same radius?
3. A pizza is cut into equal slices. What fraction of the whole pizza is one slice? What angle, at the centre, does one slice make?
4. Look at a shape made from a rectangle with a semicircle attached to one end. How many “curved” and how many “straight” edges does its outline have?

Teacher note: Question 4 is the hook — students must notice that the straight edge where the semicircle joins the rectangle is internal and is not part of the outline. Return to this explicitly in Activity 1.

Activities

Activity 1 — Explicit Instruction: Composite Shapes with Semicircles (13 min)

I do — area: A running track shape is a rectangle m by m with a semicircle of diameter m attached to each of the two short ends (together forming one full circle).

I do — perimeter: For the same shape, the outline is the two long straight sides plus the two semicircular ends (the two short straight ends of the rectangle are internal — they are not part of the outline, since the semicircles are attached there).

Explicit warning: Always sketch the shape and mark, with a highlighter, exactly which edges form the outline. Internal joining edges are never included in a perimeter calculation.

We do: A door shape is a rectangle cm wide by cm tall, topped with a semicircle of diameter cm. Find the total area and the total outline perimeter, using .

You do: A window shape is a rectangle cm by cm with a semicircle of diameter cm removed from the top edge (a semicircular notch cut out). Find the remaining area, using .

(Answer: rectangle minus semicircle , giving .)

Activity 2 — Explicit Instruction: Sectors (12 min)

A sector is a “slice” of a circle bounded by two radii and an arc. Its area and arc length are the same fraction of the full circle as the sector’s angle is of .

I do: A sector has radius cm and angle . Using :

Sector perimeter includes the two straight radii and the arc:

We do: A sector has radius cm and angle (a quarter circle). Find its area, arc length and perimeter, using .

You do:

1. Sector radius cm, angle — find area and arc length ().
2. Sector radius cm, angle — find area and arc length ().
3. Sector radius m, angle — find the perimeter ().

(Answers: 1. area , arc cm 2. area , arc cm 3. area not required; perimeter m)

Activity 3 — Inquiry Task: Designing a Sports Field Logo (10 min)

Pairs, then whole-class share.

A school wants to paint a logo on the sports field: a rectangle m by m, with a quarter-circle of radius m painted (and cut out from the paint budget) in one corner, plus a semicircle of diameter m attached to one of the short ends.

(a) Sketch the shape clearly, marking which parts are added and which are removed. (b) Find the total painted area, using . (c) Find the total outline perimeter of the shape.

Socratic scaffolding:

Prompt	Purpose
Understand: what is being asked?	The net painted area (rectangle plus semicircle, minus quarter-circle) and the outline length.
Draw a diagram. What do you label?	Rectangle dimensions, the semicircle’s diameter, the quarter-circle’s radius, and which edges are internal versus outline.
Devise a plan for the area	Break the composite region into three known shapes: rectangle semicircle quarter-circle.
Carry out the plan	Rectangle . Semicircle (, ) . Quarter-circle () . Total .
Devise a plan for the perimeter	Identify which straight edges remain, which are replaced by the arc, and which are replaced by the quarter-circle’s arc plus one remaining radius (the other radius sits along an existing edge).
Carry out the plan	This requires care and will differ by exact design choice — accept any perimeter that is correctly justified from the student’s own labelled diagram.
Looking back	Is sensible against the rectangle’s ? Yes — we added a bit more than we removed.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A shape is a rectangle cm by cm with a semicircle of diameter cm attached to one short end. Find the total area, using .
2. Find the area of a sector with radius cm and angle , using .
3. Find the arc length of a sector with radius cm and angle , using .
4. Reasoning. A student calculates the perimeter of a rectangle-plus-semicircle shape by adding all four sides of the rectangle to the semicircle’s arc. Explain their error.

Answers: 1. Rectangle + semicircle = ; 2. ; 3. cm; 4. The short end where the semicircle attaches is an internal edge, not part of the outline, so it must be excluded, not added.

Common Misconceptions

Misconception	How to pre-empt it
Including the internal “join” edge in the perimeter of a composite shape.	Always highlight the true outline on a diagram before calculating; internal joins are never counted.
Using the full circle formula for a sector without applying the fraction.	Insist the fraction is written as an explicit first line of every sector calculation.
Forgetting the straight radii when finding a sector’s perimeter (only calculating the arc).	Distinguish “arc length” (curved edge only) from “sector perimeter” (arc plus two radii) with separate vocabulary every time.
Adding areas when a shape should be subtracted (e.g. a notch or hole cut out).	Always ask: “is this circular region being added to or removed from the rectilinear shape?” before combining.
Using diameter instead of radius inside sector or semicircle area formulas.	Reinforce the same check from Lesson 76 — write "" explicitly before substituting.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A square of side cm has a quarter-circle of radius cm removed from one corner. Find the remaining area, using .

Answer

E2 (AMC Junior style). A sector has an arc length of cm and a radius of cm. Find the sector’s angle, correct to the nearest degree, using .

Answer

E3 (Challenge). Four quarter-circles of radius cm are cut from the four corners of a square. Find the remaining area, using .

Answer

The four quarter-circles together form exactly one full circle of radius cm.

Homework

1. A shape is a rectangle cm by cm with a semicircle of diameter cm attached to one short end. Find (a) the total area (b) the outline perimeter, using .
2. Find the area and arc length of a sector with radius cm and angle , using .
3. Find the perimeter of a sector with radius cm and angle , using .
4. A circular flower bed of radius m has a rectangular path m by m crossing through its centre, entirely contained within the circle. Find the remaining planted area, using .
5. Reasoning. Explain why a sector with angle has exactly half the area of the full circle, but its perimeter is not half the circle’s circumference.
6. Challenge. A stained-glass window is a semicircle of diameter cm sitting on top of a rectangle cm wide and cm tall. A sector-shaped fan (radius cm, angle ) is cut from one bottom corner of the rectangle. Find the total remaining area, using .

Answers: Q1 — (a) rectangle + semicircle , total (b) cm. Q2 — area ; arc cm. Q3 — arc cm; perimeter cm. Q4 — circle area minus path = . Q5 — the sector’s straight edge is a diameter (), which is shorter than the semicircular arc it replaces relative to what “half the circumference” alone would give — the perimeter is arc () plus two radii (), not simply , so it cannot equal half of alone. Q6 — rectangle , semicircle , quarter-circle removed ; total .