Perimeter and area of triangles activity - Free Printable

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. To find the area, we can use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is 6 cm, and we need to find the height. We can use the Pythagorean theorem in one of the right triangles formed by dropping a perpendicular from the apex to the base.

Let the height be \( h \). The base is split into two equal parts of 3 cm each. Using the Pythagorean theorem:
\[
h^2 + 3^2 = 4^2
\]
\[
h^2 + 9 = 16
\]
\[
h^2 = 7
\]
\[
h = \sqrt{7}
\]

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

#### 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 use Heron's formula since it is not a right triangle. First, calculate the semi-perimeter \( s \):
\[
s = \frac{3 + 6 + 8}{2} = 8.5 \, \text{cm}
\]

Now, apply Heron's formula:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( a = 8 \), \( b = 6 \), and \( c = 3 \):
\[
\text{Area} = \sqrt{8.5(8.5 - 8)(8.5 - 6)(8.5 - 3)}
\]
\[
= \sqrt{8.5 \times 0.5 \times 2.5 \times 5.5}
\]
\[
= \sqrt{8.5 \times 0.5 \times 2.5 \times 5.5} = \sqrt{57.8125} \approx 7.6 \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 17 \, \text{cm}, \quad \text{Area} \approx 7.6 \, \text{cm}^2
\]

---

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. To find the area, we can use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is 6 cm, and we need to find the height. We can use the Pythagorean theorem in one of the right triangles formed by dropping a perpendicular from the apex to the base.

Let the height be \( h \). The base is split into two equal parts of 3 cm each. Using the Pythagorean theorem:
\[
h^2 + 3^2 = 10^2
\]
\[
h^2 + 9 = 100
\]
\[
h^2 = 91
\]
\[
h = \sqrt{91}
\]

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

#### 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 since it is not a right triangle. First, calculate the semi-perimeter \( s \):
\[
s = \frac{11 + 12 + 13}{2} = 18 \, \text{cm}
\]

Now, apply Heron's formula:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( a = 13 \), \( b = 12 \), and \( c = 11 \):
\[
\text{Area} = \sqrt{18(18 - 13)(18 - 12)(18 - 11)}
\]
\[
= \sqrt{18 \times 5 \times 6 \times 7}
\]
\[
= \sqrt{3780} \approx 61.5 \, \text{cm}^2
\]

#### Final Answers:
\[
\text{Perimeter} = 36 \, \text{cm}, \quad \text{Area} \approx 61.5 \, \text{cm}^2
\]

---

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 given as 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 given as 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:
\[
\text{Perimeter} = 17 \, \text{cm}, \quad \text{Area} \approx 7.6 \, \text{cm}^2
\]

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

4. Pink Triangle:
\[
\text{Perimeter} = 36 \, \text{cm}, \quad \text{Area} \approx 61.5 \, \text{cm}^2
\]

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:

\[
\boxed{
\begin{array}{ll}
\text{Orange Triangle:} & \text{Perimeter} = 14 \, \text{cm}, \, \text{Area} = 3\sqrt{7} \, \text{cm}^2 \\
\text{Blue Triangle:} & \text{Perimeter} = 17 \, \text{cm}, \, \text{Area} \approx 7.6 \, \text{cm}^2 \\
\text{Green Triangle:} & \text{Perimeter} = 26 \, \text{cm}, \, \text{Area} = 3\sqrt{91} \, \text{cm}^2 \\
\text{Pink Triangle:} & \text{Perimeter} = 36 \, \text{cm}, \, \text{Area} \approx 61.5 \, \text{cm}^2 \\
\text{Red Triangle:} & \text{Perimeter} = 30 \, \text{cm}, \, \text{Area} = 40 \, \text{cm}^2 \\
\text{Purple Triangle:} & \text{Perimeter} = 27 \, \text{m}, \, \text{Area} = 36 \, \text{m}^2 \\
\end{array}
}
\]