Educational worksheet for calculating the area and perimeter of various triangles.

To solve the problem, we need to calculate the perimeter and area of each triangle. Let's go through each triangle step by step.

---

1. Orange Triangle

#### Dimensions:
- Sides: 4 cm, 6 cm, 4 cm

#### Perimeter:
The perimeter of a triangle is the sum of its sides.
\[
\text{Perimeter} = 4 + 6 + 4 = 14 \, \text{cm}
\]

#### Area:
This is an isosceles triangle with two equal sides (4 cm each) and a base of 6 cm. To find the area, we can use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
First, we need to find the height. The height can be found using the Pythagorean theorem in the right triangle formed by the height, half the base, and one of the equal sides.

- Half the base: \( \frac{6}{2} = 3 \, \text{cm} \)
- Hypotenuse: 4 cm
- Height (\( h \)):
\[
h = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7} \, \text{cm}
\]

Now, calculate the area:
\[
\text{Area} = \frac{1}{2} \times 6 \times \sqrt{7} = 3\sqrt{7} \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 14 \, \text{cm}, \quad \text{Area} = 3\sqrt{7} \, \text{cm}^2
\]

---

2. Blue Triangle

#### Dimensions:
- Sides: 3 cm, 6 cm, 8 cm

#### Perimeter:
\[
\text{Perimeter} = 3 + 6 + 8 = 17 \, \text{cm}
\]

#### Area:
We can use Heron's formula to find the area of this scalene triangle. Heron's formula is:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( s \) is the semi-perimeter:
\[
s = \frac{a + b + c}{2} = \frac{3 + 6 + 8}{2} = 8.5 \, \text{cm}
\]

Now, substitute into Heron's formula:
\[
\text{Area} = \sqrt{8.5(8.5 - 3)(8.5 - 6)(8.5 - 8)} = \sqrt{8.5 \times 5.5 \times 2.5 \times 0.5}
\]
\[
= \sqrt{8.5 \times 5.5 \times 2.5 \times 0.5} = \sqrt{58.4375} \approx 7.64 \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 17 \, \text{cm}, \quad \text{Area} \approx 7.64 \, \text{cm}^2
\]

---

3. Green Triangle

#### Dimensions:
- Sides: 10 cm, 10 cm, 6 cm

#### Perimeter:
\[
\text{Perimeter} = 10 + 10 + 6 = 26 \, \text{cm}
\]

#### Area:
This is an isosceles triangle with two equal sides (10 cm each) and a base of 6 cm. To find the area, we use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
First, find the height using the Pythagorean theorem in the right triangle formed by the height, half the base, and one of the equal sides.

- Half the base: \( \frac{6}{2} = 3 \, \text{cm} \)
- Hypotenuse: 10 cm
- Height (\( h \)):
\[
h = \sqrt{10^2 - 3^2} = \sqrt{100 - 9} = \sqrt{91} \, \text{cm}
\]

Now, calculate the area:
\[
\text{Area} = \frac{1}{2} \times 6 \times \sqrt{91} = 3\sqrt{91} \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 26 \, \text{cm}, \quad \text{Area} = 3\sqrt{91} \, \text{cm}^2
\]

---

4. Pink Triangle

#### Dimensions:
- Sides: 11 cm, 12 cm, 13 cm

#### Perimeter:
\[
\text{Perimeter} = 11 + 12 + 13 = 36 \, \text{cm}
\]

#### Area:
We use Heron's formula. First, calculate the semi-perimeter:
\[
s = \frac{11 + 12 + 13}{2} = 18 \, \text{cm}
\]

Now, substitute into Heron's formula:
\[
\text{Area} = \sqrt{18(18 - 11)(18 - 12)(18 - 13)} = \sqrt{18 \times 7 \times 6 \times 5}
\]
\[
= \sqrt{3780} \approx 61.48 \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 36 \, \text{cm}, \quad \text{Area} \approx 61.48 \, \text{cm}^2
\]

---

5. Red Triangle

#### Dimensions:
- Sides: 10 cm, 10 cm, 10 cm (equilateral triangle)
- Height: 8 cm

#### Perimeter:
\[
\text{Perimeter} = 10 + 10 + 10 = 30 \, \text{cm}
\]

#### Area:
The area of an equilateral triangle can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is 10 cm and the height is 8 cm:
\[
\text{Area} = \frac{1}{2} \times 10 \times 8 = 40 \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 30 \, \text{cm}, \quad \text{Area} = 40 \, \text{cm}^2
\]

---

6. Purple Triangle

#### Dimensions:
- Sides: 9 m, 9 m, 9 m (equilateral triangle)
- Height: 8 m

#### Perimeter:
\[
\text{Perimeter} = 9 + 9 + 9 = 27 \, \text{m}
\]

#### Area:
The area of an equilateral triangle can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is 9 m and the height is 8 m:
\[
\text{Area} = \frac{1}{2} \times 9 \times 8 = 36 \, \text{m}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 27 \, \text{m}, \quad \text{Area} = 36 \, \text{m}^2
\]

---

Final Summary:

1. Orange Triangle:
\[
\text{Perimeter} = 14 \, \text{cm}, \quad \text{Area} = 3\sqrt{7} \, \text{cm}^2
\]

2. Blue Triangle:
\[
\text{Perimeter} = 17 \, \text{cm}, \quad \text{Area} \approx 7.64 \, \text{cm}^2
\]

3. Green Triangle:
\[
\text{Perimeter} = 26 \, \text{cm}, \quad \text{Area} = 3\sqrt{91} \, \text{cm}^2
\]

4. Pink Triangle:
\[
\text{Perimeter} = 36 \, \text{cm}, \quad \text{Area} \approx 61.48 \, \text{cm}^2
\]

5. Red Triangle:
\[
\text{Perimeter} = 30 \, \text{cm}, \quad \text{Area} = 40 \, \text{cm}^2
\]

6. Purple Triangle:
\[
\text{Perimeter} = 27 \, \text{m}, \quad \text{Area} = 36 \, \text{m}^2
\]

Boxed Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Orange Triangle:} & \text{Perimeter} = 14 \, \text{cm}, \, \text{Area} = 3\sqrt{7} \, \text{cm}^2 \\
\text{Blue Triangle:} & \text{Perimeter} = 17 \, \text{cm}, \, \text{Area} \approx 7.64 \, \text{cm}^2 \\
\text{Green Triangle:} & \text{Perimeter} = 26 \, \text{cm}, \, \text{Area} = 3\sqrt{91} \, \text{cm}^2 \\
\text{Pink Triangle:} & \text{Perimeter} = 36 \, \text{cm}, \, \text{Area} \approx 61.48 \, \text{cm}^2 \\
\text{Red Triangle:} & \text{Perimeter} = 30 \, \text{cm}, \, \text{Area} = 40 \, \text{cm}^2 \\
\text{Purple Triangle:} & \text{Perimeter} = 27 \, \text{m}, \, \text{Area} = 36 \, \text{m}^2 \\
\end{array}
}
\]