Free Geometry Worksheets for homeschoolers, students, parents and ... - Free Printable

You're correct to notice that the worksheet you uploaded already contains filled-in answers for the area, perimeter, and type of each triangle. It appears to be a completed answer key rather than a blank worksheet.

However, since your request is to “solve the problem and explain the solution,” I’ll do the following:

✔ Verify the correctness of the provided answers.
✔ Explain how to calculate the area and perimeter for each triangle.
✔ Clarify the triangle types based on side lengths or angles.

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## 🔢 Key Formulas

- Perimeter (P) = sum of all three sides: `a + b + c`
- Area (A) of any triangle = `(base × height) / 2` → `A = (b × h)/2`
- Right Triangle: One angle is 90° (marked with a square). Area can also be calculated using legs as base and height: `A = (leg1 × leg2)/2`
- Isosceles Triangle: Two sides equal
- Equilateral Triangle: All three sides equal → Perimeter = `3 × side`; Area = `(√3/4) × side²`

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# 📐 Let’s go triangle by triangle

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## 🔹 Row 1, Column 1

Given:
a = 5.54 mm, b = 6.34 mm, c = 6.9 mm, h = 4.8 mm (height to side c)

Perimeter:
= a + b + c = 5.54 + 6.34 + 6.9 = 18.78 mm ✔

Area:
= (base × height)/2 = (6.9 × 4.8)/2 = 33.12 / 2 = 16.56 mm² ✔

Type: All sides different → Scalene Triangle
*(Note: Worksheet says “Common Triangle” — this is not a standard geometric term. “Scalene” is correct.)*

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## 🔹 Row 1, Column 2

Given:
a = 4.9 cm, b = 4.9 cm, c = 2.8 cm → Isosceles
h = ? (not given, but area is given as 20.7076 cm²)

Wait — we can check if the area matches using the formula.

Since it’s isosceles with two sides = 4.9 cm, and base = 2.8 cm, we can find height using Pythagoras:

Split base in half → 1.4 cm
Then: `h = √(4.9² - 1.4²) = √(24.01 - 1.96) = √22.05 ≈ 4.696 cm`

Area = (base × height)/2 = (2.8 × 4.696)/2 ≈ 13.1488 / 2 ≈ 6.57 cm²

But worksheet says 20.7076 cm² — ✘ This does NOT match.

Alternatively, maybe the height is drawn to one of the equal sides? But that would be unusual.

Let’s reverse-calculate from given area:

Area = 20.7076 = (base × height)/2 → If base = 2.8, then height = (20.7076 × 2)/2.8 ≈ 14.79 cm — which is impossible because the side is only 4.9 cm.

➡️ Conclusion: The area value 20.7076 cm² is incorrect for the given sides. The correct area should be around 6.57 cm².

Perimeter: 4.9 + 4.9 + 2.8 = 12.6 cm — worksheet says 22.5 cm ✘ — so both area and perimeter are wrong here.

*(Possibly a typo in the worksheet — maybe sides were meant to be larger?)*

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## 🔹 Row 1, Column 3

Given: Right triangle
a = 7.6 ft, b = 5.8 ft, c = 9.56 ft (hypotenuse)

Check if Pythagorean theorem holds:
a² + b² = 7.6² + 5.8² = 57.76 + 33.64 = 91.4
c² = 9.56² = 91.3936 ≈ 91.4 → ✔ So it is a right triangle.

Perimeter: 7.6 + 5.8 + 9.56 = 22.96 ft — worksheet says 22.90 ft → Close enough (rounding difference) ✔

Area: For right triangle, use legs as base and height:
A = (7.6 × 5.8)/2 = 44.08 / 2 = 22.04 ft² ✔

Type: Right Triangle ✔

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## 🔹 Row 2, Column 1

Given: Equilateral triangle, side = 8.6 inches

Perimeter: 3 × 8.6 = 25.8 inches ✔

Area: Formula for equilateral triangle = `(√3 / 4) × side²`
= (1.73205 / 4) × (8.6)² ≈ 0.4330125 × 73.96 ≈ 32.0256 in² ✔

Type: Equilateral ✔

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## 🔹 Row 2, Column 2

Given: Isosceles triangle, a = 3.2 yd, b = 3.1 yd, c = 7.79 yd — wait, that doesn’t make sense!

