The base of a regular right pyramid is an isosceles triangle with equal sides of 25 cm and a base of 30 cm. The slant height of the pyramid is 40 cm. Find the total surface area of the pyramid.
1900 cm2
A regular right pyramid's total surface area is the area of its base plus the lateral (slanted triangular) faces, so we handle the two parts separately.
The base is an isosceles triangle with equal sides 25 cm and base 30 cm. Dropping the altitude to the 30 cm base splits it into two halves of 15 cm each.
By the Pythagoras theorem the base triangle's height is \(\sqrt{25^2 - 15^2} = \sqrt{625 - 225} = \sqrt{400} = 20\) cm.
So the base area is \(\dfrac{1}{2}\times 30\times 20 = 300\text{ cm}^2\).
The lateral surface area of a regular pyramid is \(\dfrac{1}{2}\times \text{perimeter}\times \text{slant height}\), using the given slant height 40 cm.
The base perimeter is \(25 + 25 + 30 = 80\) cm, so the lateral area is \(\dfrac{1}{2}\times 80\times 40 = 1600\text{ cm}^2\).
Adding both parts, total surface area \(= 300 + 1600 = 1900\text{ cm}^2\).
The total surface area of the pyramid is 1900 cm2.
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