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

Let’s go through each triangle one by one. We’ll calculate the area and perimeter, then identify the type of triangle based on its sides or angles.

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Triangle 1 (Top Left)

Given:
a = 5.94 mm, b = 6.34 mm, c = 6.9 mm, h = 4.2 mm

- Area = (base × height) ÷ 2 → use base “c” = 6.9 mm, height = 4.2 mm
→ Area = (6.9 × 4.2) ÷ 2 = 28.98 ÷ 2 = 14.49 sq mm
*(Wait — original answer says 16.56? Let me check again. Maybe they used a different base?)*

Actually, looking at the diagram, side “c” is the base with height drawn to it. But maybe the given height corresponds to another base? Wait — let’s recalculate using the numbers provided in the image’s answer key for verification.

The image shows:
> Area: 16.56 sq mm
> Perimeter: 19.78 mm
> Type: Common Triangle

Let’s verify perimeter first:
Perimeter = a + b + c = 5.94 + 6.34 + 6.9 =
5.94 + 6.34 = 12.28; 12.28 + 6.9 = 19.18 mm — but image says 19.78? That doesn’t match.

Wait — perhaps there’s a typo in my reading? Let me recheck the values from the image:

Image says:
a=5.94 mm
b=6.34 mm
c=6.9 mm
h=4.2 mm

Perimeter should be: 5.94 + 6.34 + 6.9 = let's add carefully:

5.94
+6.34
=12.28
+6.90
=19.18 mm — but image says 19.78. Hmm.

Maybe the value of “c” is 7.5? No, image clearly says 6.9.

Alternatively, maybe the height is not for base c? Or perhaps the area calculation uses a different base?

Let’s try calculating area with base “a” = 5.94 and height 4.2? But height is drawn perpendicular to base “c”, so likely base is c.

(6.9 × 4.2)/2 = 28.98/2 = 14.49 — still not 16.56.

Wait — what if base is “b” = 6.34? Then area = (6.34 × 4.2)/2 = 26.628 / 2 = 13.314 — no.

What if height is 4.8? Not given.

Hold on — perhaps the image has a mistake? Or I’m misreading?

Looking back at the user’s uploaded image — actually, in the first box, the written answer is:

> Area: 16.56 sq mm
> Perimeter: 19.78 mm
> Type: Common Triangle

But according to given sides: 5.94 + 6.34 + 6.9 = 19.18 — not 19.78.

Unless... maybe “c” is 7.5? Let me check the image again visually — ah! In the image, under the first triangle, it says:

“c=6.9 mm” — but maybe it’s 7.5? No, looks like 6.9.

Wait — perhaps the height is 4.8? The label says “h=4.2 mm”.

This is confusing. Let’s move to the next triangle and see if we can find a pattern or correct method.

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Actually — let’s trust the process and do it correctly ourselves, ignoring the orange answers for now, since our job is to solve accurately.

We’ll recalculate all triangles properly.

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## ✔ Correct Step-by-Step Solutions:

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Triangle 1 (Top Left)

Sides: a=5.94 mm, b=6.34 mm, c=6.9 mm
Height to base c: h=4.2 mm

- Perimeter = a + b + c = 5.94 + 6.34 + 6.9 =
5.94 + 6.34 = 12.28
12.28 + 6.9 = 19.18 mm

- Area = (base × height) / 2 = (6.9 × 4.2) / 2 = 28.98 / 2 = 14.49 sq mm

- Type: All sides different → Scalene Triangle (not “Common” — that’s not a standard term; probably meant scalene)

But wait — the image says “Common Triangle” — maybe they mean “scalene”? We’ll use proper terms.

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Triangle 2 (Top Middle)

Sides: a=4.9 cm, b=2.8 cm — wait, only two sides given? Diagram shows an isosceles triangle with two equal sides labeled “a”, and base “b”.

So: a = 4.9 cm (two sides), b = 2.8 cm (base)

Height is not given? But area is calculated as 20.7076 sq cm — how?

If it’s isosceles with legs 4.9 cm and base 2.8 cm, we can find height using Pythagoras.

Half of base = 1.4 cm

Height h = √(4.9² - 1.4²) = √(24.01 - 1.96) = √22.05 ≈ 4.696 cm

Then area = (base × height)/2 = (2.8 × 4.696)/2 ≈ (13.1488)/2 ≈ 6.5744 — not matching 20.7076.

Wait — maybe “a” is the base? Diagram shows “a” on the two equal sides, “b” on base.

Perhaps the height is given implicitly? Or maybe the 20.7076 is wrong?

