Geometria Espacial: Tabela Completa de Fórmulas dos Sólidos

Cubo

\(V=a^{3}\)

\(S_{\text{total}}=6a^{2}\)

Diagonal espacial: \(d=a\sqrt{3}\)

Paralelepípedo retângulo

\(V=a\cdot b\cdot c\)

\(S_{\text{total}}=2(ab+ac+bc)\)

Diagonal espacial: \(d^{2}=a^{2}+b^{2}+c^{2}\)

Prisma regular

\(V=B\cdot h\)

Área lateral: \(A_\ell=2ph\)

Área total: \(A_t=2p(h+m)\) (com apótema da base \(m\))

Pirâmide regular

\(V=\dfrac{1}{3}B\,h\)

\(m’^{2}=h^{2}+m^{2}\) (apótema lateral)

\(A_t=p(m+m’)\)

Cilindro

\(V=\pi r^{2}h\)

\(A_\ell=2\pi r h\)

\(A_t=2\pi r(h+r)\)

Cone

\(V=\dfrac{1}{3}\pi r^{2}h\)

\(A_\ell=\pi r g\)

\(A_t=\pi r(g+r)\), com \(g^{2}=r^{2}+h^{2}\)

Tronco de cone

\(V=\dfrac{\pi h}{3}(R^{2}+Rr+r^{2})\)

\(A_\ell=\pi(R+r)g\)

\(A_t=\pi(R^{2}+r^{2})+\pi(R+r)g\)

Tronco de pirâmide regular

\(V=\dfrac{h}{3}\big(B+\sqrt{Bb}+b\big)\)

\(A_\ell=\dfrac{(P+p)m’}{2}\)

\(A_t=A_\ell+B+b\)

Esfera

\(V=\dfrac{4}{3}\pi r^{3}\)

\(A=4\pi r^{2}\)

Calota esférica

\(A=2\pi R h\)

\(V=\dfrac{\pi h^{2}}{3}(3R-h)\)

\(a^{2}=2Rh-h^{2}\)

Cunha/Fuso esférico

\(A_{\text{fuso}}=2r^{2}\alpha\) (rad) \(=\dfrac{\pi r^{2}\alpha_{\circ}}{90}\) (graus)

\(V_{\text{cunha}}=\dfrac{2\alpha}{3}r^{3}=\dfrac{\pi r^{3}\alpha_{\circ}}{270}\)

Razão de semelhança

Linear: \(k\) • Áreas: \(k^{2}\) • Volumes: \(k^{3}\)

Ex. 1 — Cilindro

Para \(r=3\) e \(h=10\):

\[ \begin{aligned} V&=\pi r^{2}h\\ &=\pi\cdot 3^{2}\cdot 10\\ &=90\pi\ \text{u}^{3} \end{aligned} \]

Ex. 2 — Tronco de cone

\(R=8,\ r=5,\ h=12\).

\[ \begin{aligned} V&=\frac{\pi h}{3}(R^{2}+Rr+r^{2})\\ &=\frac{\pi\cdot 12}{3}(64+40+25)\\ &=516\pi\ \text{u}^{3} \end{aligned} \]

Ex. 3 — Esfera

Para \(r=6\):

\[ \begin{aligned} A&=4\pi r^{2}\\ &=4\pi\cdot 36\\ &=144\pi\ \text{u}^{2} \end{aligned} \]