Answer:
l = 17 units
surface area = 736 square units
Explanation:
Slant
- The line that extends from the top of the pyramid to the middle of the square base the apothem.
- We know that all the sides of a square are congruent to each other
- Thus, if you have a line that extends from the middle of a square to the center, it's 1/2 the measure of the side of the square
- Therefore, we know that the line that forms the right angle with the apothem is 8 units since 1/2 * 16 = 8
- We see that the apothem (15 units), the line bisecting a side of the square (8 units), and the slant height form a right triangle.
- Therefore, we can find the slant height, l, using the Pythagorean theorem, which is:
a^2 + b^2 + c^2, where
- a and b are the shorter sides of the triangle (aka legs),
- and c is the longest side of the triangle, called the hypotenuse:
We must plug in 15 for a and 8 for b, allowing us to solve for c, the hypotenuse and thus the slant height:
15^2 + 8^2 = c^2
225 + 64 = c^2
289 = c^2
±17 = c
17 = c
The ± comes from the fact that taking the square root gives you both a positive and negative answer since squaring both a positive and and a negative number yields a positive answer (e.g., 2 * 2 = 4, but -2 * -2 also = 4). Because you can't have a negative measure, c (and thus our slant height, l) = 17 units.
Surface area:
One of the formulas we can use for surface area of a square pyramid is given by:
SA = B + LA, where
- SA is the surface area in square units,
- B is the area of the square base,
- and LA is the lateral area (combined area of the triangular faces)
Finding B: The formula for area of a square is given by:
A = s^2, where
- A is the area in square units,
- and s is the measure of one of the sides.
We can plug in 16 for b and simplify:
A = 16^2
A = 256 square units
Thus 256 is B in the surface area formula.
Finding LA: The formula for area of a triangle is given by:
A = 1/2bh, where
- A is the area in square units,
- b is the base,
- and h is the height
We can plug in 16 for b and 15 for h and simplify:
A = 1/2(16)(15)
A = 8 * 15
A = 120 square units
Since we know that the area of one of the triangular faces is 120, we can multiply this to find the lateral area, LA (the total area occupied by all four triangular faces:
LA = 4A
LA = 4 * 120
LA = 480 square units
Thus, 480 is LA in the surface area formula.
Combining B and LA to find SA
Since we've now found B and LA in the surface area formula, we can plug these values in and simplify:
SA = 256 + 480
SA = 736 square units
Thus, the surface area of the square pyramid is 736 square units.