Alternatively — maybe the triangle has sides a=4.9, b=2.8, and another side also 4.9? So perimeter = 4.9 + 4.9 + 2.8 = 12.6 cm — but image says 22.5 cm? That doesn’t match.

Wait — image says:

> Area: 20.7076 sq cm
> Perimeter: 22.5 cm
> Type: Isosceles Triangle

Perimeter 22.5 with two sides 4.9? 4.9 + 4.9 = 9.8, so third side would be 22.5 - 9.8 = 12.7 cm — but given b=2.8 cm? Contradiction.

I think there might be errors in the image’s provided answers. Let’s proceed with correct math.

Assume for Triangle 2: it’s isosceles with two sides = 4.9 cm, base = 2.8 cm.

Perimeter = 4.9 + 4.9 + 2.8 = 12.6 cm

To find area, need height. As above:

h = √(4.9² - (2.8/2)²) = √(24.01 - 1.96) = √22.05 ≈ 4.696 cm

Area = (2.8 × 4.696)/2 ≈ 6.5744 sq cm

But this doesn't match the image’s 20.7076. So perhaps the labels are switched?

What if “a” is the base? Suppose base a=4.9 cm, and the two equal sides are something else? But only a and b given.

This is messy. Let’s look at Triangle 3.

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Triangle 3 (Top Right) — Right Triangle

Given: a=7.6 ft, b=8.3 ft, c=9.56 ft

In right triangle, legs are a and b, hypotenuse c.

Check: 7.6² + 8.3² = 57.76 + 68.89 = 126.65
c² = 9.56² = 91.3936 — not equal! So not a right triangle? But diagram shows right angle between a and b.

Wait — 7.6² + 8.3² = 57.76 + 68.89 = 126.65
√126.65 ≈ 11.25 — but c=9.56? Doesn’t match.

Perhaps c is not hypotenuse? Diagram shows right angle at bottom left, so sides a and b are legs, c is hypotenuse.

But 7.6² + 8.3² ≠ 9.56² — so inconsistency.

Calculate actual hypotenuse: √(7.6² + 8.3²) = √(57.76 + 68.89) = √126.65 ≈ 11.25 ft — but given c=9.56? Error.

Perhaps the values are swapped? If a=7.6, c=8.3, then b=√(8.3² - 7.6²) = √(68.89 - 57.76) = √11.13 ≈ 3.34 — not 8.3.

This is not working. Let’s assume the given sides are correct and calculate area as (a*b)/2 for right triangle.

Area = (7.6 * 8.3)/2 = 63.08 / 2 = 31.54 sq ft

But image says 22.04 sq ft — not matching.

Perimeter = 7.6 + 8.3 + 9.56 = 25.46 ft — image says 22.96? No.

7.6 + 8.3 = 15.9; +9.56 = 25.46 — yes.

Image says perimeter 22.96 — difference of 2.5 — perhaps typo in c? If c=7.06, then 7.6+8.3+7.06=22.96 — possible.

But we must work with given numbers.

I think the best approach is to ignore the orange answers and calculate correctly based on given dimensions.

Let’s start over with clear calculations for each, using only the given numbers.

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## 🟩 Final Accurate Calculations:

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Triangle 1: Scalene Triangle

Sides: a=5.94 mm, b=6.34 mm, c=6.9 mm
Height to base c: h=4.2 mm

- Perimeter = 5.94 + 6.34 + 6.9 = 19.18 mm
- Area = (base c × height) / 2 = (6.9 × 4.2) / 2 = 28.98 / 2 = 14.49 mm²
- Type: All sides different → Scalene Triangle

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Triangle 2: Isosceles Triangle

Sides: two sides a=4.9 cm, base b=2.8 cm

- Perimeter = 4.9 + 4.9 + 2.8 = 12.6 cm
- To find area, need height. Half-base = 1.4 cm
Height h = √(4.9² - 1.4²) = √(24.01 - 1.96) = √22.05 ≈ 4.696 cm
Area = (2.8 × 4.696) / 2 ≈ 13.1488 / 2 = 6.5744 cm² (≈6.57 cm²)
- Type: Two sides equal → Isosceles Triangle

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Triangle 3: Right Triangle

Legs: a=7.6 ft, b=8.3 ft, Hypotenuse c=9.56 ft — but check if right triangle:

7.6² + 8.3² = 57.76 + 68.89 = 126.65
9.56² = 91.3936 — not equal, so NOT a right triangle with these sides. But diagram shows right angle, so perhaps c is not hypotenuse? Or values are approximate.

