Question

What is the total area of this figure ( _ _ _ + _ _ sqrt _)

Answer 1

Look at the picture.

We have in a base an equilateral triangle.

The formula of the area of an equilateral triangle:

$A_B=\dfrac{a^2\sqrt3}{4}$

We have a = 12. Substitute:

$A_B=\dfrac{12^2\sqrt3}{4}=\dfrac{144\sqrt3}{4}=36\sqrt3$

The lateral side is a isosceles triangle. The formula of the area of a triangle is:

$A_\triangle=\dfrac{1}{2}bh$

We must calculate the length of h using the Pythagorean theorem:

$h^2+6^2=10^2$

$h^2+36=100$              subtract 36 from both sides

$h^2=64\to h=\sqrt{64}\to h=8$

We have b = 13 and h = 8. Substitute:

$A_\triangle=\dfrac{1}{2}(12)(8)=(6)(8)=48$

The Total Area:

$T.A.=A_B+3A_\triangle$

Substitute:

$T.A.=36\sqrt3+3(48)=\boxed{144+36\sqrt3}$