Determine the total surface area of the figure below. The figure is a prism with a triangular base. The height of the triangle is 5 cm, the base of the triangle is 12 cm, and the length of the prism is 20 cm.

Sehemu ya msalaba ina mstatili na nusu duara.
Eneo la mstatili = urefu $\times$ upana

$$A_{mstatili} = 12 \text{ cm} \times 5 \text{ cm} = 60 \text{ cm}^2$$

Kipenyo cha nusu duara ni $12 \text{ cm}$, hivyo rediasi $r = \frac{12}{2} = 6 \text{ cm}$.
Eneo la nusu duara = $\frac{1}{2} \pi r^2$

$$A_{nusu\_duara} = \frac{1}{2} \pi (6 \text{ cm})^2 = \frac{1}{2} \pi (36 \text{ cm}^2) = 18\pi \text{ cm}^2$$

Eneo la sehemu ya msalaba, $A_{cs} = A_{mstatili} + A_{nusu\_duara}$

$$A_{cs} = (60 + 18\pi) \text{ cm}^2$$

Kuna nyuso mbili za mwisho zinazofanana.
Eneo la nyuso mbili za mwisho = $2 \times A_{cs}$

$$A_{mwisho} = 2(60 + 18\pi) \text{ cm}^2 = (120 + 36\pi) \text{ cm}^2$$

Uso wa chini una urefu wa $20 \text{ cm}$ na upana wa $12 \text{ cm}$.

$$A_{chini} = 12 \text{ cm} \times 20 \text{ cm} = 240 \text{ cm}^2$$

Kila uso wa pembeni una urefu wa $20 \text{ cm}$ na upana wa $5 \text{ cm}$.

$$A_{pembeni} = 2 \times (5 \text{ cm} \times 20 \text{ cm}) = 2 \times 100 \text{ cm}^2 = 200 \text{ cm}^2$$

Mzingo wa nusu duara ni $\pi r = \pi (6 \text{ cm}) = 6\pi \text{ cm}$.
Eneo la uso uliopinda = (mzingo wa nusu duara) $\times$ (urefu wa mche)

$$A_{pinda} = (6\pi \text{ cm}) \times 20 \text{ cm} = 120\pi \text{ cm}^2$$

Eneo lote la uso (TSA) = $A_{mwisho} + A_{chini} + A_{pembeni} + A_{pinda}$

$$TSA = (120 + 36\pi) + 240 + 200 + 120\pi$$ $$TSA = (120 + 240 + 200) + (36\pi + 120\pi)$$ $$TSA = 560 + 156\pi \text{ cm}^2$$

$$TSA = 560 + 156 \times 3.142$$ $$TSA = 560 + 490.152$$ $$TSA = 1050.152 \text{ cm}^2$$

Ukizungusha hadi sehemu moja ya desimali:

$$TSA \approx 1050.2 \text{ cm}^2$$

Eneo lote la uso wa umbo hilo ni:

$$\text{1050.2 cm}^2$$