Assume it is right-angled at the corner between a and b, so area = (a*b)/2 = (7.6*8.3)/2 = 63.08/2 = 31.54 ft²

Perimeter = 7.6 + 8.3 + 9.56 = 25.46 ft

Type: Has a right angle → Right Triangle (even if sides don't perfectly satisfy Pythagoras, we go by diagram)

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Triangle 4 (Middle Left): Equilateral Triangle

Side a=8.6 inches (all sides equal)

- Perimeter = 3 × 8.6 = 25.8 inches
- Area of equilateral triangle = (√3/4) × side² = (1.732/4) × 73.96 ≈ 0.433 × 73.96 ≈ 32.02 sq inches (matches image's 32.0256)
- Type: Equilateral Triangle

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Triangle 5 (Middle Middle): Isosceles Triangle

Sides: a=3.2 yds (two sides), b=3.1 yds (base), height h=7.79 yds? That can't be — height can't be larger than sides in isosceles triangle with base 3.1 and legs 3.2.

If legs are 3.2, base 3.1, then height should be less than 3.2.

h = √(3.2² - (3.1/2)²) = √(10.24 - 2.4025) = √7.8375 ≈ 2.8 yds — but given h=7.79? Impossible.

Perhaps "a" is the base? Diagram shows "a" on the two equal sides.

Maybe the height is for a different configuration. Given the confusion, let's use the formula with given height.

If base b=3.1 yds, height h=7.79 yds, then area = (3.1 * 7.79)/2 = 24.149/2 = 12.0745 sq yds — but image says 31.999? No.

Perhaps the sides are a=3.2, and the height is to the base, but base is not given? Only a and b given.

This is problematic. Let's assume the triangle has sides a=3.2, a=3.2, b=3.1, and height is calculated as above ~2.8 yds.

Area = (3.1 * 2.8)/2 = 8.68/2 = 4.34 sq yds — not matching.

Perhaps "h=7.79" is a typo, and it's 2.79? Still not.

Another possibility: the triangle is very tall, but with base b=3.1, and legs a=3.2, height can't be 7.79.

I think there's an error in the problem setup. For accuracy, let's calculate with given numbers as per diagram intention.

Suppose for Triangle 5: it's isosceles with base b=3.1 yds, and height h=7.79 yds — even though geometrically impossible with legs 3.2, we'll use it for area calculation.

Area = (base * height)/2 = (3.1 * 7.79)/2 = 24.149/2 = 12.0745 sq yds

Perimeter = 3.2 + 3.2 + 3.1 = 9.5 yds — but image says 18.6? No.

3.2+3.2+3.1=9.5, not 18.6.

Unless "a=3.2" is not the leg, but something else.

Perhaps "a" is the base, and the two equal sides are not given? But diagram shows "a" on the two sides.

I give up on this one for now — let's do the ones that are clear.

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Triangle 6 (Middle Right): Equilateral Triangle

Side a=6.5 ft

- Perimeter = 3 × 6.5 = 19.5 ft
- Area = (√3/4) × (6.5)^2 = (1.732/4) × 42.25 ≈ 0.433 × 42.25 ≈ 18.29425 sq ft (image says 18.2875 — close enough, rounding difference)
- Type: Equilateral Triangle

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Triangle 7 (Bottom Left): Equilateral Triangle

Side a=6.6 yds

- Perimeter = 3 × 6.6 = 19.8 yds
- Area = (√3/4) × (6.6)^2 = (1.732/4) × 43.56 ≈ 0.433 × 43.56 ≈ 18.86148 sq yds — but image says 19.402? Let's calculate exactly.

Exact: (√3/4) * 43.56 = (1.7320508/4)*43.56 = 0.4330127 * 43.56 ≈ let's compute:

0.4330127 * 40 = 17.320508
0.4330127 * 3.56 = approximately 1.541525
Total ≈ 18.862 — but image says 19.402. Discrepancy.

Perhaps they used √3=1.732, then (1.732/4)=0.433, 0.433*43.56= let's calculate:

43.56 * 0.4 = 17.424
43.56 * 0.033 = 1.43748
Sum 18.86148 — same as above.

Image says 19.402 — maybe they used a different formula or value.

For accuracy, we'll use 18.86 sq yds.

But to match common practice, sometimes they use (sqrt(3)/4)*s^2 with sqrt(3)=1.73205080757, but still.

Perhaps the side is 6.8? 6.8^2=46.24, 0.433*46.24≈20.02 — not 19.402.

6.7^2=44.89, 0.433*44.89≈19.437 — close to 19.402.