If it’s isosceles, two sides must be equal. Here, none are equal: 3.2, 3.1, 7.79 → Scalene.

Also, check triangle inequality:
3.2 + 3.1 = 6.3 < 7.79 → ✘ Not a valid triangle! You cannot form a triangle with these sides.

Worksheet says area = 31.998 yd² — impossible.

➡️ This triangle is invalid. The given side lengths violate the triangle inequality theorem.

*(Likely a typo — perhaps c was meant to be 3.1 or 3.2?)*

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## 🔹 Row 2, Column 3

Given: Equilateral triangle, side = 6.5 ft

Perimeter: 3 × 6.5 = 19.5 ft ✔

Area: (√3 / 4) × (6.5)² = (1.73205 / 4) × 42.25 ≈ 0.4330125 × 42.25 ≈ 18.2875 ft² ✔

Type: Equilateral ✔

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## 🔹 Row 3, Column 1

Given: Equilateral triangle, side = 6.6 yd

Perimeter: 3 × 6.6 = 19.8 yd ✔

Area: (√3 / 4) × (6.6)² = 0.4330125 × 43.56 ≈ 18.882 yd² ✔

Type: Equilateral ✔

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## 🔹 Row 3, Column 2

Given: Isosceles triangle, a = 3 m, b = 6.7 m, c = 3 m → So sides are 3, 3, 6.7

Check triangle inequality: 3 + 3 = 6 < 6.7 → ✘ Invalid triangle!

Worksheet says area = 9.7949 m² — impossible.

➡️ Again, invalid triangle — sides don't satisfy triangle inequality.

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## 🔹 Row 3, Column 3

Given: Equilateral triangle, side = 4.3 inches

Perimeter: 3 × 4.3 = 12.9 inches ✔

Area: (√3 / 4) × (4.3)² = 0.4330125 × 18.49 ≈ 7.998 in² ✔

Type: Equilateral ✔

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# 🧾 Summary of Corrections

| Triangle | Issue |
|----------|-------|
| Row 1, Col 2 | Incorrect area and perimeter. Sides suggest ~6.57 cm² and 12.6 cm. |
| Row 2, Col 2 | Invalid triangle (3.2, 3.1, 7.79 violates triangle inequality). |
| Row 3, Col 2 | Invalid triangle (3, 3, 6.7 violates triangle inequality). |

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# ✔ Final Answer (Corrected)

The worksheet contains several errors, especially in triangles that are geometrically impossible or have miscalculated areas/perimeters.

✔ Correctly Solved Triangles:

- Row 1, Col 1: Scalene, P=18.78 mm, A=16.56 mm²
- Row 1, Col 3: Right, P=22.96 ft, A=22.04 ft²
- Row 2, Col 1: Equilateral, P=25.8 in, A=32.0256 in²
- Row 2, Col 3: Equilateral, P=19.5 ft, A=18.2875 ft²
- Row 3, Col 1: Equilateral, P=19.8 yd, A=18.882 yd²
- Row 3, Col 3: Equilateral, P=12.9 in, A=7.998 in²

✘ Incorrect/Invalid Triangles:

- Row 1, Col 2: Miscalculations — likely typo in side lengths.
- Row 2, Col 2 & Row 3, Col 2: Impossible triangles — sides violate triangle inequality.

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📌 Recommendation: Use this worksheet as a learning tool to spot errors and understand why some triangles can’t exist. Always verify triangle inequality before calculating area/perimeter!

Let me know if you’d like to see corrected versions of the invalid problems!