Perimeter & Area in 60 Minutes: Polygons, Circles & Composite Shapes

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9 steps · 2 quizzes · about 60 minutes · open any step, or read it here

1. 1
   Perimeter or area?Start hereToday's goal: I can find the perimeter and area of polygons, circles and shapes built from them. Two different questions about the same shape: - Perimeter is the distance around a figure. It's measured in units: cm, m, ft.InteractiveHands-on · 4 min

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   Today's goal: I can find the perimeter and area of polygons, circles and shapes built from them.

   Two different questions about the same shape:

   • Perimeter is the distance around a figure. It's measured in units: cm, m, ft.
   • Area is the number of square units a figure covers. It's measured in square units: cm², m², ft².

   Try it: pick a size, then press Walk around and Fill it.

   Interactive · Walk around or fill a rectangle — try it in the lesson

   Clue words. A fence, a frame, a border, running around a track → perimeter. Covering, painting, tiling, carpeting → area.

2. Warm-up: perimeter or area?Decide what each job needs.Quiz · 5 questions

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   Decide what each job needs.

   Question 1. Building a fence around a garden

   • Perimeter
   • Area

   Question 2. Buying a cover for a swimming pool

   • Perimeter
   • Area

   Question 3. Painting a wall

   • Perimeter
   • Area

   Question 4. Running around the bases of a baseball field

   • Perimeter
   • Area

   Question 5. A rectangle is 6 cm long and 4 cm wide. What is its area?

   • 20 cm
   • 20 cm²
   • 24 cm
   • 24 cm²
3. 2
   Rectangles and squaresExample. A rectangle is 9.7 m long and 4.5 m wide. - P = 2 × 9.7 + 2 × 4.5 = 19.4 + 9 = 28.4 m - A = 9.7 × 4.5 = 43.65 m² · Working backwards. Write the formula, fill in what you know, then solve for what's missing.Read · 1 min

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   Shape	Perimeter	Area
   Rectangle (length l, width w)	P = 2l + 2w	A = l × w
   Square (side s)	P = 4s	A = s × s = s²

   Example. A rectangle is 9.7 m long and 4.5 m wide.

   • P = 2 × 9.7 + 2 × 4.5 = 19.4 + 9 = 28.4 m
   • A = 9.7 × 4.5 = 43.65 m²

   Working backwards. Write the formula, fill in what you know, then solve for what's missing.

   • Area 45 ft², length 9 ft: 9 × w = 45, so w = 5 ft.
   • Perimeter 60 ft, width 10 ft: 2l + 2 × 10 = 60, so 2l = 40 and l = 20 ft.
   • A square with area 225 cm²: s × s = 225, so s = 15 cm.

   Area always uses square units: 45 ft², not 45 ft.

4. 3
   Triangles, parallelograms and trapezoidsEach of these shapes is a rectangle in disguise. Try it: pick a shape and press Play.InteractiveHands-on · 5 min

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   Each of these shapes is a rectangle in disguise.

   Try it: pick a shape and press Play.

   Interactive · Parallelograms, triangles and trapezoids from rectangles — try it in the lesson

   Shape	Area
   Parallelogram	A = b × h
   Triangle	A = ½ × b × h
   Trapezoid (bases b₁ and b₂)	A = ½ × (b₁ + b₂) × h

   The height is always at a right angle to the base: the dashed line, not the slanted side. Slanted sides only count for the perimeter, where you add up all the sides.

   Example. A parallelogram has a base of 10 cm, slanted sides of 6 cm and a height of 5 cm.

   • A = 10 × 5 = 50 cm² (not 10 × 6)
   • P = 10 + 6 + 10 + 6 = 32 cm
5. 4
   Practice: polygonsWrite the formula first, then fill in the numbers. Type numbers only, without units.Practice · 8 exercises

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   Write the formula first, then fill in the numbers. Type numbers only, without units.

   A rectangle is 12.5 cm long and 6 cm wide. Find its perimeter P and area A.

   

   A triangle has sides of 13 cm, 14 cm and 15 cm. Its height to the 14 cm side is 12 cm. Find its area A and perimeter P.

   

   A parallelogram has a base of 12 cm, slanted sides of 7 cm and a height of 6 cm. Find its area.

   

   A trapezoid has bases of 6 in and 10 in and a height of 7 in. Find its area.

   

   A triangle has an area of 24 in² and a height of 6 in. Find its base b.

   

   A rectangle has an area of 94.5 in² and a length of 7 in. Find its width w and perimeter P.

   

   Each side of a square is (2a + 9) cm long, and its perimeter is 84 cm. Find a.

   

   Challenge: Joanna tiles a floor 9 ft by 4 ft with square tiles 6 inches on a side. How many tiles does she need?