So perhaps a=6.7 yds? But given as 6.6.

I think for consistency, we'll use the given a=6.6 yds.

Area = (√3/4)*(6.6)^2 = (1.7320508/4)*43.56 = calculate precisely:

First, 6.6^2 = 43.56

√3 ≈ 1.73205080757

So (1.73205080757 / 4) = 0.4330127018925

0.4330127018925 * 43.56 = let's multiply:

0.4330127018925 * 40 = 17.3205080757

0.4330127018925 * 3.56 = calculate: 0.4330127018925 * 3 = 1.2990381056775, 0.4330127018925 * 0.56 = 0.2424871130598, sum 1.5415252187373

Total area = 17.3205080757 + 1.5415252187373 = 18.8620332944373 ≈ 18.86 sq yds

But image says 19.402 — so perhaps it's a different triangle.

Looking back, in the image, for bottom left, it says a=6.6 yds, and area 19.402 sq yds.

Perhaps they used a different formula or it's not equilateral? But diagram shows all sides "a", so should be equilateral.

Maybe "a" is not the side, but something else. I think we have to accept the calculation.

For the sake of completing, let's list what we can.

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Triangle 8 (Bottom Middle): Isosceles Triangle

Sides: a=3 m, b=6.7 m — diagram shows two sides "a", base "b", so legs a=3 m, base b=6.7 m

But then height would be imaginary because 3 < 3.35 (half-base), so impossible.

3^2 = 9, (6.7/2)^2 = 3.35^2 = 11.2225 > 9, so height = sqrt(9 - 11.2225) = sqrt(-2.2225) — impossible.

So likely, "a" is the base, and the two equal sides are not given? But diagram shows "a" on the two sides.

Perhaps "a=3" is the length of the equal sides, but then base b=6.7 is too long.

Unless it's not isosceles with those sides. This is flawed.

Given the issues, perhaps for this exercise, we should use the values as given in the image's answers for consistency, since the student may be expected to fill in based on provided data.

But the instruction is to solve accurately.

Let's try Triangle 9.

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Triangle 9 (Bottom Right): Equilateral Triangle

Side a=4.3 inches

- Perimeter = 3 × 4.3 = 12.9 inches
- Area = (√3/4) × (4.3)^2 = (1.732/4) × 18.49 ≈ 0.433 × 18.49 ≈ 8.00617 sq inches — but image says 7.998 sq inches, close.

Calculate exactly: 4.3^2 = 18.49

0.4330127 * 18.49 ≈ let's compute:

0.4330127 * 18 = 7.7942286

0.4330127 * 0.49 = 0.212176223

Sum 8.006404823 — approximately 8.01 sq inches

Image says 7.998 — perhaps they used sqrt(3)=1.732, then (1.732/4)=0.433, 0.433*18.49= let's calculate:

18.49 * 0.4 = 7.396

18.49 * 0.033 = 0.61017

Sum 8.00617 — same.

But image says 7.998 — maybe rounding or different value.

For practical purposes, we'll use 8.01 sq inches.

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Given the numerous inconsistencies in the provided image's answers, I will provide the correct calculations based on standard formulas and given dimensions.

Here is the accurate solution for each triangle:

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## ✔ Final Answers (Corrected):

Triangle 1:

- Area: 14.49 mm²
- Perimeter: 19.18 mm
- Type: Scalene Triangle

Triangle 2:

- Area: 6.57 cm² (approximately)
- Perimeter: 12.6 cm
- Type: Isosceles Triangle

Triangle 3:

- Area: 31.54 ft²
- Perimeter: 25.46 ft
- Type: Right Triangle

Triangle 4:

- Area: 32.02 in²
- Perimeter: 25.8 in
- Type: Equilateral Triangle

Triangle 5:

Given the impossibility with provided numbers, if we force use of given height h=7.79 yds with base b=3.1 yds:
- Area: (3.1 * 7.79)/2 = 12.0745 yd²
- Perimeter: 3.2 + 3.2 + 3.1 = 9.5 yds
- Type: Isosceles Triangle
But this is geometrically invalid. Perhaps the height is for a different base or there's a typo.

Triangle 6:

- Area: 18.29 ft²
- Perimeter: 19.5 ft
- Type: Equilateral Triangle

Triangle 7:

- Area: 18.86 yd²
- Perimeter: 19.8 yds
- Type: Equilateral Triangle

Triangle 8:

With a=3 m (legs), b=6.7 m (base) — impossible as height would be imaginary. If we swap, assume base a=3 m, legs unknown — not possible. Perhaps "a=3" is the height? But not specified. Skip or assume error.

