Areas and Perimeters of Polygons worksheet - Free Printable

Let's solve each polygon one by one, calculating the Area, Perimeter, and identifying the Type of polygon.

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1) Rectangle

- $ a = 88 \text{ cm},\ b = 59 \text{ cm} $
- Type: Rectangle
- Area = $ a \times b = 88 \times 59 = 5192 \text{ cm}^2 $
- Perimeter = $ 2(a + b) = 2(88 + 59) = 2(147) = 294 \text{ cm} $

✔ Answer:
- Area: 5192 cm²
- Perimeter: 294 cm
- Type: Rectangle

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2) Rectangle

- $ a = 72 \text{ inches},\ b = 46 \text{ inches} $
- Type: Rectangle
- Area = $ 72 \times 46 = 3312 \text{ in}^2 $
- Perimeter = $ 2(72 + 46) = 2(118) = 236 \text{ inches} $

✔ Answer:
- Area: 3312 in²
- Perimeter: 236 inches
- Type: Rectangle

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3) Triangle (Isosceles)

- $ a = 40 \text{ ft},\ b = 72 \text{ ft} $
- Two sides equal → Isosceles triangle
- Type: Isosceles Triangle
- Perimeter = $ a + b + b = 40 + 72 + 72 = 184 \text{ ft} $
- Area: Use Heron’s formula or base-height. But we don’t have height. We can use Heron’s formula:
- $ s = \frac{a + b + b}{2} = \frac{40 + 72 + 72}{2} = \frac{184}{2} = 92 $
- Area = $ \sqrt{s(s-a)(s-b)(s-b)} = \sqrt{92(92-40)(92-72)(92-72)} $
- $ = \sqrt{92 \times 52 \times 20 \times 20} $
- $ = \sqrt{92 \times 52 \times 400} $
- First: $ 92 \times 52 = 4784 $
- Then: $ 4784 \times 400 = 1,913,600 $
- $ \sqrt{1,913,600} \approx 1383.3 \text{ ft}^2 $

✔ Answer:
- Area: ≈ 1383.3 ft²
- Perimeter: 184 ft
- Type: Isosceles Triangle

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4) Triangle (Isosceles)

- $ a = 38 \text{ yds},\ b = 75 \text{ yds} $
- Two sides equal → Isosceles triangle
- Type: Isosceles Triangle
- Perimeter = $ 38 + 75 + 75 = 188 \text{ yds} $
- Area: Use Heron’s formula
- $ s = \frac{38 + 75 + 75}{2} = \frac{188}{2} = 94 $
- Area = $ \sqrt{94(94-38)(94-75)(94-75)} = \sqrt{94 \times 56 \times 19 \times 19} $
- $ = \sqrt{94 \times 56 \times 361} $
- $ 94 \times 56 = 5264 $
- $ 5264 \times 361 \approx 1,900,000 $ (approx)
- Let's compute more accurately:
- $ 5264 \times 361 = 5264 \times (300 + 60 + 1) = 5264×300=1,579,200; 5264×60=315,840; 5264×1=5264 $
- Total = $ 1,579,200 + 315,840 + 5,264 = 1,900,304 $
- $ \sqrt{1,900,304} \approx 1378.5 \text{ yd}^2 $

✔ Answer:
- Area: ≈ 1378.5 yd²
- Perimeter: 188 yds
- Type: Isosceles Triangle

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5) Square

- $ s = 63 \text{ mm} $
- Type: Square
- Area = $ s^2 = 63^2 = 3969 \text{ mm}^2 $
- Perimeter = $ 4s = 4 \times 63 = 252 \text{ mm} $

✔ Answer:
- Area: 3969 mm²
- Perimeter: 252 mm
- Type: Square

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6) Square

- $ s = 65 \text{ yds} $
- Type: Square
- Area = $ 65^2 = 4225 \text{ yd}^2 $
- Perimeter = $ 4 \times 65 = 260 \text{ yds} $

✔ Answer:
- Area: 4225 yd²
- Perimeter: 260 yds
- Type: Square

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7) Parallelogram

- $ a = 54.96 \text{ ft},\ c = 89 \text{ ft},\ h = 52 \text{ ft} $
- Opposite sides equal → $ a = 54.96,\ c = 89 $
- Type: Parallelogram
- Area = $ \text{base} \times \text{height} = c \times h = 89 \times 52 = 4628 \text{ ft}^2 $
- Perimeter = $ 2(a + c) = 2(54.96 + 89) = 2(143.96) = 287.92 \text{ ft} $

✔ Answer:
- Area: 4628 ft²
- Perimeter: 287.92 ft
- Type: Parallelogram

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8) Trapezoid

- $ a1 = 100 \text{ cm},\ a2 = 43 \text{ cm},\ b1 = 68.28 \text{ cm},\ b2 = 58.8 \text{ cm},\ h = 56 \text{ cm} $
- Type: Trapezoid (two parallel sides: a1 and a2)
- Area = $ \frac{1}{2}(a1 + a2) \times h = \frac{1}{2}(100 + 43) \times 56 = \frac{1}{2}(143) \times 56 = 71.5 \times 56 = 4004 \text{ cm}^2 $
- Perimeter = sum of all sides = $ a1 + a2 + b1 + b2 = 100 + 43 + 68.28 + 58.8 = 270.08 \text{ cm} $

✔ Answer:
- Area: 4004 cm²
- Perimeter: 270.08 cm
- Type: Trapezoid

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9) Parallelogram

- $ a = 61 \text{ inches},\ h = 55.28 \text{ inches} $
- Since it's a parallelogram with two adjacent sides labeled as 'a', both pairs are equal.
- So sides: $ a = 61 $, and opposite side also 61, but the other pair? Wait — only one side is given.
- But since it's a parallelogram and only one side and height are given, we assume the base is 'a' = 61 inches, and height is 55.28 inches.
- But to find perimeter, we need both adjacent sides. Only one is given.

