Grade 10

Advanced

7 min

Lesson 10 of 12

Surface Area of Pyramids, Cones, and Spheres

Not Started

Calculates the surface area of three-dimensional figures using lateral area and base area components.

If you were tasked with gold-plating the Great Pyramid of Giza or a massive spherical observatory, how would you calculate the exact amount of gold needed without wasting a single gram?

Learning Objectives

1.Derive and apply formulas for the lateral and total surface area of right cones and regular pyramids
2.Calculate the total surface area of spheres and hemispheres including their bases
3.Solve algebraic problems to find missing dimensions like radius or slant height when given the total surface area

The Geometry of 'Unrolling': Pyramids and Cones

To find the surface area of a pointed solid, we look at two parts: the Base Area ( $B$ ) and the Lateral Area ( $L$ ). Imagine 'unrolling' a cone; its side forms a sector of a circle. For a right cone, the lateral area is $L = \pi r l$ , where $l$ is the slant height. For a regular pyramid, the lateral area is the sum of the areas of its congruent triangular faces: $L = \frac{1}{2} P l$ , where $P$ is the perimeter of the base. The Total Surface Area ( $S$ ) is simply the sum of the lateral area and the base: $S = L + B$ .

Find the total surface area of a regular square pyramid with a base side of $6$ cm and a slant height of $10$ cm.

1. Find the Base Area ( $B$ ): $B = 6 \times 6 = 36$ cm $^2$ . 2. Find the Perimeter ( $P$ ): $P = 4 \times 6 = 24$ cm. 3. Calculate Lateral Area ( $L$ ): $L = \frac{1}{2} P l = \frac{1}{2}(24)(10) = 120$ cm $^2$ . 4. Total Surface Area ( $S$ ): $S = L + B = 120 + 36 = 156$ cm $^2$ .

Quick Check

In the formula for the lateral area of a cone, what does the variable 'l' represent, and how does it differ from 'h'?

Think of the slant height as the hypotenuse of a right triangle inside the cone.

Answer

The variable 'l' is the slant height (the distance from the apex to the edge of the base), whereas 'h' is the vertical height (the perpendicular distance from the apex to the center of the base).

The Perfection of the Sphere

A sphere is unique because it has no 'edges' or 'vertices.' Archimedes discovered that the surface area of a sphere is exactly four times the area of its great circle. The formula is $S = 4\pi r^2$ . When dealing with a hemisphere (half a sphere), we must be careful. The curved surface is $2\pi r^2$ , but if the hemisphere is 'closed,' we must add the area of the circular base ( $\pi r^2$ ), making the total surface area $S = 3\pi r^2$ .

A solid hemisphere has a radius of $5$ meters. Calculate its total surface area in terms of $\pi$ .

1. Identify the formula for a solid hemisphere: $S = 3\pi r^2$ . 2. Substitute the radius: $S = 3\pi (5)^2$ . 3. Simplify: $S = 3\pi (25) = 75\pi$ m $^2$ .

Quick Check

If you double the radius of a sphere, by what factor does the surface area increase?

Look at the exponent of 'r' in

4\pi r^2

Answer

The surface area increases by a factor of 4, because the radius is squared in the formula ( $2^2 = 4$ ).

Advanced Application: Working Backward

In advanced geometry, you are often given the total surface area and asked to find a missing dimension. This requires algebraic manipulation. For a cone, if you know

S

and

r

, you can find the slant height

l

by rearranging

S = \pi r^2 + \pi r l

to solve for

l

l = \frac{S - \pi r^2}{\pi r}

This skill is vital in engineering and manufacturing where material constraints dictate the dimensions of a product.

A cone has a total surface area of $90\pi$ cm $^2$ and a radius of $5$ cm. Find its slant height $l$ .

1. Set up the equation: $90\pi = \pi(5)^2 + \pi(5)l$ . 2. Simplify the base area: $90\pi = 25\pi + 5\pi l$ . 3. Subtract $25\pi$ from both sides: $65\pi = 5\pi l$ . 4. Divide by $5\pi$ : $l = 13$ cm.

Key Takeaways

1The total surface area of any solid is the sum of its lateral area (sides) and its base area.
2For cones and pyramids, always distinguish between vertical height ( $h$ ) and slant height ( $l$ ).
3A closed hemisphere's surface area is $3\pi r^2$ , accounting for both the dome and the flat base.
4Surface area formulas can be rearranged algebraically to solve for unknown radii or heights.

Quick Check

Multiple Choice

What is the lateral area of a right cone with radius $r = 3$ and slant height $l = 7$ ?

Multiple Choice

A sphere has a surface area of $100\pi$ cm $^2$ . What is its radius?

True/False

The total surface area of a regular pyramid is always greater than its lateral area.

What's Next?

Review Tomorrow

In 24 hours, try to sketch a cone and a hemisphere from memory and write down their total surface area formulas without looking at your notes.

Practice Activity

Find a spherical object at home (like a ball) and a conical object (like a funnel). Measure their dimensions and calculate how much wrapping paper would be needed to cover them perfectly.

Browse

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Surface Area of Pyramids, Cones, and Spheres

Learning Objectives

The Geometry of 'Unrolling': Pyramids and Cones

The Perfection of the Sphere

Advanced Application: Working Backward

Key Takeaways

Quick Check

What's Next?

Surface Area of Pyramids, Cones, and Spheres

Learning Objectives

The Geometry of 'Unrolling': Pyramids and Cones

The Perfection of the Sphere

Advanced Application: Working Backward

Key Takeaways

Quick Check

What's Next?