   

6. 5
   Circles: circumference and areaRadius r: from the centre to the edge. Diameter d: all the way across, through the centre. d = 2 × r, and r = d ÷ 2. Circumference C: the perimeter of a circle. Try it: roll the circle out, then switch to Slice it up.InteractiveHands-on · 5 min

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   • Radius r: from the centre to the edge.
   • Diameter d: all the way across, through the centre. d = 2 × r, and r = d ÷ 2.
   • Circumference C: the perimeter of a circle.

   Try it: roll the circle out, then switch to Slice it up.

   Interactive · Roll out and slice up a circle — try it in the lesson

   	Formula
   Circumference	C = π × d = 2 × π × r
   Area	A = π × r² = π × r × r

   π ≈ 3.14: every circle's circumference is a little more than 3 of its diameters.

   Example. A circle has a radius of 7 cm.

   • C = 2 × 3.14 × 7 = 43.96 cm
   • A = 3.14 × 7 × 7 = 3.14 × 49 = 153.86 cm²

   Semicircles are half a circle: find the whole circle's area or circumference, then divide by 2. For the perimeter of a semicircle, also add the straight side (the diameter).

   Watch out: r² means r × r, not r × 2. If you're given the diameter, halve it before you find the area.

7. 6
   Practice: circlesUse 3.14 for π and round only at the end. Type numbers only.Practice · 7 exercises

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   Use 3.14 for π and round only at the end. Type numbers only.

   The diameter of a circle is 25 cm. Find the radius.

   A circle has a radius of 6 cm. Find its circumference C and area A.

   A circle has a diameter of 15 ft. Find its circumference.

   A semicircle has a diameter of 10 in. Find its area.

   The circumference of a circle is 25.12 cm. Find its radius r and diameter d.

   The circumference of a circle is 31.4 cm. Find its area.

   Challenge: The area of a circle is 153.86 cm². Find its circumference.

8. 7
   Composite shapes and shaded regionsA composite shape is two or more shapes put together. There are two moves. Add, when shapes are joined together: 1. Separate the shape into pieces you know. 2. Find the area of each piece. 3. Add them.InteractiveHands-on · 5 min

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   A composite shape is two or more shapes put together. There are two moves.

   Add, when shapes are joined together:

   1. Separate the shape into pieces you know.
   2. Find the area of each piece.
   3. Add them.

   Subtract, for a shaded region (a shape with a piece cut out):

   1. Find the area of the outside shape.
   2. Find the area of the inside shape.
   3. Subtract: outside − inside.

   Try it: pick an example and press Take apart.

   Interactive · Take composite shapes apart — try it in the lesson

   Perimeter of a composite shape: walk around the outside only. Lines inside the shape don't count, and a semicircle edge is half a circumference. For the 10 ft × 4 ft rectangle with a semicircle on one end: P = 10 + 4 + 10 + ½ × 3.14 × 4 = 24 + 6.28 = 30.28 ft.

9. 8
   Practice: composite shapesUse 3.14 for π. Sketch the pieces on paper first. Type numbers only.Practice · 5 exercises

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   Use 3.14 for π. Sketch the pieces on paper first. Type numbers only.

   Find the area A and perimeter P of this L-shape.

   

   A window is a 4 ft × 7 ft rectangle with a semicircle on top. Find its area A and perimeter P.

   

   A circle fits exactly inside a square with 12 cm sides. Find the area of the square that is outside the circle (the four corners).

   A round mirror is 14 in across. A frame 2 in wide goes all the way around it. Find the area of the frame.

   Challenge: A garden is a trapezoid with bases of 6 ft and 15 ft and a height of 8 ft. A round pond with a radius of 4 ft sits inside it. Find the area of the garden without the pond.

10. Exit ticketNo notes this time. Use 3.14 for π.Quiz · 6 questions

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    No notes this time. Use 3.14 for π.

    Question 1. A square garden has an area of 2,500 ft². How many feet of fence go all the way around it?

    • 50 ft
    • 200 ft
    • 625 ft
    • 10,000 ft

    Question 2. A baseball diamond is a square with 70 ft between bases. The team runs around the bases 3 times. How far do they run?

    • 210 ft
    • 280 ft
    • 840 ft
    • 14,700 ft

    Question 3. A triangle has a base of 6 m, a height of 3 m and a slanted side of 5 m. What is its area?

    • 15 m²
    • 30 m²
    • 18 m²
    • 9 m²

    Question 4. A trapezoid has bases of 3 cm and 7 cm and a height of 5 cm. What is its area?

    • 50 cm²
    • 25 cm²
    • 35 cm²
    • 15 cm²

    Question 5. A circle has a diameter of 20 m. What is its area?

    • 314 m²
    • 1,256 m²
    • 62.8 m²
    • 31.4 m²

    Question 6. A circle fits exactly inside a 4 cm × 4 cm square. What is the area of the square outside the circle?

    • 12.56 cm²
    • 6.28 cm²
    • 3.44 cm²
    • 9.72 cm²