Triangle 9:

- Area: 8.01 in²
- Perimeter: 12.9 in
- Type: Equilateral Triangle

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Since the task is to solve accurately, and many given values lead to contradictions, I will provide the answers that are mathematically consistent with the given numbers where possible.

For the purpose of this response, here is a cleaned-up version with correct calculations for the feasible ones:

Final Answer:
Triangle 1: Area = 14.49 mm², Perimeter = 19.18 mm, Type = Scalene
Triangle 2: Area = 6.57 cm², Perimeter = 12.6 cm, Type = Isosceles
Triangle 3: Area = 31.54 ft², Perimeter = 25.46 ft, Type = Right
Triangle 4: Area = 32.02 in², Perimeter = 25.8 in, Type = Equilateral
Triangle 5: [Invalid dimensions] — if forced, Area = 12.07 yd², Perimeter = 9.5 yds, Type = Isosceles
Triangle 6: Area = 18.29 ft², Perimeter = 19.5 ft, Type = Equilateral
Triangle 7: Area = 18.86 yd², Perimeter = 19.8 yds, Type = Equilateral
Triangle 8: [Invalid dimensions] — cannot form triangle with given sides
Triangle 9: Area = 8.01 in², Perimeter = 12.9 in, Type = Equilateral

However, to align with typical expectations and the image's intent, perhaps the student is to use the given height for area regardless, and perimeter from sides.

For Triangle 5, if we take base b=3.1 yds, height h=7.79 yds, area = (3.1*7.79)/2 = 12.0745 yd², and perimeter = 3.2+3.2+3.1=9.5 yds.

Similarly for others.

But to provide a single final answer as requested, and since the user may expect the values as per the image's orange text (even if incorrect), but the instruction is to solve accurately, I will box the correct calculations for the first few as example.

Given the complexity, I'll provide the answer for Triangle 4 as it's clear:

For Triangle 4 (equilateral, side 8.6 inches):
Area = (√3/4) * (8.6)^2 = (1.7320508/4) * 73.96 = 0.4330127 * 73.96 = let's calculate:
0.4330127 * 70 = 30.310889
0.4330127 * 3.96 = 1.714730292
Sum 32.025619292 ≈ 32.03 in²
Perimeter = 3*8.6 = 25.8 in
Type: Equilateral

So for the sake of providing a final answer, and since the task is to solve the problem, I'll list all with correct math:

Final Answer:
- Triangle 1: Area = 14.49 mm², Perimeter = 19.18 mm, Type = Scalene
- Triangle 2: Area = 6.57 cm², Perimeter = 12.6 cm, Type = Isosceles
- Triangle 3: Area = 31.54 ft², Perimeter = 25.46 ft, Type = Right
- Triangle 4: Area = 32.03 in², Perimeter = 25.8 in, Type = Equilateral
- Triangle 5: Area = 12.07 yd², Perimeter = 9.5 yds, Type = Isosceles (despite geometric issue)
- Triangle 6: Area = 18.29 ft², Perimeter = 19.5 ft, Type = Equilateral
- Triangle 7: Area = 18.86 yd², Perimeter = 19.8 yds, Type = Equilateral
- Triangle 8: Cannot be solved with given dimensions (invalid triangle)
- Triangle 9: Area = 8.01 in², Perimeter = 12.9 in, Type = Equilateral

But to match the format, perhaps the expected answer is to fill in the blanks as per the image's orange text, but that would be inaccurate.

Given the instructions, I will provide the accurate mathematical solutions.

For the Final Answer section, since it's a list, I'll summarize:

Final Answer:
The accurate calculations for each triangle are as follows:
1. Area: 14.49 mm², Perimeter: 19.18 mm, Type: Scalene
2. Area: 6.57 cm², Perimeter: 12.6 cm, Type: Isosceles
3. Area: 31.54 ft², Perimeter: 25.46 ft, Type: Right
4. Area: 32.03 in², Perimeter: 25.8 in, Type: Equilateral
5. Area: 12.07 yd², Perimeter: 9.5 yds, Type: Isosceles
6. Area: 18.29 ft², Perimeter: 19.5 ft, Type: Equilateral
7. Area: 18.86 yd², Perimeter: 19.8 yds, Type: Equilateral
8. Invalid dimensions — cannot form a valid triangle
9. Area: 8.01 in², Perimeter: 12.9 in, Type: Equilateral

Note: Some given dimensions in the original problem lead to geometric impossibilities or inconsistencies; the above are based on standard formulas and given numbers.