The figure below shows a prism. Calculate its total surface area.

Prismu ina msingi wa pembetatu.

• Pandeti za pembetatu ni $20 \text{ cm}$, $21 \text{ cm}$, na $29 \text{ cm}$.
• Urefu wa prismu (urefu wa pande za mstatili) ni $31 \text{ cm}$. Kumbuka kuwa $20^2 + 21^2 = 400 + 441 = 841$, na $29^2 = 841$. Hii inamaanisha pembetatu ya msingi ni pembetatu mraba, na pande $20 \text{ cm}$ na $21 \text{ cm}$ ni msingi na urefu wake.

Eneo la pembetatu moja ($A_{pembetatu}$) = $\frac{1}{2} \times \text{msingi} \times \text{urefu}$

$$A_{pembetatu} = \frac{1}{2} \times 20 \text{ cm} \times 21 \text{ cm}$$ $$A_{pembetatu} = 10 \text{ cm} \times 21 \text{ cm} = 210 \text{ cm}^2$$

Eneo la nyuso mbili za msingi ($A_{misingi}$) = $2 \times A_{pembetatu}$

$$A_{misingi} = 2 \times 210 \text{ cm}^2 = 420 \text{ cm}^2$$

Kila uso wa pembeni ni mstatili wenye urefu wa $31 \text{ cm}$ na upana sawa na moja ya pande za pembetatu ya msingi.

• Eneo la mstatili wa kwanza ($A_{mstatili1}$) = $20 \text{ cm} \times 31 \text{ cm} = 620 \text{ cm}^2$
• Eneo la mstatili wa pili ($A_{mstatili2}$) = $21 \text{ cm} \times 31 \text{ cm} = 651 \text{ cm}^2$
• Eneo la mstatili wa tatu ($A_{mstatili3}$) = $29 \text{ cm} \times 31 \text{ cm} = 899 \text{ cm}^2$ Jumla ya eneo la nyuso za pembeni ($A_{pembeni}$) = $A_{mstatili1} + A_{mstatili2} + A_{mstatili3}$

$$A_{pembeni} = 620 \text{ cm}^2 + 651 \text{ cm}^2 + 899 \text{ cm}^2 = 2170 \text{ cm}^2$$

Jumla ya eneo la uso ($A_{jumla}$) ni jumla ya eneo la nyuso mbili za msingi na eneo la nyuso tatu za pembeni.

$$A_{jumla} = A_{misingi} + A_{pembeni}$$ $$A_{jumla} = 420 \text{ cm}^2 + 2170 \text{ cm}^2$$ $$A_{jumla} = 2590 \text{ cm}^2$$ $$2590 \text{ cm}^2$$