Wait! The diagram shows only one side labeled 'a', and height 'h'. But in a parallelogram, you need two adjacent sides for perimeter.

But here, only one side is given. However, from the diagram, if it's a parallelogram and only one side is labeled, maybe the other side is missing?

But wait — perhaps the base is 'a' = 61 inches, and the height is 55.28 inches, but we don't know the slant side.

But in this case, we cannot compute perimeter without knowing the other side.

But let's re-express:

- In a parallelogram:
- Area = base × height = $ 61 \times 55.28 $
- Perimeter = $ 2(\text{side}_1 + \text{side}_2) $, but only one side is given.

Wait — the label says: "a = 61 inches", and "h = 55.28 inches". But no other side.

However, in many problems like this, if only one side is labeled and it's a parallelogram, and height is given, then the area can be found, but perimeter cannot unless we assume something.

But look at the diagram: It has two sides labeled 'a', so likely both sides are 61 inches? But that would make it a rhombus?

Wait — no, in a parallelogram, opposite sides are equal, but adjacent sides can differ.

But here, only one side is labeled, so unless the figure implies symmetry, we can't assume.

But the diagram shows one side labeled 'a', and another side labeled 'a'? Wait — actually, looking at the image: the bottom side is labeled 'a', and the top side is not labeled, but the left side is labeled 'a'? No — wait.

Looking again: the diagram shows:
```
a
/ \
/____\
a
```

Wait — no, it shows:
- Bottom: 'a'
- Top: not labeled
- Left: 'a'
- Right: not labeled

Wait — actually, the labels are:
- Base: 'a'
- Height: 'h'
- And the left side is labeled 'a'? Or just one 'a'?

But the problem says: "a = 61 inches, h = 55.28 inches"

And the diagram shows:
- One side labeled 'a'
- Height labeled 'h'

So likely, only one side is given, and the other side is unknown.

But that makes perimeter impossible.

Wait — but in some diagrams, if it's a rhombus, all sides are equal. But here, it's labeled as a general parallelogram.

But notice: only one value is given for side 'a', and height.

But to compute perimeter, we need the length of the non-base side.

We can't get that without more info.

But wait — perhaps the slant side is not needed? But we need it.

Alternatively, maybe the figure is implying that both pairs of sides are 'a', i.e., rhombus?

But the problem says "parallelogram", not rhombus.

But if all sides are equal, then it's a rhombus.

But we are not told that.

Wait — let's see: the diagram has only one side labeled 'a', and the base is labeled 'a' — but the vertical height is labeled 'h'.

So the base is 'a' = 61 inches, and height is 55.28 inches.

But the other side (the slant side) is not given.

So we cannot compute perimeter unless we know the other side.

But maybe the diagram implies that the other side is also 'a', meaning rhombus?

But that's not stated.

Alternatively, perhaps the label 'a' appears twice — once on the base, once on the side?

Wait — looking at the image: it says:
```
a
/ \
/___\
a
```
No — actually, the diagram shows:
- Base: 'a'
- Left side: 'a'
- So both adjacent sides are labeled 'a' → so all four sides are equal → Rhombus

Yes! Because opposite sides are equal, and both adjacent sides are labeled 'a', so it's a rhombus.

Therefore:
- All sides = 61 inches
- Height = 55.28 inches

Now:
- Area = base × height = $ 61 \times 55.28 = ? $
- $ 61 \times 55.28 = 61 \times (55 + 0.28) = 61×55 = 3355; 61×0.28 = 17.08; total = 3372.08 \text{ in}^2 $
- Perimeter = $ 4 \times 61 = 244 \text{ inches} $
- Type: Rhombus (special parallelogram)

✔ Answer:
- Area: 3372.08 in²
- Perimeter: 244 inches
- Type: Rhombus

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✔ Final Answers Summary:

| # | Shape | Area | Perimeter | Type |
|---|-------------|------------------------|------------------------|--------------------|
| 1 | Rectangle | 5192 cm² | 294 cm | Rectangle |
| 2 | Rectangle | 3312 in² | 236 inches | Rectangle |
| 3 | Triangle | ≈1383.3 ft² | 184 ft | Isosceles Triangle |
| 4 | Triangle | ≈1378.5 yd² | 188 yds | Isosceles Triangle |
| 5 | Square | 3969 mm² | 252 mm | Square |
| 6 | Square | 4225 yd² | 260 yds | Square |
| 7 | Parallelogram | 4628 ft² | 287.92 ft | Parallelogram |
| 8 | Trapezoid | 4004 cm² | 270.08 cm | Trapezoid |
| 9 | Rhombus | 3372.08 in² | 244 inches | Rhombus |

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